On a Free Piston Problem for Stokes and Navier-stokes Equations

Our goal is to model and analyze a stationary fluid flow through the junction of two pipes in the gravity field. Inside ’vertical’ pipe there is a heavy piston which can freely move along the pipe. We are interested in the equilibrium position of the piston in dependence on geometry of junction. Fluid is modeled with the Navier-Stokes equations and the piston is modeled as a rigid body. We formulate corresponding boundary value problem and prove an existence result. The problem is nonlinear even in case of the Stokes equations for fluid flow; we prove non-uniqueness of solutions and illustrate it with some numerical examples. Furthermore, derivation and analysis of the linearized problem are presented.