Navier-Stokes Equations

The Navier Stokes Equations express the mass conservation laws and momentum conservation laws for a fluid. They define how the mass and momentum are transported across the field. Hence they are called Mass transport equation and momentum transport equation respectively.

Continuity equation

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

For an incompressible fluid

$$\boxed{\nabla \cdot \mathbf{u}=0}$$

Momentum Transport Equation

$$\frac{\partial (\rho \mathbf{U})}{\partial t}+\nabla \cdot (\rho \mathbf{U}\mathbf{U})=\underset{\text{Pressure}}{\underbrace{-\nabla p}}+\underset{\text{Shear Stress}}{\underbrace{\nabla \cdot \tau }}+\underset{\text{gravity}}{\underbrace{\rho g}}$$

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References

1. Momentum The Navier-Stokes Equations