Hi, is the exact solution to axisymmetric stokes flows (steady and incompressible) in pipe found?

if yes, what's it, or where to find it?

thanks a lot

In any text book on fluid mechanics - the solutions a parabola just as in the cartaesian channel case.

If you're feeling adventurous you can also write down the unsteady solution for this problem.

thanks a lot, but I meant the solution in non-fully developed region; suppose the inform inflow is uniform.

It depends on the boundary conditions and what you assume. For general boundary conditions the answer is NO since you'll need to solve the fully nonlinear equations analytically. If your inflow profile matches the fully devolped profile and you have Neumann conditions at the outlet then the anylitic solution is the fully developed solution everywhere.

Special cases with the Reynolds number (R) <<1 and >>1 are possible. You could also get somewhere with the case where the inflow differs from the fully developed profile by a small amout (or even mixtures of these three cases).