A note on the unbounded creeping flow past a sphere for Newtonian fluids with pressure-dependent viscosity

Abstract

We investigate theoretically isothermal, incompressible, creeping Newtonian flows past a sphere, under the assumption that the shear viscosity is pressure-dependent, varying either linearly or exponentially with pressure. In particular, we consider the three-dimensional flow past a freely rotating neutrally buoyant sphere subject to shear at infinity and the axisymmetric flow past a sedimenting sphere. The method of solution is a regular perturbation scheme with the small parameter being the dimensionless coefficient which appears in the expressions for the shear viscosity. Asymptotic solutions for the pressure and the velocity field are found only for the simple shear case, while no analytical solutions could be found for the sedimentation problem. For the former flow, calculation of the streamlines around the sphere reveals that the fore-and-aft symmetry of the streamlines which is observed in the constant viscosity case breaks down. Even more importantly, the region of the closed streamlines around the sphere is absent. Last, it is revealed that the angular velocity of the sphere is not affected by the dependence of the viscosity on the pressure.