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href="https://github.com/thejinchao/ue5-style-guide" aria-label="UnrealStyleGuide" rel="noopener noreferrer" target="_blank"><!--[--><!--[--><!--]--><!--]-->UnrealStyleGuide<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="auto-link external-link vp-sidebar-item" href="https://github.com/thejinchao/AxTrace" aria-label="AxTrace" rel="noopener noreferrer" target="_blank"><!--[--><!--[--><!--]--><!--]-->AxTrace<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="auto-link external-link vp-sidebar-item" href="https://github.com/thejinchao/cyclone" aria-label="Cyclone" rel="noopener noreferrer" target="_blank"><!--[--><!--[--><!--]--><!--]-->Cyclone<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item vp-sidebar-heading collapsible">学习笔记 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">数学相关 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/symbol.html" aria-label="常用数学符号"><!--[--><!--[--><!--]--><!--]-->常用数学符号<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/group.html" aria-label="群"><!--[--><!--[--><!--]--><!--]-->群<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_1.html" aria-label="数论（一）"><!--[--><!--[--><!--]--><!--]-->数论（一）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_2.html" aria-label="数论（二）"><!--[--><!--[--><!--]--><!--]-->数论（二）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_3.html" aria-label="数论（三）"><!--[--><!--[--><!--]--><!--]-->数论（三）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/probability.html" aria-label="概率"><!--[--><!--[--><!--]--><!--]-->概率<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">密码学 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/rsa.html" aria-label="RSA"><!--[--><!--[--><!--]--><!--]-->RSA<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/flip_coin.html" aria-label="抛币协议"><!--[--><!--[--><!--]--><!--]-->抛币协议<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/mental_poker.html" aria-label="智能扑克协议"><!--[--><!--[--><!--]--><!--]-->智能扑克协议<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">图形学 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">数学基础 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_01.html" aria-label="矢量"><!--[--><!--[--><!--]--><!--]-->矢量<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_02.html" aria-label="矩阵"><!--[--><!--[--><!--]--><!--]-->矩阵<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_03.html" aria-label="立体角"><!--[--><!--[--><!--]--><!--]-->立体角<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_01.html" aria-label="几何变换(一)"><!--[--><!--[--><!--]--><!--]-->几何变换(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_02.html" aria-label="几何变换(二)"><!--[--><!--[--><!--]--><!--]-->几何变换(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_03.html" aria-label="法线变换"><!--[--><!--[--><!--]--><!--]-->法线变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_04.html" aria-label="摄像机变换"><!--[--><!--[--><!--]--><!--]-->摄像机变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">光照模型 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/classic/illumination_model_01.html" aria-label="传统光照模型"><!--[--><!--[--><!--]--><!--]-->传统光照模型<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/luminosity/luminosity.html" aria-label="光度学"><!--[--><!--[--><!--]--><!--]-->光度学<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/BRDF/BRDF.html" aria-label="双向反射分布函数(BRDF)"><!--[--><!--[--><!--]--><!--]-->双向反射分布函数(BRDF)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_1.html" aria-label="微平面理论(一)"><!--[--><!--[--><!--]--><!--]-->微平面理论(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_2.html" aria-label="微平面理论(二)"><!--[--><!--[--><!--]--><!--]-->微平面理论(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_3.html" aria-label="微平面理论(三)"><!--[--><!--[--><!--]--><!--]-->微平面理论(三)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/render_function/render_function_1.html" aria-label="光照方程"><!--[--><!--[--><!--]--><!--]-->光照方程<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">环境光渲染 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_01.html" aria-label="环境光渲染(一)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_02.html" aria-label="环境光渲染(二)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">编程语言 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">JavaScript <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/helloworld.html" aria-label="环境搭建"><!--[--><!--[--><!--]--><!--]-->环境搭建<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/grammar.html" aria-label="基本语法"><!--[--><!--[--><!--]--><!--]-->基本语法<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/function.html" aria-label="函数"><!--[--><!--[--><!--]--><!--]-->函数<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/object_and_class.html" aria-label="对象和类"><!--[--><!--[--><!--]--><!--]-->对象和类<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul><!--[--><!--]--></aside><!--]--><!--[--><main class="vp-page"><!--[--><!--]--><div vp-content><!--[--><!--]--><div><p>和其他流体力学中的数学方法类似，SPH算法同样涉及到“光滑核”的概念，可以这样理解这个概念，粒子的属性都会“扩散”到周围，并且随着距离的增加影响逐渐变小，这种随着距离而衰减的函数被称为“光滑核”函数，最大影响半径为“光滑核半径”。</p><div align="center"><table class="invisibletable"><tbody><tr><td><figure><img src="/images/2014/08/sph_21.gif" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure></td><td><figure><img src="/images/2014/08/sph_22.gif" alt="" tabindex="0" loading="lazy"><figcaption>光滑核函数一般具有的形态</figcaption></figure></td></tr></tbody></table></div><p>反过来不难理解，尽管我们将流体视为一个个分散的粒子，但流体毕竟是连续充满整个空间的，流体中每个位置参与运算的值都是由周围一组粒子累加起来的。</p><figure><img src="/images/2014/08/sph_26.gif" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>设想流体中某点<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.281em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>（此处不一定有粒子）,在光滑核半径<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></mjx-assistive-mml></mjx-container>范围内有数个粒子，位置分别是<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.444em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.444em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.444em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2026"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.413em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mn>0</mn></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mn>1</mn></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mn>2</mn></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mo>…</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>j</mi></msub><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>，则该处某项属性<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></mjx-assistive-mml></mjx-container>的累加公式为：</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:18.797em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:18.797em;" side="right"><mjx-table style="width:auto;min-width:13.085em;margin:0 2.856em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.281em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-TeXAtom texclass="ORD"><mjx-munder><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.576em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-under></mjx-row></mjx-munder><mjx-TeXAtom space="2" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.281em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.413em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-TeXAtom></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:18.797em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.525em;"><mjx-mtd id="mjx-eqn:3.1"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.525em;vertical-align:-1.329em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3.1)</mtext></mtd><mtd><mi>A</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><munder><mo data-mjx-texclass="OP">∑</mo><mi>j</mi></munder><mrow data-mjx-texclass="ORD"><msub><mi>A</mi><mi>j</mi></msub><mfrac><msub><mi>m</mi><mi>j</mi></msub><msub><mi>ρ</mi><mi>j</mi></msub></mfrac><mi>W</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>j</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>其中<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>j</mi></msub></math></mjx-assistive-mml></mjx-container>是要累加的某种属性，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>m</mi><mi>j</mi></msub></math></mjx-assistive-mml></mjx-container>和<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mi>j</mi></msub></math></mjx-assistive-mml></mjx-container>是周围粒子的质量和密度，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.281em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>是该粒子的位置，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></mjx-assistive-mml></mjx-container>是光滑核半径。