^{1,a)}and Eric Lauga

^{2,b)}

### 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 polynomialspectral 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 Péclet 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 Péclet numbers.

This work was supported in part by the US National Science Foundation (Grant CBET-0746285 to E.L.).

I. INTRODUCTION

II. NUTRIENT TRANSPORT AROUND A SWIMMING MICROORGANISM

A. Advection-diffusion of a passive scalar near a general swimming microorganism

B. The squirmer model

C. Numerical computation of the concentration field: the Legendre polynomialspectral method (LPSM)

D. Results

III. OPTIMAL FEEDING BY A STEADY SQUIRMER

A. Nutrient uptake gradient for a general swimmer

B. Nutrient uptake optimization for a squirmer

C. Results

1. Optimal squirmer for various Péclet numbers

2. Gradient near the treadmill

IV. DISCUSSION

### Key Topics

- Nutrients
- 64.0
- Microorganisms
- 10.0
- Polynomials
- 10.0
- Diffusion
- 7.0
- Hydrodynamics
- 7.0

## Figures

Squirmer model and spherical polar coordinates used in the paper. On the surface of the swimmer (*r* = 1), the fluid velocity is purely tangential . In the far-field, **u** ∼ − *U* **e** _{ x } with *U* the swimming velocity of the organism.

Squirmer model and spherical polar coordinates used in the paper. On the surface of the swimmer (*r* = 1), the fluid velocity is purely tangential . In the far-field, **u** ∼ − *U* **e** _{ x } with *U* the swimming velocity of the organism.

(Color online) Nutrient concentration around the swimmer for Pe = 1, 10 and 100 (from top to bottom) and *β* _{2}/*β* _{1} = 0, 5 and *∞* (from left to right), all the other *β _{j} * being taken equal to zero. Far from the swimmer

*c*= 0, while

*c*= 1 at the swimmer surface. The dimensionless nutrient flux

*J*is quoted for each case. On the bottom row, the streamlines are displayed for each stroke.

(Color online) Nutrient concentration around the swimmer for Pe = 1, 10 and 100 (from top to bottom) and *β* _{2}/*β* _{1} = 0, 5 and *∞* (from left to right), all the other *β _{j} * being taken equal to zero. Far from the swimmer

*c*= 0, while

*c*= 1 at the swimmer surface. The dimensionless nutrient flux

*J*is quoted for each case. On the bottom row, the streamlines are displayed for each stroke.

(Color online) Variations of the relative nutrient flux, *J*, within the (*β* _{2}, *β* _{3})-plane for (a) Pe = 5 and (b) Pe = 200 (*β* _{1} is adjusted so that ). Nutrient flux isolines are also shown for clarity and correspond to the values indicated on the right. The crosses indicate the position of the treadmill swimmer in the (*β* _{2}, *β* _{3})-plane.

(Color online) Variations of the relative nutrient flux, *J*, within the (*β* _{2}, *β* _{3})-plane for (a) Pe = 5 and (b) Pe = 200 (*β* _{1} is adjusted so that ). Nutrient flux isolines are also shown for clarity and correspond to the values indicated on the right. The crosses indicate the position of the treadmill swimmer in the (*β* _{2}, *β* _{3})-plane.

(a) Optimal stroke-induced nutrient flux *J* − 1 and (b) relative difference in nutrient flux, Δ*J*/*J*, between the optimal swimmer and the treadmill swimmer as functions of the Péclet number, Pe. Numerical results of the optimization procedure are presented for *N* = 3 (crosses) and *N* = 8 (squares). Several sets of calculations were performed for each value of Pe and *N*. In (a), the solid line corresponds to the treadmill swimmer. In (a) and (b), the dashed and dotted lines correspond to the asymptotic results for the treadmill swimmer at and obtained in Appendices B and C.

(a) Optimal stroke-induced nutrient flux *J* − 1 and (b) relative difference in nutrient flux, Δ*J*/*J*, between the optimal swimmer and the treadmill swimmer as functions of the Péclet number, Pe. Numerical results of the optimization procedure are presented for *N* = 3 (crosses) and *N* = 8 (squares). Several sets of calculations were performed for each value of Pe and *N*. In (a), the solid line corresponds to the treadmill swimmer. In (a) and (b), the dashed and dotted lines correspond to the asymptotic results for the treadmill swimmer at and obtained in Appendices B and C.

Dependence on the Péclet number, Pe, of the orientation angle in *β*-space, *t* _{opt} = cos^{− 1} *β* _{1}, of the optimal swimming stroke. As in Fig. 4, results are presented when the optimization is performed on *N* = 3 modes (crosses) and *N* = 8 modes (square). The dashed line corresponds to the prediction of the asymptotic analysis at obtained in Appendix B.

Dependence on the Péclet number, Pe, of the orientation angle in *β*-space, *t* _{opt} = cos^{− 1} *β* _{1}, of the optimal swimming stroke. As in Fig. 4, results are presented when the optimization is performed on *N* = 3 modes (crosses) and *N* = 8 modes (square). The dashed line corresponds to the prediction of the asymptotic analysis at obtained in Appendix B.

Dependence with the Péclet number, Pe, of the nutrient flux gradient *∂J*/*∂β _{n} * with respect to the first four odd modes

*n*= 1 (stars),

*n*= 3 (squares),

*n*= 5 (circles), and

*n*= 7 (triangles) and evaluated at the treadmill (the even mode gradients are equal to zero by symmetry). The power law dependence of each component is indicated by a dashed line.

Dependence with the Péclet number, Pe, of the nutrient flux gradient *∂J*/*∂β _{n} * with respect to the first four odd modes

*n*= 1 (stars),

*n*= 3 (squares),

*n*= 5 (circles), and

*n*= 7 (triangles) and evaluated at the treadmill (the even mode gradients are equal to zero by symmetry). The power law dependence of each component is indicated by a dashed line.

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