# 纳维-斯托克斯方程

（重定向自纳维－斯托克斯方程

## 基本假设

### 随質导数

${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}(\star )={\frac {\partial (\star )}{\partial t}}+(\mathbf {v} \cdot \nabla )(\star )}$

L的守恒定律在一个控制体积上的积分形式是：

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{\Omega }\mathbf {L} \,\mathrm {d} \Omega =0}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{\Omega }\mathbf {L} \,\mathrm {d} \Omega =\int _{\Omega }{\frac {\partial }{\partial t}}\mathbf {L} \,\mathrm {d} \Omega +\int _{\partial \Omega }\mathbf {L} \left(\mathbf {v} \cdot \mathbf {n} \,\mathrm {d} \partial \Omega \right)=\int _{\Omega }\left[{\frac {\partial }{\partial t}}\mathbf {L} +\nabla \cdot \left(\mathbf {L} \mathbf {v} \right)\right]\mathrm {d} \Omega =0}$

${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\mathbf {L} +\left(\nabla \cdot \mathbf {v} \right)\mathbf {L} ={\frac {\partial }{\partial t}}\mathbf {L} +\nabla \cdot \left(\mathbf {v} \mathbf {L} \right)=0}$

### 守恒定律

NS方程可以从守恒定律通过上述变换导出，并且需要用状态定律闭合

#### 连续性方程

${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \mathbf {v} )=0}$

${\displaystyle \rho }$是流体的密度。

${\displaystyle \nabla \cdot \mathbf {v} =0}$

#### 动量守恒

${\displaystyle {\frac {\partial }{\partial t}}\left(\rho \mathbf {v} \right)+\nabla (\rho \mathbf {v} \otimes \mathbf {v} )=\sum \rho \mathbf {f} }$

${\displaystyle \rho {\frac {D\mathbf {v} }{Dt}}=\sum \rho \mathbf {f} }$

## 方程组

### 一般形式

#### 方程组的形式

${\displaystyle \rho {\frac {\mathrm {D} \mathbf {v} }{\mathrm {D} t}}=\nabla \cdot \mathbb {P} +\rho \mathbf {f} }$

${\displaystyle \mathbb {P} ={\begin{pmatrix}\sigma _{xx}&\tau _{xy}&\tau _{xz}\\\tau _{yx}&\sigma _{yy}&\tau _{yz}\\\tau _{zx}&\tau _{zy}&\sigma _{zz}\end{pmatrix}}=-{\begin{pmatrix}p&0&0\\0&p&0\\0&0&p\end{pmatrix}}+{\begin{pmatrix}\sigma _{xx}+p&\tau _{xy}&\tau _{xz}\\\tau _{yx}&\sigma _{yy}+p&\tau _{yz}\\\tau _{zx}&\tau _{zy}&\sigma _{zz}+p\end{pmatrix}}}$

${\displaystyle \sigma _{xx}+\sigma _{yy}+\sigma _{zz}}$在流体处于平衡态时为0。这等价于流体粒子上的法向力的积分为0。

${\displaystyle {\frac {\mathrm {D} \rho }{\mathrm {D} t}}+\rho \nabla \cdot \mathbf {v} =0}$

p是压强

${\displaystyle \rho {\frac {\mathrm {D} \mathbf {v} }{\mathrm {D} t}}=-\nabla p+\nabla \cdot \mathbb {T} +\rho \mathbf {f} }$

#### 闭合问题

${\displaystyle \mathbb {P} }$的分量是流体的一个无穷小元上面的约束。它们代表垂直和剪切约束。${\displaystyle \mathbb {P} }$对称的，除非存在非零的自旋密度

## 特殊形式

### 牛顿流体

${\displaystyle \tau _{ij}=\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}-{\frac {2}{3}}\delta _{ij}\nabla \cdot \mathbf {v} \right)}$

${\displaystyle \mu }$是液体的粘滞度
${\displaystyle \rho \left({\frac {\partial \mathbf {v} }{\partial t}}+\nabla _{\mathbf {v} }\mathbf {v} \right)=\rho \mathbf {f} -\nabla p+\mu \left(\Delta \mathbf {v} +{\frac {1}{3}}\nabla \left(\nabla \cdot \mathbf {v} \right)\right)}$
${\displaystyle \rho \left({\frac {\partial v_{i}}{\partial t}}+v_{j}{\frac {\partial v_{i}}{\partial x_{j}}}\right)=\rho f_{i}-{\frac {\partial p}{\partial x_{i}}}+\mu \left({\frac {\partial ^{2}v_{i}}{\partial x_{j}\partial x_{j}}}+{\frac {1}{3}}{\frac {\partial ^{2}v_{j}}{\partial x_{i}\partial x_{j}}}\right)}$

