Axially symmetric steady motion of a viscous incompressible fluid: some numerical experiments

This note examines an axially symmetric steady motion of a viscous incompressible fluid in a circular cylinder of constant radius. A boundary layer type assumption is made which reduces the Navier-Stokes equations from elliptic to parabolic and renders the solution independent of downstream data. Explicit methode of solution are shown to be unsatisfactory and in practice the fully implicit method has advantages over the Crank-Nicolson method. Hagen-Poiseuille now is used as a test case. Two examples are solved with initial nows of arbitrary form, which correctly tend to Hagen-Poiseuille flow as the numerical solution is continued downstream. The more general problem of motion in an arbitrary cylindrical region r = R(x) (either an actual boundary or a surface in the flow) should be soluble by the same method.