函数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44A TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math></mjx-assistive-mml></mjx-container>就是光滑核函数。<br> 光滑核函数两个重要属性，首先一定是偶函数，也就是<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo stretchy="false">(</mo><mo>−</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mi>W</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>，第二，是“规整函数”，也就是</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-mstyle></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mi>W</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mi>d</mi><mi>r</mi></mrow></mrow><mo>=</mo><mn>1</mn></mstyle></math></mjx-assistive-mml></mjx-container><hr><h4 id="sph推导过程" tabindex="-1"><a class="header-anchor" href="#sph推导过程"><span><strong>SPH推导过程</strong></span></a></h4><p>我们假设流体中一个位置为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.389em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>的点，此处的密度为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>、压力为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo stretchy="false">(</mo><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>、速度为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">(</mo><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>，那么我们可以根据上一篇的公式2.8，可以推导出此处的加速度<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">(</mo><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>为</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:19.825em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:19.825em;" side="right"><mjx-table style="width:auto;min-width:14.113em;margin:0 2.856em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.266em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mfrac space="3"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-TeXAtom></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mfrac space="3"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D707 TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:19.825em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.515em;"><mjx-mtd id="mjx-eqn:3.2"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.515em;vertical-align:-0.96em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3.2)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">(</mo><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mrow data-mjx-texclass="ORD"><mover><mi>g</mi><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mfrac><mrow><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="ORD"><mi>p</mi><mo stretchy="false">(</mo><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mrow><mrow><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>μ</mi><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">(</mo><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>对于SPH算法来说，基本流程就是这样，根据光滑核函数逐个推出流体中某点的密度，压力，速度相关的累加函数，进而推导出此处的加速度，从而模拟流体的运动趋势，下面我们逐个来分析</p><hr><h4 id="密度" tabindex="-1"><a class="header-anchor" href="#密度"><span><strong>密度</strong></span></a></h4><p>根据公式3.1，用密度<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ρ</mi></math></mjx-assistive-mml></mjx-container>代替<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></mjx-assistive-mml></mjx-container>，可以得到</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:28.854em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:28.854em;" side="right"><mjx-table style="width:auto;min-width:23.142em;margin:0 2.856em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-munder space="4"><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.576em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-under></mjx-row></mjx-munder><mjx-TeXAtom space="2" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.389em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.413em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-munder space="4"><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.576em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-under></mjx-row></mjx-munder><mjx-TeXAtom space="2" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.389em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.413em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-TeXAtom></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:28.854em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.525em;"><mjx-mtd id="mjx-eqn:3.3"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.525em;vertical-align:-1.329em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3.3)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><munder><mo data-mjx-texclass="OP">∑</mo><mi>j</mi></munder><mrow data-mjx-texclass="ORD"><msub><mi>ρ</mi><mi>j</mi></msub><mfrac><msub><mi>m</mi><mi>j</mi></msub><msub><mi>ρ</mi><mi>j</mi></msub></mfrac><mi>W</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>j</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><munder><mo data-mjx-texclass="OP">∑</mo><mi>j</mi></munder><mrow data-mjx-texclass="ORD"><msub><mi>m</mi><mi>j</mi></msub><mi>W</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>j</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>计算使用的光滑核函数称为Poly6函数，具体形式为:</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:31.248em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:31.248em;" side="right"><mjx-table style="width:auto;min-width:25.536em;margin:0 2.856em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44A TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.104em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D443 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D459 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mn class="mjx-n"><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.281em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-s4"><mjx-c class="mjx-c7B TEX-S4"></mjx-c></mjx-mo><mjx-mtable style="min-width:13.099em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:left;padding-right:0.5em;padding-bottom:0.53em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.04em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D443 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D459 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mn class="mjx-n"><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msup space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-bottom:0.53em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2264"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2264"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:left;padding-right:0.5em;padding-top:0.53em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-top:0.53em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mtext