Cauchy Momentum Equation

Statement

The Cauchy Momentum Equation is the general form of the momentum conservation law for any continuous medium (solid or fluid).

Mathematical Form

$\rho\frac{D\mathbf{v}}{Dt} = \nabla\cdot\boldsymbol{\sigma} + \rho\mathbf{g}$ where:

• $\boldsymbol{\sigma}$: Cauchy stress tensor.
• $\mathbf{g}$: Body force vector.

Relationships

• Special Case: Navier-Stokes Equation (for Newtonian fluids)
• Component: Stress Tensor