One of the Millennium Prize Problems is the namesake existence and smoothness of solutions to these equations.
A formal averaging procedure can be used on these to obtain Darcy's Law. By using these equations and applying Newton's first law to a case with zero acceleration, the Hagen-Poiseuille equation can be derived. When heat conductivity and another quantity are zero, these equations simplify to the Euler equations. One of the Millennium Prize Problems is the namesake existence and smoothness of solutions to these equations. This set of equations is useful only at low Knudsen numbers, and solutions are easier to find when the Reynolds number is small. For 10 points, name this set of nonlinear partial differential equations that describe the motion of fluids.
A formal averaging procedure can be used on these to obtain Darcy's Law.
Navier-Stokes equations