Accurate solutions for steady plane flow in the driven cavity. I. Stokes flow

摘要

The problem considered here is the steady, incompressible plane Stokes flow in a rectangular cavity generated by uniform translation of the upper wall, An exact analytical solution of the governing biharmonic equation is derived which not only contains the leading term of the required singularities at the upper corners, but also approximately Hutislies the boundary conditions at all four walls. A standard numerical algorithm is employed to correct the small deviations in the boundary conditions satisfied by the analytical solution. This tochnkjUL* enables accurate compulation of the solution uniformly through-out the domain; in particular, near the singular corners and in those regions where the flow is extremely weak, for example, in the secondary vortex regions of the deep cavity. The method is illustrated for the square cavity and also for a deep cavity with a deplh-to-width ratio of live, and the results are compared with previous analytical and numerical solutions.