On the performance of quadrilateral finite elements in the solution to the Stokes equations in periodic structures

The use of the finite element method in solving the problem of flow of a Newtonian fluid in periodically constricted tubes is explored. The performance of eight node serendipity and nine node Lagrangian elements is compared. It was found that the Lagrangian element results in unstable velocity fields when stagnant or recirculation regions are present. This is characteristic of tubes with large expansion zones. The eight node element does not exhibit instabilities. Both elements give accurate pressure fields. This behaviour is contrary to traditional results obtained for flow problems with similar geometrical characteristics. This suggests that the periodicity of the boundary conditions might be the cause of the instabilities in the numerical solution. The use of the continuity equation to simplify the viscous terms in the Stokes equations resulted, in this particular case, in a deterioration of the rate of convergence of the algorithm.