The problem considered is the evolutionary Stokes flow through a system of two pipes in a gravity field; inside vertical pipe there is a heavy rigid body (piston) which can freely move along the vertical pipe and which interacts with the fluid. Although the Stokes equations are linear, the full problem in unknowns fluid pressure, fluid velocity and motion of the piston, is nonlinear. On the lateral boundary of the “vertical” pipe Navier's slip boundary conditions are imposed. We formulate corresponding initial-boundary value problem and prove the existence result. We compare obtained results with corresponding results for potential ideal fluid flow.