Generalized squirming motion of a sphere

Abstract

A number of swimming microorganisms, such as ciliates (Opalina) and multicellular colonies of flagellates (Volvox), are approximately spherical in shape and swim using beating arrays of cilia or short flagella covering their surfaces. Their physical actuation on the fluid may be mathematically modeled as the generation of surface velocities on a continuous spherical surface—a model known in the literature as squirming, which has been used to address various aspects of the biological physics of locomotion. Previous analyses of squirming assumed axisymmetric fluid motion and hence required all swimming kinematics to take place along a line. In this paper we generalize squirming to three spatial dimensions. We derive analytically the flow field surrounding a spherical squirmer with arbitrary surface motion and use it to derive its three-dimensional translational and rotational swimming kinematics. We then use our results to physically interpret the flow field induced by the swimmer in terms of fundamental flow singularities up to terms decaying spatially as ${\sim } 1/r^3$. Our results will make it possible to develop new models in biological physics, in particular in the area of hydrodynamic interactions and collective locomotion.

Appendices

Appendix A: Fundamental flow singularities in spherical coordinates

1.1 Stokeslets

The solution to Stokes equations due to a point force $f \varvec{\alpha } \delta ({\mathbf {r}})$ of magnitude $f$, and the direction $\varvec{\alpha }$ at the origin is given by ${\mathbf {u}}= f {\mathbf {G}}(\varvec{\alpha })/ (8\pi \eta )$, where the vectorial representation of a Stokeslet is given by Eq. (36) in the main text.

Note that $\varvec{\alpha }$ denotes the direction of the Stokeslet. Stokeslets in different Cartesian directions are expressed in spherical coordinates as

$$\begin{aligned} {\mathbf {G}}({\mathbf {e}}_x)&= \frac{1}{r} \left[ 2 \sin \theta \cos \phi \ {\mathbf {e}}_r + \cos \theta \cos \phi \ {\mathbf {e}}_\theta -\sin \phi \ {\mathbf {e}}_\phi \right] , \end{aligned}$$

(94)

$$\begin{aligned} {\mathbf {G}}({\mathbf {e}}_y)&= \frac{1}{r} \left[ 2 \sin \theta \sin \phi \ {\mathbf {e}}_r + \cos \theta \sin \phi \ {\mathbf {e}}_\theta + \cos \phi \ {\mathbf {e}}_\phi \right] ,\end{aligned}$$

(95)

$$\begin{aligned} {\mathbf {G}}({\mathbf {e}}_z)&= \frac{1}{r} \left[ 2 \cos \theta \ {\mathbf {e}}_r - \sin \theta \ {\mathbf {e}}_\theta \right] . \end{aligned}$$

(96)

1.2 A general Stokes dipole

A general force dipole is obtained by taking the derivative of a Stokeslet along the direction of interest. The vectorial representation of a force dipole is given by Eq. (44) in the main text, where $\varvec{\alpha }$ and $\varvec{\beta }$ denote respectively the direction of the Stokeslet and the direction along which the derivative is taken. The expressions of Stokes dipoles of different configurations in spherical coordinates are given by

$$\begin{aligned} {\mathbf {G}}_\mathrm{D}({\mathbf {e}}_x, {\mathbf {e}}_x)&= \frac{1}{r^2} \left[ -\frac{1}{4}(1+3 \cos 2\theta ) + \frac{3}{4} (1-\cos 2\theta ) \cos 2\phi \right] {\mathbf {e}}_r, \end{aligned}$$

(97)

$$\begin{aligned} {\mathbf {G}}_\mathrm{D}({\mathbf {e}}_y, {\mathbf {e}}_y)&= \frac{1}{r^2} \left[ -\frac{1}{4} (1+3 \cos 2\theta ) -\frac{3}{4} (1-\cos 2\theta ) \cos 2\phi \right] {\mathbf {e}}_r, \end{aligned}$$

(98)

$$\begin{aligned} {\mathbf {G}}_\mathrm{D}({\mathbf {e}}_z, {\mathbf {e}}_z)&= \frac{1}{2r^2} (1+3 \cos 2\theta ) {\mathbf {e}}_r, \end{aligned}$$

(99)

$$\begin{aligned} {\mathbf {G}}_\mathrm{D}({\mathbf {e}}_y, {\mathbf {e}}_x)&= \frac{1}{r^2} \left[ \frac{3}{4} (1-\cos 2\theta ) \sin 2\phi \ {\mathbf {e}}_r - \sin \theta \ {\mathbf {e}}_\phi \right] ,\end{aligned}$$

