Variable viscosity effects for the steady flow past a sphere

Abstract

The effect of a spatially dependent viscosity in the unbounded flow around a rigid spherical particle that translates with constant velocity is investigated theoretically. The variable viscosity emulates the effect of a variable concentration of an additive material in a simple solvent. The analysis is performed utilizing a smooth function for the viscosity of the additive material, which can describe qualitatively both depletion and accumulation phenomena around the particle. Assuming steady state, creeping and isothermal conditions, and no external forces and torques, the momentum and mass balances are a generalization of the classical Stokes equations for which the linearity is preserved. Manipulating suitably the governing partial differential equations, a single ordinary integrodifferential equation for the radial part of the radial velocity component is derived. This equation is solved either numerically using a Chebyshev pseudospectral method or analytically using an asymptotic technique. A decrease in the total drag on the particle as the additive material increases is predicted in the depletion case. In the accumulation case, the total drag may increase or decrease in comparison to the simple Newtonian fluid. Analysis of the total drag to its individual contributions reveals that the friction drag (due to viscous forces) is affected substantially by the change of the viscosity, while the form drag (due to pressure) varies much smoother and milder. Finally, we investigate under which conditions the variable viscosity fluid can be approximated as a constant viscosity fluid with Navier type slip at the wall.