Designing a swimming rheometer to measure the linear and non-linear properties of a viscoelastic fluid

Abstract

At low Reynolds numbers, “swirlers” – swimmers with an axisymmetric “head” and “tail” counterrotating about the axis of symmetry – generate no net propulsion in a Newtonian fluid as a consequence of the “scallop theorem”. Viscoelasticity in the suspending fluid breaks the time-reversibility and allows swirlers to propel themselves, with the swim speed being a function of swimmer geometry, fluid elasticity, and swimming gait. Using analytical theory and numerical simulations, we study the unsteady motion of a freely-suspended self-propelled swirler though viscoelastic fluids described by the Giesekus model, allowing for general axisymmetric geometry and time-dependent tail rotation rate. We show the steady swim speed can be calculated for general arbitrary axisymmetric geometries at low Deborah number via the reciprocal theorem and the solution of two Newtonian flow problems. In this “weak flow” limit, we analytically determine the swim speed and its dependence on the parameters of the Giesekus fluid which in turn are related to the primary and secondary normal stress coefficients ${\Psi }_1$ and ${\Psi }_2$. Furthermore, at low De, we derive the unsteady swim speed as a function of a specified unsteady tail rotation rate and the material properties of the suspending fluid. We show that for a particular tail rotation rate, the unsteady swim speed can be analyzed to recover the spectrum of fluid relaxation times, analogous to small-amplitude oscillatory shear measurements on a benchtop rheometer. This study expands upon the design space for a “swimming rheometer” by increasing its functionality to make and interpret rheological measurements.