(100)

$$\begin{aligned} {\mathbf {G}}_\mathrm{D}({\mathbf {e}}_z, {\mathbf {e}}_x)&= \frac{1}{r^2} \left( \frac{3}{2} \sin 2\theta \cos \phi \ {\mathbf {e}}_r + \cos \phi \ {\mathbf {e}}_\theta - \cos \theta \sin \phi \ {\mathbf {e}}_\phi \right) , \end{aligned}$$

(101)

$$\begin{aligned} {\mathbf {G}}_\mathrm{D}({\mathbf {e}}_x, {\mathbf {e}}_y)&= \frac{1}{r^2} \left[ \frac{3}{4} (1-\cos 2\theta ) \sin 2\phi \ {\mathbf {e}}_r + \sin \theta \ {\mathbf {e}}_\phi \right] ,\end{aligned}$$

(102)

$$\begin{aligned} {\mathbf {G}}_\mathrm{D}({\mathbf {e}}_z, {\mathbf {e}}_y)&= \frac{1}{r^2} \left( \frac{3}{2} \sin 2\theta \sin \phi \ {\mathbf {e}}_r + \sin \phi \ {\mathbf {e}}_\theta + \cos \theta \cos \phi \ {\mathbf {e}}_\phi \right) ,\end{aligned}$$

(103)

$$\begin{aligned} {\mathbf {G}}_\mathrm{D}({\mathbf {e}}_x, {\mathbf {e}}_z)&= \frac{1}{r^2}\left( \frac{3}{2} \sin 2\theta \cos \phi \ {\mathbf {e}}_r -\cos \phi \ {\mathbf {e}}_\theta + \cos \theta \sin \phi \ {\mathbf {e}}_\phi \right) ,\end{aligned}$$

(104)

$$\begin{aligned} {\mathbf {G}}_\mathrm{D}({\mathbf {e}}_y, {\mathbf {e}}_z)&= \frac{1}{r^2} \left( \frac{3}{2} \sin 2\theta \sin \phi \ {\mathbf {e}}_r - \sin \phi \ {\mathbf {e}}_\theta - \cos \theta \cos \phi \ {\mathbf {e}}_\phi \right) . \end{aligned}$$

(105)

1.3 Stresslets

The vectorial representation of a stresslet is given by Eq. (45) in the main text, where $\varvec{\alpha }$ and $\varvec{\beta }$ denote respectively the direction of the Stokeslet and the direction along which the derivative is taken. The expressions of stresslets of different configurations in spherical coordinates are given by

$$\begin{aligned} {\mathbf {S}}({\mathbf {e}}_x, {\mathbf {e}}_x)&= {\mathbf {G}}_\mathrm{D}({\mathbf {e}}_x, {\mathbf {e}}_x) = \frac{1}{r^2} \left[ -\frac{1}{4}(1+3 \cos 2\theta ) + \frac{3}{4} (1-\cos 2\theta ) \cos 2\phi \right] {\mathbf {e}}_r, \end{aligned}$$

(106)

$$\begin{aligned} {\mathbf {S}}({\mathbf {e}}_y, {\mathbf {e}}_y)&= {\mathbf {G}}_\mathrm{D}({\mathbf {e}}_y, {\mathbf {e}}_y) = \frac{1}{r^2} \left[ -\frac{1}{4} (1+3 \cos 2\theta ) -\frac{3}{4} (1-\cos 2\theta ) \cos 2\phi \right] {\mathbf {e}}_r, \end{aligned}$$

(107)

$$\begin{aligned} {\mathbf {S}}({\mathbf {e}}_z, {\mathbf {e}}_z)&= {\mathbf {G}}_\mathrm{D}({\mathbf {e}}_z,{\mathbf {e}}_z) = \frac{1}{2r^2} (1+3 \cos 2\theta ) {\mathbf {e}}_r, \end{aligned}$$

(108)

$$\begin{aligned} {\mathbf {S}}({\mathbf {e}}_y, {\mathbf {e}}_x)&= {\mathbf {S}}({\mathbf {e}}_x, {\mathbf {e}}_y)= \frac{3}{4r^2} (1-\cos 2\theta ) \sin 2\phi \ {\mathbf {e}}_r ,\end{aligned}$$

