The governing equations for a Newtonian fluid satisfying Fourier's law are called the Navier-Stokes equations.
Compressible flow:
The students can write them out themselves.
Incompressible flow:
Assuming constant viscosity, the net stress force per unit volume simplifies a bit:
Conservation form:
Nonconservation form:
Vector form:
Bottom line: for incompressible flow, the net pressure force is the gradient of the pressure (as always) and the net viscous force the Laplacian of the velocity.
Exercise:
What happened to the energy equation?
