Optimal Shapes of Surface Slip Driven Self-Propelled Microswimmers

We study the efficiency of self-propelled swimmers at low Reynolds numbers, assuming that the local energetic cost of maintaining a propulsive surface slip velocity is proportional to the square of that velocity. We determine numerically the optimal shape of a swimmer such that the total power is minimal while maintaining the volume and the swimming speed. The resulting shape depends strongly on the allowed maximum curvature. When sufficient curvature is allowed the optimal swimmer exhibits two protrusions along the symmetry axis. The results show that prolate swimmers such as Paramecium have an efficiency that is $\sim 20\%$ higher than that of a spherical body, whereas some microorganisms have shapes that allow even higher efficiency.

Figure 1

We parametrize the axisymmetric body as $N$ rings, each at height $z_i$ with radius ${\rho }_i$. The fluid velocity at each ring has the radial component $v_{\rho ,i}$ and the axial component $v_{z,i}$.

Figure 2

Optimal shapes of bodies with minimum radius of curvature ${\widehat{r}}_{\min }=0.1,\dots ,1$.

Figure 3

(a) Color coded tangential propulsion velocity $v_t/u$ of the optimal optimal swimmer for ${\widehat{r}}_{\min }=0.1$ (left) and ${\widehat{r}}_{\min }=0.5$ (right). (b) Streamlines of the same two swimmers in a body-fixed frame.

Figure 4

Optimal normalized dissipation $\widehat{P}$ (dissipated power, divided by the value for a sphere of the same volume) as a function of the minimum curvature radius ${\widehat{r}}_{\min }=0.1,\dots ,1$. Optimal shapes are shown as insets at four points.

Figure 5

(a) Three-dimensional shapes of optimal swimmers for ${\widehat{r}}_{\min }=0.1$, 0.2, 0.5, and 0.7. (b) Some ciliated protozoa: Litonotus cygnus, Amphileptus pleurosigma (drawn after Ref. 24) and Paramecium caudatum.