A Novel Boundary Integral Method for Slow Free Surface Flows

Abstract

The present article introduces a novel boundary integral method (BIM), adapted from an earlier method of Kelmanson (1992, 1994) and suitable for the solution of creeping flow boundary value problems where the boundary presents singularities in the stresses. We use the new BIM to solve the problem of the planar extrusion of a Newtonian fluid at zero Reynolds number and, in particular, to determine the shape of the free surface in the immediate neighbourhood of the separation point for a range of capillary numbers. The proposed method incorporates the singular solution near the separation point, thus overcoming one limitation of a classical BIM to the problem (see, for example, ). In a recent article, Owens (2022) also incorporated the singular solution into his BIM formulation. However, since the integration path used in the present BIM passes directly through the separation point this leads to an important improvement on the method of Owens (2022), who was obligated to skirt the singularity due to the non-integrability there of the normal derivative of the vorticity. Results presented for the extrudate swell ratio, the angle of separation and the leading exponent in the asymptotic expression for the stream function are shown to be in convincing agreement with others in the theoretical, numerical and experimental literature.