class="mjx-n" space="2"><mjx-c class="mjx-c6F"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D464 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n" style="vertical-align:0.25em;"></mjx-mo></mjx-mrow><mjx-mspace></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c77"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c68"></mjx-c></mjx-mtext><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.281em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:31.248em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:3.189em;"><mjx-mtd id="mjx-eqn:3.4"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:3.189em;vertical-align:-1.344em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3.4)</mtext></mtd><mtd><msub><mi>W</mi><mrow data-mjx-texclass="ORD"><mi>P</mi><mi>o</mi><mi>l</mi><mi>y</mi><mn>6</mn></mrow></msub><mtext> </mtext><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mtable columnalign="left left" columnspacing="1em" rowspacing="1.06em 0.2em"><mtr><mtd><msub><mi>K</mi><mrow data-mjx-texclass="ORD"><mi>P</mi><mi>o</mi><mi>l</mi><mi>y</mi><mn>6</mn></mrow></msub><mtext> </mtext><mo stretchy="false">(</mo><msup><mi>h</mi><mn>2</mn></msup><mo>−</mo><msup><mi>r</mi><mn>2</mn></msup><msup><mo stretchy="false">)</mo><mn>3</mn></msup></mtd><mtd><mo>,</mo><mn>0</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><mi>h</mi></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>,</mo><mtext>o</mtext><mi>t</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>w</mi><mi>i</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo></mrow><mspace linebreak="newline"></mspace><mtext>with</mtext><mtext> </mtext><mi>r</mi><mo>=</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-texclass="ORD" stretchy="false">|</mo></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>其中<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.04em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D443 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D459 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mn class="mjx-n"><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>K</mi><mrow data-mjx-texclass="ORD"><mi>P</mi><mi>o</mi><mi>l</mi><mi>y</mi><mn>6</mn></mrow></msub></math></mjx-assistive-mml></mjx-container>是一个固定的系数，根据光滑核的规整属性，通过积分计算出这个系数的具体值，在2D情况下，在极坐标中计算积分：</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:24.807em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:24.807em;" side="right"><mjx-table style="width:auto;min-width:19.095em;margin:0 2.856em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.04em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D443 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D459 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mn class="mjx-n"><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-TeXAtom texclass="ORD"><mjx-msubsup><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-TeXAtom size="s" texclass="ORD" style="margin-left:0.647em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-spacer style="margin-top:1.506em;"></mjx-spacer><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msubsup><mjx-msubsup space="2"><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-mi class="mjx-i" size="s" style="margin-left:0.647em;"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-spacer style="margin-top:1.506em;"></mjx-spacer><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msubsup><mjx-TeXAtom space="2" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msup space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-TeXAtom><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c38"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:24.807em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.491em;"><mjx-mtd id="mjx-eqn:3.5"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c35"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.491em;vertical-align:-0.912em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3.5)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><msub><mi>K</mi><mrow data-mjx-texclass="ORD"><mi>P</mi><mi>o</mi><mi>l</mi><mi>y</mi><mn>6</mn></mrow></msub><mo>=</mo><mn>1</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mrow data-mjx-texclass="ORD"><msubsup><mo data-mjx-texclass="OP">∫</mo><mn>0</mn><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>π</mi></mrow></msubsup><msubsup><mo data-mjx-texclass="OP">∫</mo><mn>0</mn><mi>h</mi></msubsup><mrow data-mjx-texclass="ORD"><mi>r</mi><mo stretchy="false">(</mo><msup><mi>h</mi><mn>2</mn></msup><mo>−</mo><msup><mi>r</mi><mn>2</mn></msup><msup><mo stretchy="false">)</mo><mn>3</mn></msup></mrow><mi>d</mi><mi>r</mi><mtext> </mtext><mi>d</mi><mi>θ</mi></mrow><mo>=</mo><mfrac><mn>4</mn><mrow><mi>π</mi><msup><mi>h</mi><mn>8</mn></msup></mrow></mfrac></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>3D情况下，在球坐标中计算:</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:32.179em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:32.179em;" side="right"><mjx-table style="width:auto;min-width:26.467em;margin:0 2.856em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.04em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D443 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D459 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mn class="mjx-n"><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-TeXAtom texclass="ORD"><mjx-msubsup><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-TeXAtom size="s" texclass="ORD" style="margin-left:0.647em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-spacer style="margin-top:1.506em;"></mjx-spacer><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msubsup><mjx-msubsup space="2"><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-TeXAtom size="s" texclass="ORD" style="margin-left:0.647em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-spacer style="margin-top:1.506em;"></mjx-spacer><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msubsup><mjx-msubsup space="2"><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-mi class="mjx-i" size="s" style="margin-left:0.647em;"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-spacer style="margin-top:1.506em;"></mjx-spacer><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msubsup><mjx-TeXAtom space="2" texclass="ORD"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D711 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msup space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-TeXAtom><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D711 TEX-I"></mjx-c></mjx-mi><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c36"></mjx-c><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c39"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:32.179em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.491em;"><mjx-mtd id="mjx-eqn:3.6"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c36"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.491em;vertical-align:-0.912em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3.6)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><msub><mi>K</mi><mrow data-mjx-texclass="ORD"><mi>P</mi><mi>o</mi><mi>l</mi><mi>y</mi><mn>6</mn></mrow></msub><mo>=</mo><mn>1</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mrow data-mjx-texclass="ORD"><msubsup><mo