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. In convective (or Lagrangian) form it is written:
where ρ is the density at the point considered in the continuum (for which the continuity equation holds), σ is the stress tensor, and g contains all of the body forces per unit mass (often simply gravitational acceleration). u is the flow velocity vector field, which depends on time and space.
Notably, it can be written, through an appropriate change of variables, also in conservation (or Eulerian) form:
where j is the momentum density at the point considered in the continuum (for which the continuity equation holds), F is the flux associated to the momentum density, and s contains all of the body forces per unit volume.
Applying Newton's second law (ith component) to a control volume in the continuum being modeled gives:
and basing on the Reynolds transport theorem and on the material derivative notation:
where Ω 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 step (not done above) in deriving this equation is establishing that the derivative of the stress tensor is one of the forces that constitutes Fi.
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