${\displaystyle \rho \cdot \left({\partial u \over \partial t}+u{\partial u \over \partial x}+v{\partial u \over \partial y}+w{\partial u \over \partial z}\right)=k_{x}-{\partial p \over \partial x}+{\partial \over \partial x}\left[\mu \cdot \left(2\cdot {\partial u \over \partial x}-{\frac {2}{3}}\cdot (\nabla \cdot {\vec {v}}\right)\right]+{\partial \over \partial y}\left[\mu \cdot \left({\partial u \over \partial y}+{\partial v \over \partial x}\right)\right]+{\partial \over \partial z}\left[\mu \cdot \left({\partial w \over \partial x}+{\partial u \over \partial z}\right)\right]}$
${\displaystyle \rho \cdot \left({\partial v \over \partial t}+u{\partial v \over \partial x}+v{\partial v \over \partial y}+w{\partial v \over \partial z}\right)=k_{y}-{\partial p \over \partial y}+{\partial \over \partial y}\left[\mu \cdot \left(2\cdot {\partial v \over \partial y}-{\frac {2}{3}}\cdot (\nabla \cdot {\vec {v}}\right)\right]+{\partial \over \partial z}\left[\mu \cdot \left({\partial v \over \partial z}+{\partial w \over \partial y}\right)\right]+{\partial \over \partial x}\left[\mu \cdot \left({\partial u \over \partial y}+{\partial v \over \partial x}\right)\right]}$
${\displaystyle \rho \cdot \left({\partial w \over \partial t}+u{\partial w \over \partial x}+v{\partial w \over \partial y}+w{\partial w \over \partial z}\right)=k_{z}-{\partial p \over \partial z}+{\partial \over \partial z}\left[\mu \cdot \left(2\cdot {\partial w \over \partial z}-{\frac {2}{3}}\cdot (\nabla \cdot {\vec {v}}\right)\right]+{\partial \over \partial x}\left[\mu \cdot \left({\partial w \over \partial x}+{\partial u \over \partial z}\right)\right]+{\partial \over \partial y}\left[\mu \cdot \left({\partial v \over \partial z}+{\partial w \over \partial y}\right)\right]}$

${\displaystyle {\partial \rho \over \partial t}+{\partial (\rho \cdot u) \over \partial x}+{\partial (\rho \cdot v) \over \partial y}+{\partial (\rho \cdot w) \over \partial z}=0}$

${\displaystyle \rho \left({\partial e \over \partial t}+u{\partial e \over \partial x}+v{\partial e \over \partial y}+w{\partial e \over \partial z}\right)=\left({\partial \over \partial x}\left(\lambda \cdot {\partial T \over \partial x}\right)+{\partial \over \partial y}\left(\lambda \cdot {\partial T \over \partial y}\right)+{\partial \over \partial z}\left(\lambda \cdot {\partial T \over \partial z}\right)\right)-p\cdot \left(\nabla \cdot {\vec {v}}\right)+{\vec {k}}\cdot {\vec {v}}+\rho \cdot {\dot {q}}_{s}+\mu \cdot \Phi }$

${\displaystyle \Phi =2\cdot \left[\left({\partial u \over \partial x}\right)^{2}+\left({\partial v \over \partial y}\right)^{2}+\left({\partial w \over \partial z}\right)^{2}\right]+\left({\partial v \over \partial x}+{\partial u \over \partial y}\right)^{2}+\left({\partial w \over \partial y}+{\partial v \over \partial z}\right)^{2}+\left({\partial u \over \partial z}+{\partial w \over \partial x}\right)^{2}-{\frac {2}{3}}\cdot \left({\partial u \over \partial x}+{\partial v \over \partial y}+{\partial w \over \partial z}\right)^{2}}$

${\displaystyle e=c_{p}\cdot T-{\frac {p}{\rho }}}$

### 宾汉（Bingham）流体

${\displaystyle \tau _{ij}=\tau _{0}+\mu {\frac {\partial v_{i}}{\partial x_{j}}},\;{\frac {\partial v_{i}}{\partial x_{j}}}>0}$

