Reciprocity Relations Between Stokes Flows of Viscous and Viscoelastic Fluids

Linear response theory of thermal fluctuations or driven motion of tracers provides a basis for exploring viscous, elastic and compressible properties of condensed matter. Applications range from atomic physics [1, 24] where the methods were first developed, to recent applications in microrheology [13, 18, 7, 5, 11, 12, 2]. The emphasis here is on hydrodynamics and deformations of incompressible viscoelastic materials for various geometries and driving conditions, as determined from known viscous behavior by straightforward prescriptions, called reciprocity relations. Linear response theory can be formulated to yield an explicit correspondence in the governing equations of Stokes flow between a viscous fluid and any linear viscoelastic material, valid for an arbitrary prescribed source: of force, flow, displacement or stress; local or nonlocal; steady or oscillatory. Upon specification of the geometry and source, non-inertial and inertial viscous Stokes solutions (known as Stokes singularities [8, 17]) transfer to exact solutions for linear viscoelastic fluids. Reciprocity relations inform elasticity-induced contrasts in flow or displacement fields for prescribed forces or stresses; conversely, one may infer sources necessary to achieve identical responses in viscous and viscoelastic materials. Two special Stokes singularities form the basis of microrheology experiments and their interpretation: a prescribed velocity on a translating sphere [24, 13, 7] and a stationary point source of force [5, 11, 12, 2]. We revisit and amplify these examples as an illustration of the reciprocity relations, focusing on measurable non-inertial and inertial features. Next, we illustrate the generality in source type and geometry of this correspondence principle by analyzing the linear response for a nonlocal, planar source of unsteady stress.