May 2022

Tags: CFD 220507 Transport Equation Conservation of Momentum 220507 Navier-Stokes Equations

Momentum Transport equation

The momentum transport equation defines how the momentum is transported across the field. Essentially, it expresses the Newton’s Second Law applied to a fluid.

$$\text{Force}\propto \text{Rate of change of momentum}$$

For a fluid parcel of volume $V$, Newton’s II law can be expressed as follows

$$F=\frac{D}{Dt}(m\mathbf{U})$$

The 220430 Material, Substantial or Lagrangian Derivative is used here as the velocity is function of both time and space. When applied to a fluid parcel, the force per unit volume $f$ is given by

$$\frac{D}{Dt}(\rho \mathbf{U})=f$$

Simplifying

$$\frac{\partial (\rho \mathbf{U})}{\partial t}+\mathbf{U}\cdot \nabla (\rho \mathbf{U})=f$$

It can be shown that

$$\frac{\partial (\rho \mathbf{U})}{\partial t}+\nabla \cdot (\rho \mathbf{U}\mathbf{U})=f$$

The net forces acting on the fluid parcel can be expressed as follows

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

• The pressure term is -ve as the velocity increases in the direction of decreasing pressure.

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References

1. Momentum The Navier-Stokes Equations