Navier-Stokes Equations

Continuity Equation

$$\frac{\partial \rho }{\partial t}+\rho \nabla \cdot \vec{u}=0$$ $$\frac{\partial \rho }{\partial t}+\rho \left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)=0$$

Momentum Equations

In vector notation

$$\rho \underset{\text{Acceleration }}{\underbrace{\left(\frac{\partial \vec{u}}{\partial t}+\vec{u}\cdot \nabla \vec{u}\right)}}=\underset{\text{Pressure }}{\underbrace{-\nabla p}}+\underset{\text{Viscosity }}{\underbrace{\mu \Delta \vec{u}}}$$

or

$$\rho \underset{\text{Acceleration }}{\underbrace{\left(\frac{\partial \vec{u}}{\partial t}+\vec{u}\cdot \nabla \vec{u}\right)}}=\underset{\text{Pressure }}{\underbrace{-\nabla p}}+\underset{\text{Shear Stress}}{\underbrace{\nabla \cdot \tau }}$$

where $\tau =\left(\begin{array}{ccc}{\tau }_{xx}& {\tau }_{xy}& {\tau }_{xz}\\ {\tau }_{xy}& {\tau }_{yy}& {\tau }_{yz}\\ {\tau }_{xz}& {\tau }_{yz}& {\tau }_{zz}\end{array}\right)$

In Leibniz notation (2D)

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