The stability of a plane Couette-Poiseuille flow consisting of two layers of different fluids is analyzed. The fluids have different viscosities, densities and relaxation times and there is surface tension at the interface. The center manifold reduction method is used to derive the final amplitude evolution equation. The nonlinear calculations are carried out with two alternative approaches. One approach is to keep the combined volume flux fixed, and the other is to keep the pressure gradient in the horizontal direction fixed. Numerical results are presented for the one-fluid Poiseuille flow, and some two-layer Poiseuille flows with reference to co-extrusion through a slit die. Situations with subcritical and supercritical bifurcations are described.