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
Famous quotes containing the word equation:
“Jail sentences have many functions, but one is surely to send a message about what our society abhors and what it values. This week, the equation was twofold: female infidelity twice as bad as male abuse, the life of a woman half as valuable as that of a man. The killing of the woman taken in adultery has a long history and survives today in many cultures. One of those is our own.”
—Anna Quindlen (b. 1952)