data-mjx-texclass="OP">∫</mo><mn>0</mn><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>π</mi></mrow></msubsup><msubsup><mo data-mjx-texclass="OP">∫</mo><mn>0</mn><mrow data-mjx-texclass="ORD"><mi>π</mi></mrow></msubsup><msubsup><mo data-mjx-texclass="OP">∫</mo><mn>0</mn><mi>h</mi></msubsup><mrow data-mjx-texclass="ORD"><msup><mi>r</mi><mn>2</mn></msup><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>φ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>h</mi><mn>2</mn></msup><mo>−</mo><msup><mi>r</mi><mn>2</mn></msup><msup><mo stretchy="false">)</mo><mn>3</mn></msup></mrow><mi>d</mi><mi>r</mi><mtext> </mtext><mi>d</mi><mi>φ</mi><mtext> </mtext><mi>d</mi><mi>θ</mi></mrow><mo>=</mo><mfrac><mn>315</mn><mrow><mn>64</mn><mi>π</mi><msup><mi>h</mi><mn>9</mn></msup></mrow></mfrac></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>由于所有粒子的质量相同都是<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></mjx-assistive-mml></mjx-container>，所以在3D情况下，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.389em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>处的密度计算公式最终为：</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:22.202em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:22.202em;" side="right"><mjx-table style="width:auto;min-width:16.49em;margin:0 2.856em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c36"></mjx-c><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c39"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-munder space="2"><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.576em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-under></mjx-row></mjx-munder><mjx-TeXAtom space="2" texclass="ORD"><mjx-msup><mjx-mrow><mjx-mo class="mjx-sop"><mjx-c class="mjx-c28 TEX-S1"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.389em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.413em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-sop"><mjx-c class="mjx-c29 TEX-S1"></mjx-c></mjx-mo></mjx-mrow><mjx-script style="vertical-align:0.611em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-TeXAtom></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:22.202em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.671em;"><mjx-mtd id="mjx-eqn:3.7"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c37"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.671em;vertical-align:-1.329em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3.7)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>m</mi><mfrac><mn>315</mn><mrow><mn>64</mn><mi>π</mi><msup><mi>h</mi><mn>9</mn></msup></mrow></mfrac><munder><mo data-mjx-texclass="OP">∑</mo><mi>j</mi></munder><mrow data-mjx-texclass="ORD"><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>h</mi><mn>2</mn></msup><mo>−</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>j</mi></msub><mo stretchy="false">→</mo></mover></mrow><msup><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mn>2</mn></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mn>3</mn></msup></mrow></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><hr><h4 id="压力" tabindex="-1"><a class="header-anchor" href="#压力"><span><strong>压力</strong></span></a></h4><p>根据上一节的结论，在位置<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container>之处的由压力产生的作用力的计算公式为</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:27.453em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:27.453em;" side="right"><mjx-table style="width:auto;min-width:21.741em;margin:0 2.856em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-msup><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.485em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:0.81em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.389em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-munder space="2"><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.576em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-under></mjx-row></mjx-munder><mjx-TeXAtom space="2" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.389em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.413em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-TeXAtom></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:27.453em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.525em;"><mjx-mtd id="mjx-eqn:3.8"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c38"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.525em;vertical-align:-1.329em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3.8)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><msup><mrow data-mjx-texclass="ORD"><mover><msub><mi>F</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mrow data-mjx-texclass="ORD"><mi>p</mi><mi>r</mi><mi>e</mi><mi>s</mi><mi>s</mi><mi>u</mi><mi>r</mi><mi>e</mi></mrow></msup><mo>=</mo><mo>−</mo><mi mathvariant="normal">∇</mi><mi>p</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><munder><mo data-mjx-texclass="OP">∑</mo><mi>j</mi></munder><mrow data-mjx-texclass="ORD"><msub><mi>p</mi><mi>j</mi></msub><mfrac><msub><mi>m</mi><mi>j</mi></msub><msub><mi>ρ</mi><mi>j</mi></msub></mfrac><mi mathvariant="normal">∇</mi><mi>W</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>j</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>不过不幸的是，这个公式是“不平衡”的，也就是说，位于不同压强区的两个粒子之间的作用力不等，所以计算中一般使用双方粒子压强的算术平均值代替单个粒子的压力<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container>之处的由压力产生的作用力的计算公式为</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:25.28em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:25.28em;" side="right"><mjx-table style="width:auto;min-width:19.568em;margin:0 2.856em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-msup><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.485em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:0.81em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-munder space="2"><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.576em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-under></mjx-row></mjx-munder><mjx-TeXAtom space="2" texclass="ORD"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.389em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.413em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-TeXAtom></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:25.28em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.833em;"><mjx-mtd id="mjx-eqn:3.9"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c39"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.833em;vertical-align:-1.329em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3.9)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><msup><mrow data-mjx-texclass="ORD"><mover><msub><mi>F</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mrow data-mjx-texclass="ORD"><mi>p</mi><mi>r</mi><mi>e</mi><mi>s</mi><mi>s</mi><mi>u</mi><mi>r</mi><mi>e</mi></mrow></msup><mo>=</mo><mo>−</mo><munder><mo data-mjx-texclass="OP">∑</mo><mi>j</mi></munder><mrow data-mjx-texclass="ORD"><mfrac><mrow><msub><mi>m</mi><mi>j</mi></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mi>i</mi></msub><mo>+</mo><msub><mi>p</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><msub><mi>ρ</mi><mi>j</mi></msub></mrow></mfrac><mi mathvariant="normal">∇</mi><mi>W</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>j</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>对于单个粒子产生的压力<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></mjx-assistive-mml></mjx-container>,可以用理想气体状态方程计算</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:12.909em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:12.909em;" side="right"><mjx-table style="width:auto;min-width:6.197em;margin:0 3.356em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D43E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:12.909em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:1em;"><mjx-mtd id="mjx-eqn:3.10"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:1em;vertical-align:-0.25em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3.10)</mtext></mtd><mtd><mi>p</mi><mo>=</mo><mi>K</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo>−</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>其中<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mn>0</mn></msub></math></mjx-assistive-mml></mjx-container>是流体的静态密度，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43E TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>K</mi></math></mjx-assistive-mml></mjx-container>是和流体相关的常数，只跟温度相关。