Local behavior near triple point in a laminar two-phase flow in an arbitrary tube cross-section

Abstract

In this study, we propose a simple approach for examining the local behavior of the velocity and shear stresses in the vicinity of a triple point (TP) in a two-phase flow. We assume a laminar steady, fully developed, stratified two-phase flow with any contact angle, α, independent of the shape of the tube’s cross-section. The axial component of the fluid’s velocity can be treated as a scalar function in two dimensions that obeys Poisson’s equation, together with the no-slip condition on the boundary (tube walls) and appropriate conditions on the interface between the phases. In our opinion, the exact solutions described in previous studies (for the special case of a cylindrical tube) are complicated and difficult to follow. Moreover, the approximated interface presented in previous studies predicts a geometric contact angle (αgeometry) that might not coincide with the physical contact angle, α, which is located at the real interface’s shape. The exact solutions numerically solve the global problem for a circular tube (i.e., they find the velocity and shear stresses) and analytically obtain results in the vicinity of the TP to solve the problem with the aid of the residue theorem. However, we begin in the vicinity of the TP and our entire analysis focuses within this domain. To simplify the calculations, three steps are suggested: 1) zooming in on the TP allows us to approximate the curved arcs of the walls and the interface as straight lines (the tangents of the arcs of the wall and the outward ray of the interface)11Intuitively, for a local observation up to first order in distance, a regular curve can be approximated as a straight line; i.e., an arc can be approximated to a cord. Up to second order, a curve can be approximated as a circle. For a formal proof that Laplace’s equation for the region is bounded by the wall circle (radius Rwall), the interface circle (radius Ri), and our radius R defined in Fig 2.2, a possible option is to Möbius-transform the wall circle and the interface circle into straight lines concurrent in the angle α (so the other meeting points of these circles are mapped onto the point at infinity), and to obtain the Laplace solution, thereby proving that when R/(min(Rwall,Rinterface))<<1, the deviation tends to zero.; 2) attaching a polar coordinate system where the origin of this coordinate system is located at the TP; and 3) suppressing the local driving forces and expressing distant driving forces as a function on the boundary, thereby leaving the velocity as a harmonic function.The proposed method can be converted for electromagnetism near a junction of several dielectric materials. Furthermore, the electromagnetic parallel problem can be extended to three dimensions; i.e., a junction point of prisms with different dielectric constants.