On the wall drag term in the averaged momentum equation for dispersed flows

When fluid particles such as bubbles and droplets are not in contact with the wall, one probably neglects the wall drag term in the one-dimensional momentum equation for the dispersed phase. This treatment however leads to an unphysical prediction of the motion of the dispersed phase. In the framework of the conventional two-fluid model, how to apply the wall drag to the dispersed phase is disputable. The interface force acting on a fluid particle results from the interaction between the fluid particle and the surrounding continuous fluid. To clarify the contributions to the forces acting on the dispersed phase, the volume-averaged momentum equations are formulated based on the equation of a single fluid particle motion. After that, one-dimensional momentum equations are newly obtained from the averaged equations. It is shown that the wall drag term in the dispersed phase is associated with the spatial gradient of the volume-averaged viscous stress of the continuous phase. The magnitude of the wall drag term for a phase is its volume fraction multiplied by the total two-phase pressure drop induced by the wall shear of the continuous phase.