Problem 4.21: Consider the steady laminar flow of two immiscible incompressible fluids in...

Problem 4.21: Consider the steady laminar flow of two immiscible incompressible fluids in a region between two parallel stationary plates under the influence of a pressure gradient. The fluid rates are adjusted such that the lower half of the region is filled with Fluid I (the denser and more viscous fluid) and the upper half is filled with Fluid II (the less dense and less viscous fluid), as shown in Fig. P4.21. We wish to determine the velocity distributions in each region using the finite element method.

The governing equations for the two fluids are

where f0 (P0 - PL)/2b is the pressure gradient. The boundary conditions are

Solve the problem using four linear elements, and compare the finite element solutions with the exact solution at the nodes