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The unsteady axisymmetric incompressible flow between two concentric spheres was discussed in this paper. It is useful to most astrophysical, geophysical and engineering applications. In order to get the existence and uniqueness of weak solution of this flow with the stream-velocity form, firstly, the relations among the nonlinear terms in this equation is found; then, the existence is proved by an auxiliary semi-discrete scheme and a compactness argument.

A special penalty method is presented to improve the accuracy of the standard penalty method for solving Stokes equation with nonconforming finite element. It is shown that this method with a larger penalty parameter can achieve the same accuracy as the standard method with a smaller penalty parameter. The convergence rate of the standard method is just half order of this penalty method when using the same penalty parameter, while the extrapolation method proposed by Faik et al can not yield so high accuracy of convergence. At last, we also get the super-convergence estimates for total flux.