Steady sphere translation in weakly viscoelastic UCM/Oldroyd-B fluids with perfect slip on the sphere

Abstract

We study analytically the influence of perfect slip boundary conditions to the steady translation of a spherical particle in a viscoelastic fluid which consists of a Newtonian solvent and a polymer solute. The viscoelastic fluid is modeled with the Oldroyd-B constitutive equation with the ratio of polymer zero shear rate viscosity to the total viscosity, η, considered as a fluid parameter and varying from zero (Newtonian limit) to 1 (Upper Convected Maxwell (UCM) limit). The solution for all the dependent flow variables is expanded as asymptotic power series with the small parameter being the Weissenberg number, Wi=λU/R, where λ is the single relaxation time of the fluid, R is the radius of the particle and U is its steady translational velocity. The resulting sequence of equations is solved analytically up to eight-order in Wi. Hence, the current work is a substantial expansion of the previous work by Gkormpatsis et al., (2020) which has examined the most general case of Navier slip, but the perturbation series has been found only up to fourth order in Wi. The current work confirms the results obtained in that study in the limit of perfect slip conditions. Moreover, the higher-order analytical solution allows for the better investigation of the convergence of the results with techniques that accelerate the convergence of series. Under perfect slip conditions, it is shown that the relative drag force on the sphere increases monotonically with viscoelasticity, as opposed to the cases of no-slip or partial slip where the drag can also decrease or attain a local minimum. This difference in behavior is attributed to the higher dominance of extensional deformations close to the stagnation points of the flow, while the shear deformations prevail away from these points. Finally, evidence is provided that the UCM/Oldroyd-B models become singular at a finite Weissenberg number, Wiu, which depends weakly on η. Although the exact value of Wiu could not be calculated, a sequence of approximations which results from the eight-order perturbation solution reveals that 0.5<Wiu<1. Wiu appears to decrease as η increases, with its minimum value being observed for the UCM model.