Navier-Stokes Equations for Incompressible Flows

Recall the equations of motion for a fluid element:

$$\begin{array}{rl}x:& \frac{\partial {\sigma }_{xx}}{\partial x}+\frac{\partial {\tau }_{yx}}{\partial y}+\frac{\partial {\tau }_{zx}}{\partial z}+\rho g_x=\rho \left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)\\ y:& \frac{\partial {\tau }_{xy}}{\partial x}+\frac{\partial {\sigma }_{yy}}{\partial y}+\frac{\partial {\tau }_{zy}}{\partial z}+\rho g_y=\rho \left(\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\right)\\ z:& \frac{\partial {\tau }_{xz}}{\partial x}+\frac{\partial {\tau }_{yz}}{\partial y}+\frac{\partial {\sigma }_{zz}}{\partial z}+\rho g_z=\rho \left(\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}\right)\end{array}$$

In the case of a viscous fluid, internal friction (viscosity) creates normal stresses $\sigma$ and shear stresses $\tau$:

$$\begin{array}{r}{\sigma }_{xx}=-p+2\mu \frac{\partial u}{\partial x}\\ {\sigma }_{yy}=-p+2\mu \frac{\partial v}{\partial y}\\ {\sigma }_{zz}=-p+2\mu \frac{\partial w}{\partial z}\end{array}$$ $$\begin{array}{r}{\tau }_{xy}={\tau }_{yx}=\mu \left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)\\ {\tau }_{yz}={\tau }_{zy}=\mu \left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\right)\\ {\tau }_{zx}={\tau }_{xz}=\mu \left(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}\right)\end{array}$$

• $p$ is the pressure
• $\mu$ is the dynamic viscosity of the fluid
• These terms account for both pressure forces and viscous effects.

Substituting these, we arrive at the Navier-Stokes Equations for incompressible flows:

$$\begin{array}{rl}\rho \left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)& =-\frac{\partial p}{\partial x}+\rho g_x+\mu \left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\partial }^{2}u}{\partial z^2}\right)\\ \rho \left(\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\right)& =-\frac{\partial p}{\partial y}+\rho g_y+\mu \left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}+\frac{{\partial }^{2}v}{\partial z^2}\right)\\ \rho \left(\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}\right)& =-\frac{\partial p}{\partial z}+\rho g_z+\mu \left(\frac{{\partial }^{2}w}{\partial x^2}+\frac{{\partial }^{2}w}{\partial y^2}+\frac{{\partial }^{2}w}{\partial z^2}\right)\end{array}$$

These are a generalization of Euler’s Equations of Motion for Inviscid Flows; we just set $\mu =0$ for no viscosity for the inviscid case.

There is a fully generalized form of the Navier-Stokes equations that applies to compressible flows as well:

$$\rho \left(\frac{\partial \mathbf{V}}{\partial t}+(\mathbf{V}\cdot \nabla )\mathbf{V}\right)=-\nabla p+\nabla \cdot \mathbf{T}+\rho \mathbf{g}$$

where $\mathbf{T}$ is a viscous stress tensor.