<br> 压力计算中使用的光滑核函数称为Spiky函数</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:31.041em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:31.041em;" side="right"><mjx-table style="width:auto;min-width:24.329em;margin:0 3.356em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44A TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.104em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D446 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.281em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-s3"><mjx-c class="mjx-c7B TEX-S3"></mjx-c></mjx-mo><mjx-mtable style="min-width:12.211em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:left;padding-right:0.5em;padding-bottom:0.1em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.04em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D446 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-bottom:0.1em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2264"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2264"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:left;padding-right:0.5em;padding-top:0.1em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-top:0.1em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mtext class="mjx-n" space="2"><mjx-c class="mjx-c6F"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D464 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n" style="vertical-align:0.25em;"></mjx-mo></mjx-mrow><mjx-mspace></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c77"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c68"></mjx-c></mjx-mtext><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.281em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:31.041em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.399em;"><mjx-mtd id="mjx-eqn:3.11"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.399em;vertical-align:-0.949em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3.11)</mtext></mtd><mtd><msub><mi>W</mi><mrow data-mjx-texclass="ORD"><mi>S</mi><mi>p</mi><mi>i</mi><mi>k</mi><mi>y</mi></mrow></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mtable columnalign="left left" columnspacing="1em" rowspacing=".2em"><mtr><mtd><msub><mi>K</mi><mrow data-mjx-texclass="ORD"><mi>S</mi><mi>p</mi><mi>i</mi><mi>k</mi><mi>y</mi></mrow></msub><mtext> </mtext><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mi>r</mi><msup><mo stretchy="false">)</mo><mn>3</mn></msup></mtd><mtd><mo>,</mo><mn>0</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><mi>h</mi></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>,</mo><mtext>o</mtext><mi>t</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>w</mi><mi>i</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo></mrow><mspace linebreak="newline"></mspace><mtext>with</mtext><mtext> </mtext><mi>r</mi><mo>=</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-texclass="ORD" stretchy="false">|</mo></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>在3D情况下，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.04em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D446 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>K</mi><mrow data-mjx-texclass="ORD"><mi>S</mi><mi>p</mi><mi>i</mi><mi>k</mi><mi>y</mi></mrow></msub><mo>=</mo><mn>15</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>π</mi><msup><mi>h</mi><mn>6</mn></msup><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:28.995em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:28.995em;" side="right"><mjx-table style="width:auto;min-width:22.283em;margin:0 3.356em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44A TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.104em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D446 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.281em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.281em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:28.995em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.084em;"><mjx-mtd id="mjx-eqn:3.12"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.084em;vertical-align:-0.731em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3.12)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∇</mi><msub><mi>W</mi><mrow data-mjx-texclass="ORD"><mi>S</mi><mi>p</mi><mi>i</mi><mi>k</mi><mi>y</mi></mrow></msub><mtext> </mtext><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>15</mn><mrow><mi>π</mi><msup><mi>h</mi><mn>6</mn></msup></mrow></mfrac><mi mathvariant="normal">∇</mi><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mi>r</mi><msup><mo stretchy="false">)</mo><mn>3</mn></msup><mo>=</mo><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow><mfrac><mn>45</mn><mrow><mi>π</mi><msup><mi>h</mi><mn>6</mn></msup><mi>r</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mi>r</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>将公式3.12带入3.9，可以整理出公式3.2中压力产生的加速度部分</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:40.871em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:40.871em;" side="right"><mjx-table style="width:auto;min-width:34.159em;margin:0 3.356em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-msup><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.428em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:0.572em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.389em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-munder space="2"><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.576em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-under></mjx-row></mjx-munder><mjx-TeXAtom space="2" texclass="ORD"><mjx-mrow><mjx-mo class="mjx-s4"><mjx-c class="mjx-c28 TEX-S4"></mjx-c></mjx-mo><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.389em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.413em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-s4"><mjx-c class="mjx-c29 TEX-S4"></mjx-c></mjx-mo></mjx-mrow></mjx-TeXAtom></mjx-TeXAtom></mjx-mstyle><mjx-mspace></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c77"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c68"></mjx-c></mjx-mtext><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.389em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.413em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:40.871em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:3.078em;"><mjx-mtd id="mjx-eqn:3.13"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:3.078em;vertical-align:-1.329em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3.13)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><msup><mrow