On an algorithm of search for periodic points (Q2771584)

Two-dimensional flow of incompressible viscous fluid in a semicircle \(0\leq r\leq a\), \(0\leq \theta\leq \pi\), caused by uniform alternate motion of its circular and straight-line boundaries with constant tangential velocities is considered. In Stokes approximation for stationary flow the stream function \(\psi(x,y)\) satisfies the biharmonic equation. Closed-form expression for the stream function in terms of elementary functions was found earlier by the second author. Motion of a passive particle with coordinates \([x(t), y(t)]\) in the velocity field \([V_x(x,y), V_y(x,y)]\) is determined from the Cauchy problem for first-order differential equations \(\dot{X}= V_x(x,y) =\partial\psi/\partial y\), \(\dot{Y}= V_y(x,y) =-\partial\psi/\partial x\) under initial conditions \(X(0)=X^0\), \(Y(0)=Y^0\). An algorithm for searching periodic points of flow is constructed. Numerical examples are considered, the corresponding Poincaré sections are presented.