On the correspondence between creeping flows of viscous and viscoelastic fluids

From the wealth of exact solutions for Stokes flow of simple viscous fluids [C. Pozrikidis, Introduction to Theoretical and Computational Fluid Dynamics, Oxford University Press, Oxford, 1997, pp. 222-311], the classical “viscous-viscoelastic correspondence” between creeping flows of viscous and linear viscoelastic materials yields exact viscoelastic creeping flow solutions. The correspondence is valid for an arbitrary prescribed source: of force, flow, displacement or stress; local or nonlocal; steady or oscillatory. Two special Stokes singularities, extended to viscoelasticity in this way, form the basis of modern microrheology [T.G. Mason, D.A. Weitz, Optical measurements of the linear viscoelastic moduli of complex fluids, Phys. Rev. Lett. 74 (1995) 1250-1253]: the Stokeslet (for a stationary point source of force) and the solution for a driven sphere. We amplify these viscoelastic creeping flow solutions with a detailed focus on experimentally measurable signatures: of elastic and viscous responses to steady and time-periodic driving forces; and of unsteady (inertial) effects. We also assess the point force approximation for micron-size driven beads. Finally, we illustrate the generality in source geometry by analyzing the linear response for a nonlocal, planar source of unsteady stress.