Symplectic solution for low Reynolds number flow problems in a wedge cavity

We present a symplectic analytical method for the study of two-dimensional low Reynolds number flow in a wedge-shaped cavity caused by constant unit tangential velocity of the curved wall. Taking velocities and their dual variables as the basic variables, the Hamiltonian formulation can be introduced into low Reynolds number flow problems and a direct method is put forward. In the symplectic space, the problem can be solved via the method of separation of variables and eigenvector expansion. The flow is anti-symmetric with respect to the polar axis, and all values towards the wedge vertex are necessarily finite, so its expansion is composed of the anti-symmetric solution with eigenvalues whose real parts are positive. The direct method is employed to solve the flow problem in the wedge cavity with special corner angle values. Results show that the wedge cavity consists of a sequence of eddies separated by the zero-value streamlines which intersect the wedge lateral wall, and the adjacent eddies rotate in the opposite sense. Numerical examples show that the Hamiltonian method is effective for low Reynolds number flow problems in polar coordinates.