data-mjx-texclass="ORD"><mover><msub><mi>a</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mrow data-mjx-texclass="ORD"><mi>p</mi><mi>r</mi><mi>e</mi><mi>s</mi><mi>s</mi><mi>u</mi><mi>r</mi><mi>e</mi></mrow></msup><mo>=</mo><mo>−</mo><mfrac><mrow><mi mathvariant="normal">∇</mi><mi>p</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow><msub><mi>ρ</mi><mi>i</mi></msub></mfrac><mo>=</mo><mi>m</mi><mfrac><mn>45</mn><mrow><mi>π</mi><msup><mi>h</mi><mn>6</mn></msup></mrow></mfrac><munder><mo data-mjx-texclass="OP">∑</mo><mi>j</mi></munder><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mfrac><mrow><msub><mi>p</mi><mi>i</mi></msub><mo>+</mo><msub><mi>p</mi><mi>j</mi></msub></mrow><mrow><mn>2</mn><msub><mi>ρ</mi><mi>i</mi></msub><msub><mi>ρ</mi><mi>j</mi></msub></mrow></mfrac><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mi>r</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mfrac><mrow><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>j</mi></msub><mo stretchy="false">→</mo></mover></mrow></mrow><mi>r</mi></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mrow></mstyle><mspace linebreak="newline"></mspace><mtext>with</mtext><mtext> </mtext><mi>r</mi><mo>=</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>j</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-texclass="ORD" stretchy="false">|</mo></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><hr><h4 id="粘度" tabindex="-1"><a class="header-anchor" href="#粘度"><span><strong>粘度</strong></span></a></h4><p>现在把注意力集中到公式3.2中最后一部分，由粘度产生的作用力</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:29.138em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:29.138em;" side="right"><mjx-table style="width:auto;min-width:22.426em;margin:0 3.356em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-msup><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.485em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:0.81em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D707 TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D707 TEX-I"></mjx-c></mjx-mi><mjx-munder space="2"><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.576em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-under></mjx-row></mjx-munder><mjx-TeXAtom space="2" texclass="ORD"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.473em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-msup><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.389em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.413em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-TeXAtom></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:29.138em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.606em;"><mjx-mtd id="mjx-eqn:3.14"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.606em;vertical-align:-1.329em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3.14)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><msup><mrow data-mjx-texclass="ORD"><mover><msub><mi>F</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mrow data-mjx-texclass="ORD"><mi>v</mi><mi>i</mi><mi>s</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>i</mi><mi>t</mi><mi>y</mi></mrow></msup><mo>=</mo><mi>μ</mi><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">(</mo><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>μ</mi><munder><mo data-mjx-texclass="OP">∑</mo><mi>j</mi></munder><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><msub><mi>u</mi><mi>j</mi></msub><mo stretchy="false">→</mo></mover></mrow><mfrac><msub><mi>m</mi><mi>j</mi></msub><msub><mi>ρ</mi><mi>j</mi></msub></mfrac><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mi>W</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>j</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>这个公式同样有不平衡的问题，考虑到公式中的速度其实并不是绝对速度，而是粒子间的相对速度，所以正确写法应该是：</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:25.926em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:25.926em;" side="right"><mjx-table style="width:auto;min-width:19.214em;margin:0 3.356em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-msup><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.485em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:0.81em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D707 TEX-I"></mjx-c></mjx-mi><mjx-munder space="2"><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.576em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-under></mjx-row></mjx-munder><mjx-TeXAtom space="2" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.473em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.449em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-msup><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.389em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.413em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-TeXAtom></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:25.926em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.928em;"><mjx-mtd id="mjx-eqn:3.15"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.928em;vertical-align:-1.329em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3.15)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><msup><mrow data-mjx-texclass="ORD"><mover><msub><mi>F</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mrow data-mjx-texclass="ORD"><mi>v</mi><mi>i</mi><mi>s</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>i</mi><mi>t</mi><mi>y</mi></mrow></msup><mo>=</mo><mi>μ</mi><munder><mo data-mjx-texclass="OP">∑</mo><mi>j</mi></munder><mrow data-mjx-texclass="ORD"><msub><mi>m</mi><mi>j</mi></msub><mfrac><mrow><mrow data-mjx-texclass="ORD"><mover><msub><mi>u</mi><mi>j</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>u</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow></mrow><msub><mi>ρ</mi><mi>j</mi></msub></mfrac><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mi>W</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>j</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>其中的光滑核函数形式如下：</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:41.295em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:41.295em;" side="right"><mjx-table style="width:auto;min-width:34.583em;margin:0 3.356em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44A TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.104em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.281em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.599em;vertical-align:-1.55em;" class="mjx-c7B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-mid><mjx-c></mjx-c></mjx-mid><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:21.102em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:left;padding-right:0.5em;padding-bottom:0.1em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.04em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mrow space="2"><mjx-mo class="mjx-s3"><mjx-c class="mjx-c28 TEX-S3"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-mpadded><mjx-block style="margin:0.86em 0 0.3em;"><mjx-mrow></mjx-mrow></mjx-block></mjx-mpadded><mjx-mstyle style="font-size:141.4%;"><mjx-TeXAtom texclass="ORD"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-TeXAtom></mjx-mstyle></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-mpadded><mjx-block style="margin:0.86em 0 0.3em;"><mjx-mrow></mjx-mrow></mjx-block></mjx-mpadded><mjx-mstyle style="font-size:141.4%;"><mjx-TeXAtom texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-TeXAtom></mjx-mstyle></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mfrac