Flexibility, stroke, and dimensionless parameters: the importance of telling the whole story for swimming micro-organisms in complex fluids
Becca Thomases, UC Davis

Low Reynolds number swimming of microorganisms in Newtonian fluids is an extensively studied classical problem, and the underlying physics is well understood. However, many biological fluids such as mucus are mixtures of water and polymers and are more appropriately described as viscoelastic fluids.  The question of how fluid elasticity affects the swimming performance of micro-organisms is complicated and has been the subject of many recent experimental and theoretical studies. The Deborah number is typically used to characterize the strength of the fluid elasticity in these studies, and is expressed as the product of the elastic relaxation time and the frequency of the swimmer stroke. In recent simulations of undulatory flexible swimmers in an Oldroyd-B-type fluid, we find that varying the frequency of the stroke OR varying the relaxation time results in significantly different dependence of swimming speed for the same range of Deborah number. Thus the elastic effects on swimming cannot be characterized by a single dimensionless number.
The Weissenberg number is different dimensionless number used to quantify fluid elasticity and it is  defined as the product of elastic relaxation time and characteristic strain rate. When the swimmer is sufficiently flexible increasing the frequency of the stroke will eventually result in a leveling off of the strain-rate and it is in this regime where the Deborah number is no longer equal to the Weissenberg number.  This difference is due to the introduction of another time-scale, the body elasticity time-scale. We conjecture that the different swimming speeds for the same range of Deborah number is a consequence of a "stroke-stabilizing" effect that fluid elasticity has for flexible swimmers, in addition to a Weissenberg-number transition in the fluid which depends on the amplitude of the swimmer stroke.