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:

or, with the material derivative expanded out,

where is the density of the continuum, is the stress tensor, and contains all of the body forces per unit volume (often simply density times gravity). is the velocity vector field, which depends on time and space.

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

where is the identity matrix and the deviatoric stress tensor. The divergence of the stress tensor can be written as

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.

Read more about Cauchy Momentum Equation:  Derivation

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