space="3"><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-mpadded><mjx-block style="margin:0.86em 0 0.3em;"><mjx-mrow></mjx-mrow></mjx-block></mjx-mpadded><mjx-mstyle style="font-size:141.4%;"><mjx-TeXAtom texclass="ORD"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-TeXAtom></mjx-mstyle></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-mpadded><mjx-block style="margin:0.86em 0 0.3em;"><mjx-mrow></mjx-mrow></mjx-block></mjx-mpadded><mjx-mstyle style="font-size:141.4%;"><mjx-TeXAtom texclass="ORD"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-TeXAtom></mjx-mstyle></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mfrac space="3"><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-mpadded><mjx-block style="margin:0.86em 0 0.3em;"><mjx-mrow></mjx-mrow></mjx-block></mjx-mpadded><mjx-mstyle style="font-size:141.4%;"><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-mstyle></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-mpadded><mjx-block style="margin:0.86em 0 0.3em;"><mjx-mrow></mjx-mrow></mjx-block></mjx-mpadded><mjx-mstyle style="font-size:141.4%;"><mjx-TeXAtom texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-mstyle></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-s3"><mjx-c class="mjx-c29 TEX-S3"></mjx-c></mjx-mo></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-bottom:0.1em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2264"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2264"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:left;padding-right:0.5em;padding-top:0.1em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-top:0.1em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mtext class="mjx-n" space="2"><mjx-c class="mjx-c6F"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D464 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n" style="vertical-align:0.25em;"></mjx-mo></mjx-mrow><mjx-mspace></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c77"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c68"></mjx-c></mjx-mtext><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.281em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:41.295em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:3.599em;"><mjx-mtd id="mjx-eqn:3.16"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c36"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:3.599em;vertical-align:-1.55em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3.16)</mtext></mtd><mtd><msub><mi>W</mi><mrow data-mjx-texclass="ORD"><mi>v</mi><mi>i</mi><mi>s</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>i</mi><mi>t</mi><mi>y</mi></mrow></msub><mtext> </mtext><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mtable columnalign="left left" columnspacing="1em" rowspacing=".2em"><mtr><mtd><msub><mi>K</mi><mrow data-mjx-texclass="ORD"><mi>v</mi><mi>i</mi><mi>s</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>i</mi><mi>t</mi><mi>y</mi></mrow></msub><mtext> </mtext><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mo>−</mo><mfrac><mrow><mpadded height="8.6pt" depth="3pt" width="0"><mrow></mrow></mpadded><mstyle displaystyle="false" scriptlevel="0"><mrow data-mjx-texclass="ORD"><msup><mi>r</mi><mn>3</mn></msup></mrow></mstyle></mrow><mrow><mpadded height="8.6pt" depth="3pt" width="0"><mrow></mrow></mpadded><mstyle displaystyle="false" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mn>2</mn><msup><mi>h</mi><mn>3</mn></msup></mrow></mstyle></mrow></mfrac><mo>+</mo><mfrac><mrow><mpadded height="8.6pt" depth="3pt" width="0"><mrow></mrow></mpadded><mstyle displaystyle="false" scriptlevel="0"><mrow data-mjx-texclass="ORD"><msup><mi>r</mi><mn>2</mn></msup></mrow></mstyle></mrow><mrow><mpadded height="8.6pt" depth="3pt" width="0"><mrow></mrow></mpadded><mstyle displaystyle="false" scriptlevel="0"><mrow data-mjx-texclass="ORD"><msup><mi>h</mi><mn>2</mn></msup></mrow></mstyle></mrow></mfrac><mo>+</mo><mfrac><mrow><mpadded height="8.6pt" depth="3pt" width="0"><mrow></mrow></mpadded><mstyle displaystyle="false" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mi>h</mi></mrow></mstyle></mrow><mrow><mpadded height="8.6pt" depth="3pt" width="0"><mrow></mrow></mpadded><mstyle displaystyle="false" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>r</mi></mrow></mstyle></mrow></mfrac><mo>−</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd><mtd><mo>,</mo><mn>0</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><mi>h</mi></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>,</mo><mtext>o</mtext><mi>t</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>w</mi><mi>i</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo></mrow><mspace linebreak="newline"></mspace><mtext>with</mtext><mtext> </mtext><mi>r</mi><mo>=</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-texclass="ORD" stretchy="false">|</mo></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>在3D情况下，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.04em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>K</mi><mrow data-mjx-texclass="ORD"><mi>v</mi><mi>i</mi><mi>s</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>i</mi><mi>t</mi><mi>y</mi></mrow></msub><mo>=</mo><mn>15</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mi>h</mi><mn>3</mn></msup><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:36.976em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:36.976em;" side="right"><mjx-table style="width:auto;min-width:30.264em;margin:0 3.356em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-msup><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44A TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.104em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.281em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mrow><mjx-mo class="mjx-s3"><mjx-c class="mjx-c28 TEX-S3"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mfrac space="3"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mfrac space="3"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-s3"><mjx-c class="mjx-c29 TEX-S3"></mjx-c></mjx-mo></mjx-mrow><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:36.976em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.459em;"><mjx-mtd id="mjx-eqn:3.17"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c37"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.459em;vertical-align:-0.949em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3.17)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><msub><mi>W</mi><mrow data-mjx-texclass="ORD"><mi>v</mi><mi>i</mi><mi>s</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>i</mi><mi>t</mi><mi>y</mi></mrow></msub><mtext> </mtext><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mfrac><mn>15</mn><mrow><mn>2</mn><mi>π</mi><msup><mi>h</mi><mn>3</mn></msup></mrow></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mo>−</mo><mfrac><msup><mi>r</mi><mn>3</mn></msup><mrow><mn>2</mn><msup><mi>h</mi><mn>3</mn></msup></mrow></mfrac><mo>+</mo><mfrac><msup><mi>r</mi><mn>2</mn></msup><msup><mi>h</mi><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mi>h</mi><mrow><mn>2</mn><mi>r</mi></mrow></mfrac><mo>−</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mfrac><mn>45</mn><mrow><mi>π</mi><msup><mi>h</mi><mn>6</mn></msup></mrow></mfrac><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>由此可得到公式3.2的粘度部分</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:32.437em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:32.437em;" side="right"><mjx-table