(109)

$$\begin{aligned} {\mathbf {S}}({\mathbf {e}}_z, {\mathbf {e}}_x)&= {\mathbf {S}}({\mathbf {e}}_x, {\mathbf {e}}_z)= \frac{3}{2r^2} \sin 2\theta \cos \phi \ {\mathbf {e}}_r , \end{aligned}$$

(110)

$$\begin{aligned} {\mathbf {S}}({\mathbf {e}}_z, {\mathbf {e}}_y)&= {\mathbf {S}}({\mathbf {e}}_y, {\mathbf {e}}_z) = \frac{3}{2r^2} \sin 2\theta \sin \phi \ {\mathbf {e}}_r . \end{aligned}$$

(111)

1.4 Rotlets

The vectorial representation of a rotlet is given by Eq. (46) in the main text, where $\varvec{\zeta } = \varvec{\beta } \times \varvec{\alpha }$ denotes the direction of the rotlet. The expressions of rotlets in different directions in spherical coordinates are given by

$$\begin{aligned} {\mathbf {R}}({\mathbf {e}}_x)&= \frac{1}{r^2}\left( - \sin \phi \ {\mathbf {e}}_\theta -\cos \theta \cos \phi \ {\mathbf {e}}_\phi \right) , \end{aligned}$$

(112)

$$\begin{aligned} {\mathbf {R}}({\mathbf {e}}_y)&= \frac{ 1}{r^2} \left( \cos \phi \ {\mathbf {e}}_\theta -\cos \theta \sin \phi \ {\mathbf {e}}_\phi \right) ,\end{aligned}$$

(113)

$$\begin{aligned} {\mathbf {R}}({\mathbf {e}}_z)&= \frac{1}{r^2} \left( \sin \theta \ {\mathbf {e}}_\phi \right) . \end{aligned}$$

(114)

1.5 A general Stokes quadrupole

A higher-order singularity, the force quadrupole, is obtained by taking the derivative of a force dipole along different directions and is given by Eq. (50) in the main text, where $\varvec{\beta }, \varvec{\gamma }$ are the directions along which each derivative is taken. The expressions of Stokes quadrupoles of different configurations in spherical coordinates are given by

$$\begin{aligned}&{\mathbf {G}}({\mathbf {e}}_x,{\mathbf {e}}_x,{\mathbf {e}}_x) = \frac{1}{4r^3} \left[ - (5 \sin \theta +9\sin 3\theta )\cos \phi + 3 (3\sin \theta -\sin 3\theta )\cos 3\phi \right] {\mathbf {e}}_r \nonumber \\&\qquad \qquad \qquad \qquad \, + \frac{1}{16r^3} \left[ (7 \cos \theta +9\cos 3\theta )\cos \phi -3 (\cos \theta -\cos 3\theta ) \cos 3\phi \right] {\mathbf {e}}_\theta \nonumber \\&\qquad \qquad \qquad \qquad \, + \frac{1}{8r^3} \left[ -(5+3\cos 2\theta )\sin \phi + 3(1-\cos 2\theta )\sin 3\phi \right] {\mathbf {e}}_\phi ,\end{aligned}$$

(115)

$$\begin{aligned}&{\mathbf {G}}({\mathbf {e}}_y,{\mathbf {e}}_x,{\mathbf {e}}_x)={\mathbf {G}}({\mathbf {e}}_x,{\mathbf {e}}_y,{\mathbf {e}}_x) \nonumber \\&\qquad \qquad \qquad \quad = \ \frac{1}{4r^3} \left[ (\sin \theta -3\sin 3\theta )\sin \phi +3(3\sin \theta -\sin 3\theta ) \sin 3\phi \right] {\mathbf {e}}_r \nonumber \\&\quad \quad \qquad \qquad \qquad \, + \frac{1}{16r^3} \left[ (13 \cos \theta +3\cos 3\theta )\sin \phi + 3(\cos 3\theta -\cos \theta )\sin 3\phi \right] {\mathbf {e}}_\theta \nonumber \\&\qquad \qquad \qquad \quad \quad \, + \frac{1}{8r^3} \left[ (9\cos 2\theta -1)\cos \phi -3(1-\cos 2\theta ) \cos 3\phi \right] {\mathbf {e}}_\phi ,\end{aligned}$$

