Laminar flow of two immiscible fluids in a plane pipe

The author considers the laminar flow of two immiscible fluids either two liquids or a liquid and a gas - between two flat walls represented by two unbounded planes with constant distance between them. The equations of motion are approximate and take into account the average velocity across a section. The boundary conditions require that the velocity at the walls be zero, that the velocities of the two fluids at the phase boundary be equal, and that the forces with which the two fluids act upon one another be equal and opposite in direction. The equations were solved by the Laplace transform method, and after some transformations, rather lengthy and unwieldy expressions are obtained for the velocity fields of the two fluids and the rate of change of pressure in the two fluid regions.