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Optimal feeding is optimal swimming for all Péclet numbers
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View: Figures


Image of FIG. 1.
FIG. 1.

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.

Image of FIG. 2.
FIG. 2.

(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 βj 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.

Image of FIG. 3.
FIG. 3.

(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.

Image of FIG. 4.
FIG. 4.

(a) Optimal stroke-induced nutrient flux J − 1 and (b) relative difference in nutrient flux, ΔJ/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.

Image of FIG. 5.
FIG. 5.

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.

Image of FIG. 6.
FIG. 6.

Dependence with the Péclet number, Pe, of the nutrient flux gradient ∂J/∂βn 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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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Optimal feeding is optimal swimming for all Péclet numbers