A squirmer across Reynolds numbers

Abstract

The self-propulsion of a spherical squirmer – a model swimming organism that achieves locomotion via steady tangential movement of its surface – is quantified across the transition from viscously to inertially dominated flow. Specifically, the flow around a squirmer is computed for Reynolds numbers ($Re$) between 0.01 and 1000 by numerical solution of the Navier–Stokes equations. A squirmer with a fixed swimming stroke and fixed swimming direction is considered. We find that fluid inertia leads to profound differences in the locomotion of pusher (propelled from the rear) versus puller (propelled from the front) squirmers. Specifically, pushers have a swimming speed that increases monotonically with $Re$, and efficient convection of vorticity past their surface leads to steady axisymmetric flow that remains stable up to at least $Re=1000$. In contrast, pullers have a swimming speed that is non-monotonic with $Re$. Moreover, they trap vorticity within their wake, which leads to flow instabilities that cause a decrease in the time-averaged swimming speed at large $Re$. The power expenditure and swimming efficiency are also computed. We show that pushers are more efficient at large $Re$, mainly because the flow around them can remain stable to much greater $Re$ than is the case for pullers. Interestingly, if unstable axisymmetric flows at large $Re$ are considered, pullers are more efficient due to the development of a Hill’s vortex-like wake structure.