Derivation Notes of Entropy Conservation

Page last edited 297 days ago
From Wei-Liang Qian
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<math>\begin{align}

n_pTdS_{N_p}=d \varepsilon - \frac{(P+\varepsilon)}{n_p}dn_p \end{align}</math>

其中 <math>n_p=\frac{N_p}{V}</math> ,而 <math>S_{N_p}=\frac{S}{N_p}</math> 为比熵,即单位守恒荷的熵.我们将在最后附上该公式的简单证明.利用此表达式,单位守恒荷的熵随固有时的演化为

<math>\begin{align}

n_pT DS_{N_p}=n_pT\frac{dS_{N_p}}{d\tau}=\frac{d \varepsilon}{d\tau} - \frac{(P+\varepsilon)}Template:N p\frac{d n_p}{d\tau}=D \varepsilon - \frac{(P+\varepsilon)}{n_p}Dn_p \end{align}</math>

注意到等式的右边正是体系的演化方程

<math>\begin{align}

u\cdot \partial \varepsilon - (P+\varepsilon)\frac{u \cdot \partial n_p}{n_p}=u\cdot \partial \varepsilon - (P+\varepsilon)\partial \cdot u=0 \end{align}</math>

这里我们证明这个关系式,其实只要注意到

<math>\begin{align}

n_pTDS_{N_p}=n_pTu\cdot \partial\left(\frac{s}{n_p}\right)=T \partial \cdot\left(\left(\frac{s}{n_p}\right)(n_pu)\right)=T \partial \cdot(su) \end{align}</math>

借助上面的推导过程即得所需表达式

<math>\begin{align}

&n_pTDS_{N_p}=T \partial \cdot(su) \\ &=TDs+Ts\partial\cdot u \\ &=TDs+Ts\partial\cdot u+\sum_p\mu_p\partial\cdot (n_pu) \\ &=\left(TDs+\sum_p\mu_pDn_p\right)+\left(Ts+\sum_p\mu_pn_p\right)\partial\cdot u \\ &=D\varepsilon+(\varepsilon+P)\partial\cdot u=0 \end{align}</math>