Cauchy momentum equation

The Cauchy momentum equation describes how the velocity of a fluid or deformable solid changes over time due to forces acting on it. It is a statement of Newton’s second law applied to continuous material.

Differential form

$$\frac{\partial }{\partial t}\mathbf{u}+(\mathbf{u}\cdot \nabla )\mathbf{u}=\frac{1}{\rho }\nabla \cdot \mathbf{\sigma }+\mathbf{f}$$

where
$\mathbf{u}(\mathbf{x},t)$	flow velocity vector field, measured in meters per second $\frac{\text{m}}{\text{s}}$
$\mathbf{x}$	position vector in three-dimensional space, typically expressed as $\mathbf{x}=(x,y,z)$ and measured in meters $\text{m}$
$t$	time, measured in seconds $\text{s}$
$\nabla$	nabla symbol which denotes the three-dimensional gradient operator
$\rho$	density of the continuum, measured in kilograms per cubic meter $\frac{\text{kg}}{{\text{m}}^3}$
$\mathbf{\sigma }(\mathbf{x})$	stress tensor, measured in pascals $\text{Pa}$
$\nabla \cdot$	divergence operator, which when applied to $\mathbf{\sigma }$ is: $\nabla \cdot \mathbf{\sigma }=\left[\begin{array}{c}\frac{\partial }{\partial x}{\sigma }_{xx}+\frac{\partial }{\partial y}{\sigma }_{yx}+\frac{\partial }{\partial z}{\sigma }_{zx}\\ \frac{\partial }{\partial x}{\sigma }_{xy}+\frac{\partial }{\partial y}{\sigma }_{yy}+\frac{\partial }{\partial z}{\sigma }_{zy}\\ \frac{\partial }{\partial x}{\sigma }_{xz}+\frac{\partial }{\partial y}{\sigma }_{yz}+\frac{\partial }{\partial z}{\sigma }_{zz}\end{array}\right],$ measured in pascals per cubic meter $\frac{\text{Pa}}{{\text{m}}^3}$
$\mathbf{f}$	vector containing all accelerations caused by body forces, measured in newtons $\text{N}$