### 幂律流体

${\displaystyle \tau =K\left({\frac {\partial u}{\partial y}}\right)^{n}}$

### 不可壓缩流體

${\displaystyle \overbrace {\rho {\Big (}\underbrace {\frac {\partial \mathbf {v} }{\partial t}} _{\begin{smallmatrix}{\text{Unsteady}}\\{\text{acceleration}}\end{smallmatrix}}+\underbrace {(\mathbf {v} \cdot \nabla )\mathbf {v} } _{\begin{smallmatrix}{\text{Convective}}\\{\text{acceleration}}\end{smallmatrix}}{\Big )}} ^{\text{Inertia}}=\underbrace {-\nabla p} _{\begin{smallmatrix}{\text{Pressure}}\\{\text{gradient}}\end{smallmatrix}}+\underbrace {\mu \nabla ^{2}\mathbf {v} } _{\text{Viscosity}}+\underbrace {\mathbf {f} } _{\begin{smallmatrix}{\text{Other}}\\{\text{forces}}\end{smallmatrix}}}$

${\displaystyle \nabla \cdot \mathbf {v} =0}$

${\displaystyle e_{ij}={\frac {1}{2}}\left({\frac {\partial u_{i}}{\partial x_{j}}}+{\frac {\partial u_{j}}{\partial x_{i}}}\right)}$;
${\displaystyle \Delta =e_{ii}}$散度
${\displaystyle \delta _{ij}}$克罗内克记号

${\displaystyle \mu }$在整个流体上均匀，动量方程简化为

${\displaystyle \rho {\frac {Du_{i}}{Dt}}=\rho f_{i}-{\frac {\partial p}{\partial x_{i}}}+\mu \left({\frac {\partial ^{2}u_{i}}{\partial x_{j}\partial x_{j}}}+{\frac {1}{3}}{\frac {\partial \Delta }{\partial x_{i}}}\right)}$

（若${\displaystyle \mu =0}$这个方程称为欧拉方程；那里的重点是可压缩流冲击波）。

${\displaystyle \rho \left({\partial v_{x} \over \partial t}+v_{x}{\partial v_{x} \over \partial x}+v_{y}{\partial v_{x} \over \partial y}+v_{z}{\partial v_{x} \over \partial z}\right)=\mu \left[{\partial ^{2}v_{x} \over \partial x^{2}}+{\partial ^{2}v_{x} \over \partial y^{2}}+{\partial ^{2}v_{x} \over \partial z^{2}}\right]-{\partial p \over \partial x}+\rho g_{x}}$
${\displaystyle \rho \left({\partial v_{y} \over \partial t}+v_{x}{\partial v_{y} \over \partial x}+v_{y}{\partial v_{y} \over \partial y}+v_{z}{\partial v_{y} \over \partial z}\right)=\mu \left[{\partial ^{2}v_{y} \over \partial x^{2}}+{\partial ^{2}v_{y} \over \partial y^{2}}+{\partial ^{2}v_{y} \over \partial z^{2}}\right]-{\partial p \over \partial y}+\rho g_{y}}$
${\displaystyle \rho \left({\partial v_{z} \over \partial t}+v_{x}{\partial v_{z} \over \partial x}+v_{y}{\partial v_{z} \over \partial y}+v_{z}{\partial v_{z} \over \partial z}\right)=\mu \left[{\partial ^{2}v_{z} \over \partial x^{2}}+{\partial ^{2}v_{z} \over \partial y^{2}}+{\partial ^{2}v_{z} \over \partial z^{2}}\right]-{\partial p \over \partial z}+\rho g_{z}}$

${\displaystyle {\partial v_{x} \over \partial x}+{\partial v_{y} \over \partial y}+{\partial v_{z} \over \partial z}=0}$
N-S方程的简化版本。采用《不可压缩流》，Ronald Panton所著第二版

## 参考文献

1. ^ Fluid Mechanics（Schaum's Series）, M. Potter, D.C. Wiggert, Schaum's Outlines, McGraw-Hill (USA), 2008, ISBN 978-0-07-148781-8
2. ^ Vectors, Tensors, and the basic Equations of Fluid Mechanics, R. Aris, Dover Publications, 1989, ISBN (10) 0-486-66110-5
3. ^ Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN（VHC Inc.）0-89573-752-3
• Inge L. Rhyming Dynamique des fluides, 1991 PPUR.
• Polyanin A.D., Kutepov A.M., Vyazmin A.V., Kazenin D.A., Hydrodynamics, Mass and Heat Transfer in Chemical Engineering, Taylor & Francis, London, 2002. ISBN 0-415-27237-8.