Perturbation solution for the viscoelastic 3D flow around a rigid sphere subject to simple shear

Abstract

We study the steady, three-dimensional creeping, and viscoelastic flow around a freely rotating rigid sphere subject to simple shear flow imposed at infinity. The viscoelasticity of the ambient fluid is modeled using the second-order-fluid model, the Upper Convected Maxwell, the exponential affine Phan-Thien-Tanner, and the Giesekus constitutive equations. A spherical coordinate system with origin at the center of the sphere is used to describe the flow field. The solution of the governing equations is expanded as a series for small values of the Deborah number. The resulting sequence of differential equations is solved analytically up to second order and numerically up to fourth order in Deborah number by employing fully spectral representations for all the primary variables. In particular, Chebyshev polynomials are used in the radial coordinate and the double Fourier series in the longitudinal and latitudinal coordinates. The numerical results up to second-order agree within machine accuracy with the available analytical solutions clearly indicating the correctness and accuracy of the numerical method developed here. Analytical expressions for the angular velocity of the rigid sphere up to fourth order, which show the slowdown of the rotation of the sphere with respect to the Newtonian creeping case, are also derived. For small Deborah numbers, these expressions, along with those presented in a recent letter [Housiadas and Tanner, Phys. Fluids 23, 051702 (2011)] are in agreement with the few available experimental data and numerical results.