(116)

$$\begin{aligned}&{\mathbf {G}}({\mathbf {e}}_z, {\mathbf {e}}_x,{\mathbf {e}}_x) ={\mathbf {G}}({\mathbf {e}}_x, {\mathbf {e}}_z,{\mathbf {e}}_x) \nonumber \\&\quad \qquad \qquad \qquad = \ \frac{1}{2r^3} \left[ - (\cos \theta +3\cos 3\theta )+3 (\cos \theta -\cos 3\theta )\cos 2\phi \right] {\mathbf {e}}_r \nonumber \\&\quad \quad \qquad \qquad \qquad \, + \frac{1}{8r^3} \left[ (\sin \theta -3\sin 3\theta )+3 (3\sin \theta -\sin 3\theta )\cos 2\phi \right] {\mathbf {e}}_\theta ,\end{aligned}$$

(117)

$$\begin{aligned}&{\mathbf {G}}({\mathbf {e}}_y,{\mathbf {e}}_y,{\mathbf {e}}_x) = \ \frac{1}{4r^3} \left[ - (7 \sin \theta +3\sin 3\theta )\cos \phi -3 (3\sin \theta -\sin 3\theta )\cos 3\phi \right] {\mathbf {e}}_r \nonumber \\&\qquad \qquad \qquad \qquad \quad + \frac{1}{16r^3} \left[ (3\cos 3\theta -19\cos \theta )\cos \phi + 3(\cos \theta -\cos 3\theta )\cos 3\phi \right] {\mathbf {e}}_\theta \nonumber \\&\qquad \qquad \qquad \qquad \quad + \frac{1}{8r^3} \left[ (15\cos 2\theta -7)\sin \phi -3(1-\cos 2\theta ) \sin 3\phi \right] {\mathbf {e}}_\phi ,\end{aligned}$$

(118)

$$\begin{aligned}&{\mathbf {G}}({\mathbf {e}}_z,{\mathbf {e}}_y,{\mathbf {e}}_x) = \ {\mathbf {G}}({\mathbf {e}}_y,{\mathbf {e}}_z,{\mathbf {e}}_x) \nonumber \\&\qquad \qquad \qquad \quad = \ \frac{1}{r^3} \left[ \frac{3}{2}(\cos \theta -\cos 3\theta )\sin 2\phi \ {\mathbf {e}}_r + \frac{3}{8} (3\sin \theta -\sin 3\theta )\sin 2\phi \ {\mathbf {e}}_\theta -\frac{3}{2}\sin 2\theta \ {\mathbf {e}}_\phi \right] ,\end{aligned}$$

(119)

$$\begin{aligned}&{\mathbf {G}}({\mathbf {e}}_z,{\mathbf {e}}_z,{\mathbf {e}}_x) = \ \frac{1}{r^3} \left[ (3\sin 3\theta -\sin \theta ) \cos \phi \ {\mathbf {e}}_r + \frac{1}{4} (11\cos \theta -3\cos 3\theta )\cos \phi \ {\mathbf {e}}_\theta \right. \nonumber \\&\qquad \qquad \qquad \qquad \qquad \quad -\left. \frac{1}{2}(1+3\cos 2\theta )\sin \phi \ {\mathbf {e}}_\phi \right] ,\end{aligned}$$

(120)

$$\begin{aligned}&{\mathbf {G}}({\mathbf {e}}_x,{\mathbf {e}}_x,{\mathbf {e}}_y) = \frac{1}{4r^3} \left[ -(7 \sin \theta + 3\sin 3\theta ) \sin \phi +3 (3\sin \theta -\sin 3\theta ) \sin 3\phi \right] {\mathbf {e}}_r \nonumber \\&\qquad \qquad \qquad \qquad \, + \frac{1}{16r^3} \left[ \cos \theta (-19+3\cos 2\theta )\sin \phi -3(\cos \theta -\cos 3\theta )\sin 3\phi \right] {\mathbf {e}}_\theta \nonumber \\&\qquad \qquad \qquad \qquad \, + \frac{1}{8r^3} \left[ (7-15 \cos 2\theta )\cos \phi - 3(1-\cos 2\theta )\cos 3\phi \right] {\mathbf {e}}_\phi ,\end{aligned}$$

