Flapping motion and force generation in a viscoelastic fluid

TL;DR

This paper demonstrates that reciprocal flapping motion can generate net forces in viscoelastic fluids, challenging Purcell's scallop theorem and revealing new mechanisms for force generation in biological and synthetic systems.

Contribution

It shows that in polymeric fluids, reciprocal motions produce net forces, unlike in Newtonian fluids, expanding understanding of force generation in complex biological and synthetic environments.

Findings

Reciprocal flapping induces net forces in viscoelastic fluids.

Scallop theorem does not hold in polymeric fluids.

Viscoelastic forces can be harnessed for propulsion.

Abstract

In a variety of biological situations, swimming cells have to move through complex fluids. Similarly, mucociliary clearance involves the transport of polymeric fluids by beating cilia. Here, we consider the extent to which complex fluids could be exploited for force generation on small scales. We consider a prototypical reciprocal motion (i.e. identical under time-reversal symmetry): The periodic flapping of a tethered semi-infinite plane. In the Newtonian limit, such motion cannot be used for force generation according to Purcell's scallop theorem. In a polymeric fluid (Oldroyd-B, and its generalization), we show that this is not the case and calculate explicitly the forces on the flapper for small-amplitude sinusoidal motion. Three setups are considered: a flapper near a wall, a flapper in a wedge, and a two-dimensional scallop-like flapper. In all cases, we show that at quadratic order in the oscillation amplitude, the tethered flapping motion induces net forces but no average flow. Our results demonstrate therefore that the scallop theorem is not valid in polymeric fluids. The reciprocal component of the movement of biological appendages such as cilia can thus generate nontrivial forces in polymeric fluid such as mucus, and normal-stress differences can be exploited as a pure viscoelastic force generation and propulsion method.