Wall and interfacial shear stress in pressure driven two-phase laminar stratified pipe flow

Abstract

Two-phase pressure driven laminar stratified pipe flow is studied with emphasis on the wall and interfacial shear stresses. The basic solution of the Navier–Stokes equations is recast into a simpler form, alleviating physical interpretation and constituting a convenient basis for further developments. Utilizing two-phase symmetry facilitates writing the velocity field for each phase as one generic expression, which is then split into a linear combination of two terms. The first term equals the single-phase free surface flow of either phase due to the given driving forces. The second term links the phases together and represents the shear flow given by the interfacial drag from the opposite phase. The corresponding expression for the interfacial shear stress now emerges naturally as part of the solution of the boundary value problem. The wall shear stresses are obtained by formal differentiation. The limiting behaviour of the wall and interfacial shear stresses in the triple points, where the fluid–fluid interface meets the pipe wall, is obtained by application of residue calculus. Surprisingly, it proves possible to integrate out the Fourier integrals in the expressions for the mean wall and interfacial shear stresses. The expressions for the mean wall shear stresses are demonstrated to be equivalent to the fluid momentum balances, thus confirming consistency. The new and remarkably simple closed form expression for the mean interfacial shear stress, however, represents the local solution of the boundary value problem at the interface, not obtainable from a regular force balance. It thus complements the momentum balances, facilitating a simple computation of the exact general solution for the mean wall and interfacial shear stresses for a given holdup and pressure drop in a given pipe containing a given pair of fluids. The equation system developed here forms the basis for the inversion of the laminar stratified pipe flow problem in terms of flow rates. The corresponding, however, considerably more transparent, channel flow solution is utilized as a guide to the pipe flow problem.