  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.
