Optimal feeding is optimal swimming for all P\'eclet numbers

TL;DR

This paper demonstrates that for a spherical swimmer in viscous fluids, the optimal feeding strategy at zero Reynolds number involves treadmill motion, which is essentially independent of the Péclet number, linking optimal feeding and swimming.

Contribution

The study introduces an adjoint-based numerical optimization approach to determine optimal feeding strategies, revealing their equivalence to optimal swimming across all Péclet numbers.

Findings

Optimal feeding is achieved by treadmill motion, maximizing energy efficiency.

The optimal feeding stroke is nearly independent of the Péclet number.

Optimal feeding and swimming are equivalent for all Péclet numbers.

Abstract

Cells swimming in viscous fluids create flow fields which influence the transport of relevant nutrients, and therefore their feeding rate. We propose a modeling approach to the problem of optimal feeding at zero Reynolds number. We consider a simplified spherical swimmer deforming its shape tangentially in a steady fashion (so-called squirmer). Assuming that the nutrient is a passive scalar obeying an advection-diffusion equation, the optimal use of flow fields by the swimmer for feeding is determined by maximizing the diffusive flux at the organism surface for a fixed rate of energy dissipation in the fluid. The results are obtained through the use of an adjoint-based numerical optimization implemented by a Legendre polynomial spectral method. We show that, to within a negligible amount, the optimal feeding mechanism consists in putting all the energy expended by surface distortion into swimming - so-called treadmill motion - which is also the solution maximizing the swimming efficiency. Surprisingly, although the rate of feeding depends strongly on the value of the P\'eclet number, the optimal feeding stroke is shown to be essentially independent of it, which is confirmed by asymptotic analysis. Within the context of steady actuation, optimal feeding is therefore found to be equivalent to optimal swimming for all P\'eclet numbers.