Ke Xu, University of North Carolina at Chapel Hill


Reciprocity Relations Between Creeping Flows of Viscous and Viscoelastic Fluids 

Abstract:  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 to microrheology. The emphasis presented 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 of a viscous fluid and the creeping flow of any linear viscoelastic material, valid for an arbitrary prescribed source: of force, flow, displacement or stress; local or nonlocal; quasi-steady or non-quasi-steady. Upon specification of the geometry and source, quasi-steady and non-quasi-steady viscous Stokes solutions (known as Stokes singularities) transfer to exact solutions for linear viscoelastic fluids. Reciprocity relations inform the role of storage and loss moduli in response data: e.g., flow or displacement fields for prescribed forces or stresses; and sources necessary to achieve identical responses in viscosity-matched, elasticity-contrasted, materials. Two special Stokes singularities form the basis of microrheology experiments and data interpretation: a prescribed velocity on a translating sphere and a stationary point source of force. We amplify these specialized reciprocity relations to make further predictions of typical active linear microrheology experiments, focusing on measurable quasi-steady and non-quasi-steady features. Next, we illustrate the generality in source type and geometry of these relations by analyzing the linear response for a nonlocal, planar source of unsteady stress.

Advisor: Greg Forest (UNC)