Cauchy momentum equation

The Cauchy momentum equation is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum: [cite book
last = Acheson
first = D. J.
title = Elementary Fluid Dynamics
publisher = Oxford University Press
year = 1990
isbn = 0198596790]

:$ho\; frac\{d\; mathbf\{v\{d\; t\}\; =\; abla\; cdot\; sigma\; +\; mathbf\{f\}$

or, with the derivative expanded out,

:$ho\; left(frac\{partial\; mathbf\{v\{partial\; t\}\; +\; mathbf\{v\}\; cdot\; abla\; mathbf\{v\}\; ight)\; =\; abla\; cdot\; sigma\; +\; mathbf\{f\}$

where $ho$ is the density of the continuum, $sigma$ is the stress tensor, and $mathbf\{f\}$ contains all of the body forces (normally just gravity). $mathbf\{v\}$ is the velocity vector field, which depends on time and space.

The stress tensor is sometimes split into pressure and the deviatoric stress tensor:

:$sigma\; =\; -pmathbb\{I\}\; +\; mathbb\{T\}$

where $scriptstyle\; mathbb\{I\}$ is the $scriptstyle\; 3\; imes\; 3$ identity matrix and $scriptstyle\; mathbb\{T\}$ the deviatoric stress tensor. The divergence of the stress tensor can be written as

:$abla\; cdot\; sigma\; =\; -\; abla\; p\; +\; abla\; cdotmathbb\{T\}$

All non-relativistic momentum conservation equations, such as the Navier-Stokes equation, can be derived by beginning with the Cauchy momentum equation and specifying the stress tensor through a constitutive relation.

Derivation

Applying Newton's second law ($i^\{th\}$ component) to a control volume in the continuum being modeled gives:

:$m\; a\_i\; =\; F\_i,$

:$ho\; int\_\{Omega\}\; frac\{d\; u\_i\}\{d\; t\}\; ,\; dV\; =\; int\_\{Omega\}\; frac\{partial\; sigma\_\{ij\{partial\; x\_j\}\; ,\; dV\; +\; int\_\{Omega\}\; f\_i\; ,\; dV$

:$ho\; int\_\{Omega\}\; (frac\{d\; u\_i\}\{d\; t\}\; -\; frac\{partial\; sigma\_\{ij\{partial\; x\_j\}\; -\; f\_i\; ),\; dV\; =\; 0$

where $Omega$ represents the control volume. Since this equation must hold for any control volume, it must be true that the integrand is zero, from this the Cauchy momentum equation follows. The main challenge in deriving this equation is establishing that the derivative of the the stress tensor is one of the forces that constitutes $F\_i$.

References