Spherical particle sedimenting in weakly viscoelastic shear flow

TL;DR

This paper provides analytical solutions for the motion of a spherical particle in weakly viscoelastic shear flow, revealing shear-induced lift, modified drag, and their dependence on flow orientation, with implications for particle settling velocities.

Contribution

It introduces a second-order perturbation theory for particle dynamics in viscoelastic fluids, including a new tensor basis for tensorial calculations and solutions to the inhomogeneous Stokes equations.

Findings

Identification of shear-induced lift at O(Wi)

Modified drag effects at O(De^2) and O(Wi^2)

Orthogonal second lift at O(Wi^2)

Abstract

We consider the dynamics of a small spherical particle driven through an unbounded viscoelastic shear flow by an external force. We give analytical solutions to both the mobility problem (velocity of forced particle) and the resistance problem (force on fixed particle), valid to second order in the dimensionless Deborah and Weissenberg numbers, which represent the elastic relaxation time of the fluid relative to the rate of translation and the imposed shear rate. We find a shear-induced lift at $O({\rm Wi})$, a modified drag at $O({\rm De}^2)$ and $O({\rm Wi}^2)$, and a second lift that is orthogonal to the first, at $O({\rm Wi}^2)$. The relative importance of these effects depends strongly on the orientation of the forcing relative to the shear. We discuss how these forces affect the terminal settling velocity in an inclined shear flow. We also describe a new basis set of symmetric Cartesian tensors, and demonstrate how they enable general tensorial perturbation calculations such as the present theory. In particular this scheme allows us to write down a solution to the inhomogenous Stokes equations, required by the perturbation expansion, by a sequence of algebraic manipulations well suited to computer implementation.