style="width:auto;min-width:25.725em;margin:0 3.356em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-msup><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.428em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:0.572em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msup><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.485em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:0.81em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msup></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D707 TEX-I"></mjx-c></mjx-mi><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-munder space="2"><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.576em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-under></mjx-row></mjx-munder><mjx-mfrac space="2"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.473em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.449em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2223"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.389em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.413em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2223"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:32.437em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:3.282em;"><mjx-mtd id="mjx-eqn:3.18"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c38"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:3.282em;vertical-align:-1.329em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3.18)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><msup><mrow data-mjx-texclass="ORD"><mover><msub><mi>a</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mrow data-mjx-texclass="ORD"><mi>v</mi><mi>i</mi><mi>s</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>i</mi><mi>t</mi><mi>y</mi></mrow></msup><mo>=</mo><mfrac><msup><mrow data-mjx-texclass="ORD"><mover><msub><mi>F</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mrow data-mjx-texclass="ORD"><mi>v</mi><mi>i</mi><mi>s</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>i</mi><mi>t</mi><mi>y</mi></mrow></msup><msub><mi>ρ</mi><mi>i</mi></msub></mfrac><mo>=</mo><mi>m</mi><mi>μ</mi><mfrac><mn>45</mn><mrow><mi>π</mi><msup><mi>h</mi><mn>6</mn></msup></mrow></mfrac><munder><mo data-mjx-texclass="OP">∑</mo><mi>j</mi></munder><mfrac><mrow><mrow data-mjx-texclass="ORD"><mover><msub><mi>u</mi><mi>j</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>u</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow></mrow><mrow><msub><mi>ρ</mi><mi>i</mi></msub><msub><mi>ρ</mi><mi>j</mi></msub></mrow></mfrac><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mo>∣</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>j</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>∣</mo><mo stretchy="false">)</mo></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><hr><p>把公式3.13和3.17带入3.2，可以得到，对于粒子<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></mjx-assistive-mml></mjx-container>，它的加速度可以由下面的公式计算</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:48.591em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:48.591em;" side="right"><mjx-table style="width:auto;min-width:41.879em;margin:0 3.356em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.266em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-munder space="2"><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.576em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-under></mjx-row></mjx-munder><mjx-mrow space="2"><mjx-mo class="mjx-s4"><mjx-c class="mjx-c28 TEX-S4"></mjx-c></mjx-mo><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.389em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.413em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-s4"><mjx-c class="mjx-c29 TEX-S4"></mjx-c></mjx-mo></mjx-mrow><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D707 TEX-I"></mjx-c></mjx-mi><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-munder space="2"><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.576em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-under></mjx-row></mjx-munder><mjx-mfrac space="2"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.473em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.449em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-TeXAtom></mjx-mstyle><mjx-mspace></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c77"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c68"></mjx-c></mjx-mtext><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.389em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.413em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:48.591em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:3.078em;"><mjx-mtd id="mjx-eqn:3.19"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c39"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:3.078em;vertical-align:-1.329em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3.19)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">(</mo><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mrow data-mjx-texclass="ORD"><mover><mi>g</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><mi>m</mi><mfrac><mn>45</mn><mrow><mi>π</mi><msup><mi>h</mi><mn>6</mn></msup></mrow></mfrac><munder><mo data-mjx-texclass="OP">∑</mo><mi>j</mi></munder><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mfrac><mrow><msub><mi>p</mi><mi>i</mi></msub><mo>+</mo><msub><mi>p</mi><mi>j</mi></msub></mrow><mrow><mn>2</mn><msub><mi>ρ</mi><mi>i</mi></msub><msub><mi>ρ</mi><mi>j</mi></msub></mrow></mfrac><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mi>r</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mfrac><mrow><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>j</mi></msub><mo stretchy="false">→</mo></mover></mrow></mrow><mi>r</mi></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mi>m</mi><mi>μ</mi><mfrac><mn>45</mn><mrow><mi>π</mi><msup><mi>h</mi><mn>6</mn></msup></mrow></mfrac><munder><mo data-mjx-texclass="OP">∑</mo><mi>j</mi></munder><mfrac><mrow><mrow data-mjx-texclass="ORD"><mover><msub><mi>u</mi><mi>j</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>u</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow></mrow><mrow><msub><mi>ρ</mi><mi>i</mi></msub><msub><mi>ρ</mi><mi>j</mi></msub></mrow></mfrac><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></mstyle><mspace linebreak="newline"></mspace><mtext>with</mtext><mtext> </mtext><mi>r</mi><mo>=</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>r</mi><mi>j</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-texclass="ORD" stretchy="false">|</mo></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>好了，我们似乎推导出一大推复杂的公式，不用担心，你已经过了最困难的部分，下一节我们来点真的，让这些公式运行起来看看</p></div><!--[--><!--]--></div><footer class="vp-page-meta"><!----><div class="vp-meta-item git-info"><div class="vp-meta-item last-updated"><span class="meta-item-label">Last Updated: </span><!----></div><div class="vp-meta-item contributors"><span class="meta-item-label">Contributors: </span><span class="meta-item-info"><!--[--><!--[--><span class="contributor" title="email: thejinchao@gmail.com">thejinchao</span><!----><!--]--><!--]--></span></div></div></footer><nav class="vp-page-nav" aria-label="page navigation"><a class="route-link auto-link prev" href="/blog/2025/02/SPH002.html" aria-label="Part2"><!--[--><div class="hint"><span class="arrow left"></span> Prev</div><div class="link"><span>Part2</span></div><!--]--></a><a class="route-link auto-link next" 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