Well-posedness of a viscoelastic resistive force theory and applications to swimming

Abstract

We propose and analyze a simple model for the evolution of an immersed, inextensible filament which incorporates linear viscoelastic effects of the surrounding fluid. The model is a closed-form system of equations along the curve only which includes a `memory' term due to viscoelasticity. For a planar filament, given a forcing in the form of a preferred curvature, we prove well-posedness of the fiber evolution as well as the existence of a unique time-periodic solution in the case of time-periodic forcing. Moreover, we obtain an expression for the swimming speed of the filament in terms of the preferred curvature. The swimming speed depends in a complicated way on the viscoelastic parameters corresponding to the fluid relaxation time and additional polymeric viscosity. We study this expression in detail, accompanied by numerical simulations, and show that this simple model can capture complex effects of viscoelasticity on swimming. In particular, the viscoelastic swimmer is shown to be faster than its Newtonian counterpart in some situations and slower in others. Strikingly, we even find an example where viscoelastic effects may lead to a reversal in swimming direction from the Newtonian setting, although this occurs when the displacement for both the Newtonian and viscoelastic swimmers is practically negligible.