Rheological effects in the 3D creeping flow past a sedimenting sphere subject to orthogonal shear

Abstract

The effects of the rheological parameters of nonlinear differential constitutive models in the isothermal, steady, creeping, flow past a sedimenting sphere, in an incompressible viscoelastic matrix fluid, subject to simple shear in a plane perpendicular to the direction of sedimentation are studied analytically. The viscoelasticity of the ambient fluid is modeled using the Upper Convected Maxwell, Oldroyd-B, exponential Phan-Thien-Tanner, Giesekus, and FENE-P constitutive equations. The solution of the governing equations is expanded as a regular perturbation series for small values of the Deborah number, and the resulting sequence of three-dimensional partial differential equations is solved analytically up to fourth order. Approximate analytical expressions for the angular velocity of the rigid sphere, as well as for the drag force on the sphere, are derived and discussed. The solutions reveal both the increase of the drag in case of Boger-type fluids (modeled with the FENE-P model) and the decrease of the drag in case of elastic fluids (modeled with the Giesekus model).