This Demonstration displays the velocity profile for the flow of two immiscible fluids between two flat plates under the influence of a pressure gradient
. The fluid rates are adjusted so that the slit is half filled with the more dense phase (lower fluid) and half filled with the less dense phase (upper fluid). The governing equation is the momentum balance
, and you also have to use Newton's law of viscosity
. Analytical expressions are derived in R. B. Bird, W. Stewart, and E. N. Lightfoot, Transport Phenomena
, New York: Wiley, 1960. You can see how the velocity profile in the channel changes as you vary the viscosities of the fluids.