Stratified laminar flows in a circular pipe: New analytical solutions in terms of elementary functions

Laminar stratified two-phase flows in circular pipes are studied. Under the assumption that the wall-liquid wettability properties of the phases are identical, new exact analytical solutions in terms of elementary functions are constructed. The solutions satisfy the Navier-Stokes equations exactly in fluids and the boundary conditions on the pipe walls and the interface for the two cases: the pipe is horizontal and the capillarity forces dominate the gravity ones, the pipe is inclined and the volumetric quantities of liquids in the pipe are the same. For the second case, the capillarity and gravity forces can be arbitrary, but if the gravity forces dominate the capillarity ones, the assumption about the equal wall-liquid tensions of the phases can be withdrawn.