Optimal translational swimming of a sphere at low Reynolds number

Swimming velocity and rate of dissipation of a sphere with surface distortions are discussed on the basis of the Stokes equations of low-Reynolds-number hydrodynamics. At first the surface distortions are assumed to cause an irrotational axisymmetric flow pattern. The efficiency of swimming is optimized within this class of flows. Subsequently, more general axisymmetric polar flows with vorticity are considered. This leads to a considerably higher maximum efficiency. An additional measure of swimming performance is proposed based on the energy consumption for given amplitude of stroke.

Figure 1

Plot of one-half the maximum eigenvalue $\frac{1}{2}{\lambda }_{\max }(1,L)$ for sets of multipoles $\{{\mu }_{ls},{\mu }_{lc}\}$ with $1\le l\le L$ as a function of $L$ for $L=2,\cdots ,30$. The values tend to unity as $L\to \infty$.

Figure 2

Plot of the components of the eigenvector with largest eigenvalue, normalized to unity, for $L=8$. The corresponding multipoles $\{{\mu }_{ls},{\mu }_{lc}\}$ with $1\le l\le 8$ follow from Eq. (4.2). The values ${\mu }_{ls}$ for $l$ even and the values ${\mu }_{lc}$ for $l$ odd vanish.

Figure 3

Plot of the reduced speed ${\widehat{U}}_{1L}^A$ for fixed maximum amplitude of the displacement at $\theta =\pi /2$ as a function of $L$. At each value of $L$ the most efficient set of multipoles $\{{\mu }_{1s},{\mu }_{1c},\cdots ,{\mu }_{Ls},{\mu }_{Lc}\}$ for swimming via irrotational flow is considered.

Figure 4

Plot of the reduced power ${\widehat{\mathcal{D}}}_{1L}^A$ for fixed maximum amplitude of the displacement at $\theta =\pi /2$ as a function of $L$. At each value of $L$ the most efficient set of multipoles $\{{\mu }_{1s},{\mu }_{1c},\cdots ,{\mu }_{Ls},{\mu }_{Lc}\}$ for swimming via irrotational flow is considered.

Figure 5

Plot of the end of the displacement vector $\mathit{\xi }\left(t\right)$ at $\theta =3\pi /12,5\pi /12,\pi /2,7\pi /12$, and $9\pi /12$ for maximum amplitude of the displacement at $\theta =\pi /2$ equal to $0.1a$ for the optimum eigenvector for $L=8$ with complex multipoles $\left\{{\mu }_l^c\right\}$. The motion is depicted with start at $t=0$ and finish at $t=\frac{7}{8}T$, where $T=2\pi /\omega$. The end point is denoted by a small circle.

Figure 6

Plot of the radial displacement for maximum amplitude $0.1a$ as a function of polar angle $\theta$ for $t=0$ (solid curve), $t=T/8$ (long dashes), and $t=T/4$ (short dashes). A running wave can be discerned.

Figure 7

Plot of the ratio $U_2\left(t\right)/{\overline{U}}_2$ as a function of time for swimming motion corresponding to the optimal set of multipoles $\{{\mu }_{ls},{\mu }_{lc}\}$ with $1\le l\le 8$ (solid curve). We also plot the ratio ${\mathcal{D}}_2\left(t\right)/{\overline{\mathcal{D}}}_2$ for the same swimming motion. This equals unity within numerical accuracy.

Figure 8

Plot of one-half the maximum eigenvalue $\frac{1}{2}{\lambda }_{\max }(1,L)$ for sets of complex multipoles $\{{\kappa }_l,{\mu }_l\}$ with $1\le l\le L$ and ${\kappa }_1=0$ as a function of $L$ for $L=3,\cdots ,40$. The values tend to $\sqrt{2}$ as $L\to \infty$.

Figure 9

Plot of the reduced speed ${\widehat{U}}_{1L}^B$ for fixed maximum amplitude of the displacement at $\theta =\pi /2$ as a function of $L$. At each value of $L$ the most efficient set of multipoles $\{{\mu }_{1s},{\mu }_{1c},\cdots ,{\kappa }_{Ls},{\kappa }_{Lc},{\mu }_{Ls},{\mu }_{Lc}\}$ is considered.

Figure 10

Plot of the reduced power ${\widehat{\mathcal{D}}}_{1L}^B$ for fixed maximum amplitude of the displacement at $\theta =\pi /2$ as a function of $L$. At each value of $L$ the most efficient set of multipoles $\{{\mu }_{1s},{\mu }_{1c},\cdots ,{\kappa }_{Ls},{\kappa }_{Lc},{\mu }_{Ls},{\mu }_{Lc}\}$ is considered.

Figure 11

Plot of the nonvanishing components of the eigenvector with largest eigenvalue, normalized to unity, for a set of complex multipoles $\{{\kappa }_l,{\mu }_l\}$ with $1\le l\le 7$ and ${\kappa }_1=0$. The absolute values of the $\left\{{\kappa }_l\right\}$ are indicated by squares and those of the $\left\{{\mu }_l\right\}$ are indicated by dots.

Figure 12

Plot of the end of the displacement vector $\mathit{\xi }\left(t\right)$ at $\theta =3\pi /12,5\pi /12,\pi /2,7\pi /12$, and $9\pi /12$ for maximum amplitude of the displacement at $\theta =\pi /2$ equal to $0.1a$ for the optimum eigenvector for $L=7$ with complex multipoles $\{{\kappa }_l,{\mu }_l\}$. The motion is depicted with start at $t=0$ and finish at $t=\frac{7}{8}T$, where $T=2\pi /\omega$. The end point is marked by a small circle.