Three-Dimensional Mixed Formulation for Creeping Flow Problems

3 Three-Dimensional Mixed Formulation for Creeping Flow Problems 

In this chapter, we have derived the two-dimensional finite element penalty formulation for creeping flows where the pressure was eliminated by assuming a compressible flow. Here, we will use a mixed formulation, where the pressure is included among the unknown variables. In the mixed formulation, we use different order of approximation for the pressure as we will for the velocity. For instance, if tetrahedral elements are used, we can use a quadratic representation for the velocity (10 nodes) and a linear representation for the pressure (4 nodes). Hence, we must use different shape functions for the velocity and pressure. For such a formulation we can write 

Similar to the two-dimensional formulation for creeping flow, the terms in the above equation are defined by 

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