Slip boundary condition near the wall-interface contact line in axial stratified two-phase flow

Abstract

The behavior of the velocity and shear stresses in two-phase laminar stratified flow near a triple point (TP), formed in the flow cross-section by the intersection of the interface with the conduit wall, is reconsidered. Differently from the no-slip boundary condition that showed a possibility of diverging shear stresses upon approaching the TP, we allow a slip of the fluids at the wall that can be prominent in the TP region. The Navier's boundary condition is used, which considers a linear relationship between the slip velocity and the strain rate component perpendicular to the surface. The solution for the velocity field in the TP region is obtained in terms of a power series of the distance from the TP having a discrete set of powers. Similarly to the solution obtained with no-slip B.C., a so-called ‘characteristic equation’ is obtained. Its solution yields a discrete set of non-integer powers that are dependent on the dynamic viscosity ratio and the contact angle. However, contrary to the no-slip solution, with Navier's B.C, each of these powers introduces a tower of terms resulting from the relation between the slip velocity and the velocity derivative at the wall. In addition, the solution also includes a tower of integer powers. The solution yields the velocity and the shear stress fields, as well as the flux of the force per length that acts on an arbitrary surface in the flow field. The latter is obtained by considering an equivalent 2D representation of the stress tensor. These are examined in comparison to the corresponding results obtained when no-slip at the wall is imposed. The relaxation of the wall shear stresses in cases where the no-slip condition predicts diverging wall shear stresses at the TP is demonstrated. It is shown that if the local velocity in the TP region is a limit of a global no-slip solution, the scaled velocity field in the TP region is independent of the details of the global flow field.