Wall effects on Reiner-Rivlin liquid spheroid

Abstract

An analysis is carried out to study the flow characteristics of creeping motion of an inner non-Newtonian Reiner-Rivlin liquid spheroid r = 1+ ∑_k=2^∞α_kG_k(cos θ), here α_k is very small shape factor and G_k is Gegenbauer function of first kind of order k, at the instant it passes the centre of a rigid spherical container filled with a Newtonian fluid. The shape of the liquid spheroid is assumed to depart a bit at its surface from the shape a sphere. The analytical expression for stream function solution for the flow in spherical container is obtained by using Stokes equation. While for the flow inside the Reiner-Rivlin liquid spheroid, the expression for stream function is obtained by expressing it in a power series of S, characterizing the cross-viscosity of Reiner-Rivlin fluid. Both the flow fields are then determined explicitly by matching the boundary conditions at the interface of Newtonian fluid and non-Newtonian fluid and also the condition of impenetrability and no-slip on the outer surface to the first order in the small parameter ε, characterizing the deformation of the liquid sphere. As an application, we consider an oblate liquid spheroid r = 1+2εG₂(cos θ) and the drag and wall effects on the body are evaluated. Their variations with regard to separation parameter, viscosity ratio λ, cross-viscosity, i.e., S and deformation parameter are studied and demonstrated graphically. Several well-noted cases of interest are derived from the present analysis. Attempts are made to compare between Newtonian and Reiner-Rivlin fluids which yield that the cross-viscosity μ_c is to decrease the wall effects K and to increase the drag D_N when deformation is comparatively small. It is  observed that drag not only varies with λ, but as η increases, the rate of change in behavior of drag force increases also.