A Thin Film Flow Rupture Model for Newtonian and Viscoelastic Fluids

Abstract

The problem of specifying adequate boundary conditions in two-dimensional steady incompressible flow between a cylinder and a plane for Newtonian fluids has been studied for many years. Unfortunately, Newtonian boundary conditions cannot be readily transposed to the case of viscoelastic fluids. In the present paper we consider the case of the second-order Rivlin-Ericksen fluid in prescribing suitable boundary conditions. The crux of the issue is that there are at least three possible stresses to be considered: (1) the transverse normal stress σy, (2) the longitudinal normal stress σx, and (3) the mechanical pressure p. The former, σy, is constant across the film and this variable is the one which carries load and measured by an experimental “pressure” tap. In the process of developing the viscoelastic model, a Newtonian model is developed which uses the Birkhoff-Hays separation conditions, considers surface tension as the Coyne-Elrod model but invokes the thin film assumption. The equations of motion are satisfied along and across the separated interface. Variation of the transverse normal stress profile with a minimum film thickness ratio, a surface tension parameter and Deborah number is shown. An earlier paper [6] shows that inconsistencies arise in applying simple rupture models (Gumbel or Swift-Stieber) to viscoelastic fluids. A relatively complicated model of this type is necessary to portray the basic behavior of film rupture in viscoelastic fluids.