On a free piston problem for potential ideal fluid flow

摘要

Our goal was to model and analyze a stationary and evolutionary potential ideal fluid flow through the junction of two pipes in the gravity field. Inside the 'vertical' pipe, there is a heavy piston that can freely move along the pipe. In the stationary case, we are interested in the equilibrium position of the piston in dependence on the geometry of junction, and in the evolutionary case, we study motion of the piston also in dependence on geometry. We formulate corresponding initial and boundary value problems and prove the existence results. The problem is nonlinear because the domain is unknown. Furthermore, we study some qualitative properties of the solutions and compare them with the qualitative properties of a free piston problem for Newtonian fluid flow. All theoretical results are illustrated with numerical experiments.