Numerical analysis of a free piston problem

The problem considered is the Stokes and Navier-Stokes flow through a system of two pipes in the gravity field; inside a vertical pipe there is a free heavy piston. Theoretical analysis, the existence and non-uniqueness of solution, has been completed recently by the authors. Here we present numerical analysis, using finite elements methods, of the stationary state with respect to the angle between the two pipes, diameters of the pipes, search for solution of a full problem and search for bifurcation points. The analysis is carried out for both, 2D and 3D, and for Stokes and Navier-Stokes case.