Closure relations for the shear stresses in two-fluid models for laminar stratified flow

New closure relations for the wall and interfacial shear stresses are suggested for the two-fluid model for stratified laminar pipe flows. The new closure relations are formulated in terms of the single-phase-based expressions, which are augmented by the two-phase interaction factors. o-fluid model is tested against the exact solution of fully developed laminar pipe flow with a flat and smooth interface. The predictions of the two-fluid model for the pressure gradient and holdup favorably compare with those obtained by the exact solution for a wide range of dimensionless parameters in co-current and counter-current laminar flows. The two-fluid model is also capable of handling the change in the direction of the wall shear-stress when gravity driven backflow in either of the phases is encountered. Additionally, it also correctly predicts the multiple holdups and the operational conditions for their possible occurrence.