Cauchy momentum equation
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.
where 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 stress tensor is one of the forces that constitutes .
By expressing the shear stress in terms of viscosity and fluid velocity, and assuming constant density and viscosity, the Cauchy momentum equation will lead to the Navier–Stokes equations. By assuming inviscid flow, the Navier–Stokes equations can further simplify to the Euler equations.