Stratified laminar flow of two immiscible fluids

We consider the steady co-current flow of two immiscible viscous liquids in a horizontal pipe, the fluid interface being ripple-free and plane. It is shown that if the cross-section of the duct is symmetric with respect to the interface, the velocity distribution may be expressed in terms of two separate pipe-flow solutions. One corresponds to the flow of a single fluid occupying the whole of the pipe, and the second to a similar flow in a pipe whose cross-section coincides with that of the region occupied by one fluid in the two-phase motion. Flow rates and other quantities of engineering interest are evaluated, and several particular flows are discussed.