(121)

$$\begin{aligned}&{\mathbf {G}}({\mathbf {e}}_y, {\mathbf {e}}_x, {\mathbf {e}}_y) ={\mathbf {G}}({\mathbf {e}}_x, {\mathbf {e}}_y, {\mathbf {e}}_y) \nonumber \\&\qquad \qquad \qquad \quad = \ \frac{1}{4r^3} \left[ (\sin \theta -3\sin 3\theta ) \cos \phi - 3(3\sin \theta -\sin 3\theta ) \cos 3\phi \right] {\mathbf {e}}_r \nonumber \\&\qquad \qquad \qquad \qquad + \frac{1}{16r^3} \left[ (13 \cos \theta +3\cos 3\theta )\cos \phi +3(\cos \theta -\cos 3\theta )\cos 3\phi \right] {\mathbf {e}}_\theta \nonumber \\&\qquad \qquad \qquad \qquad + \frac{1}{8r^3} \left[ (1-9\cos 2\theta )\sin \phi -3(1-\cos 2\theta ) \sin 3\phi \right] {\mathbf {e}}_\phi , \end{aligned}$$

(122)

$$\begin{aligned}&{\mathbf {G}}({\mathbf {e}}_z,{\mathbf {e}}_x,{\mathbf {e}}_y) = {\mathbf {G}}({\mathbf {e}}_x,{\mathbf {e}}_z,{\mathbf {e}}_y) \nonumber \\&\qquad \qquad \qquad \quad = \frac{1}{r^3} \left[ \frac{3}{2} (\cos \theta -\cos 3\theta )\sin 2\phi \ {\mathbf {e}}_r + \frac{3}{8}(3\sin \theta -\sin 3\theta )\sin 2\phi \ {\mathbf {e}}_\theta + \frac{3}{2}\sin 2\theta \ {\mathbf {e}}_\phi \right] ,\end{aligned}$$

(123)

$$\begin{aligned}&{\mathbf {G}}({\mathbf {e}}_y,{\mathbf {e}}_y,{\mathbf {e}}_y) = \ \frac{1}{4r^3} \left[ -(5\sin \theta +9\sin 3\theta )\sin \phi -3(3\sin \theta -\sin 3\theta )\sin 3\phi \right] {\mathbf {e}}_r \nonumber \\&\qquad \qquad \qquad \qquad \, + \frac{1}{16r^3} \left[ (7\cos \theta +9\cos 3\theta ) \sin \phi +3(\cos \theta -\cos 3\theta )\sin 3\phi \right] {\mathbf {e}}_\theta \nonumber \\&\qquad \qquad \qquad \qquad \,+ \frac{1}{8r^3} \left[ (5+3\cos 2\theta )\cos \phi + 3 (1-\cos 2\theta ) \cos 3\phi \right] {\mathbf {e}}_\phi ,\end{aligned}$$

(124)

$$\begin{aligned}&{\mathbf {G}}({\mathbf {e}}_z, {\mathbf {e}}_y, {\mathbf {e}}_y)= {\mathbf {G}}({\mathbf {e}}_y, {\mathbf {e}}_z, {\mathbf {e}}_y) \nonumber \\&\qquad \qquad \qquad \quad = \ \frac{1}{2r^3} \left[ - (\cos \theta +3\cos 3\theta )- 3(\cos \theta -\cos 3\theta ) \cos 2\phi \right] {\mathbf {e}}_r \nonumber \\&\qquad \qquad \qquad \qquad \, + \frac{1}{8r^3} \left[ (\sin \theta -3\sin 3\theta )-3 (3\sin \theta -\sin 3 \theta )\cos 2\phi \right] {\mathbf {e}}_\theta ,\end{aligned}$$

(125)

$$\begin{aligned}&{\mathbf {G}}({\mathbf {e}}_z,{\mathbf {e}}_z,{\mathbf {e}}_y) = \frac{1}{r^3} \left[ (3\sin \theta -\sin \theta )\sin \phi \ {\mathbf {e}}_r+ \frac{1}{4}(11\cos \theta -3\cos 3\theta )\sin \phi \ {\mathbf {e}}_\theta \right. \nonumber \\&\qquad \qquad \qquad \qquad \qquad \quad +\left. \frac{1}{2}(1+3\cos 2\theta )\cos \phi \ {\mathbf {e}}_\phi \right] ,\end{aligned}$$

(126)

$$\begin{aligned}&{\mathbf {G}}({\mathbf {e}}_x,{\mathbf {e}}_x,{\mathbf {e}}_z) = \frac{1}{2r^3}\left[ -(5 \cos \theta +3 \cos 3\theta )+ 3(\cos \theta -\cos 3\theta )\cos 2\phi \right] {\mathbf {e}}_r \nonumber \\&\qquad \qquad \qquad \qquad \, - \frac{1}{8r^3} \left[ (7\sin \theta +3\sin 3\theta )+ 3(5\sin \theta +\sin 3\theta )\cos 2\phi \right] {\mathbf {e}}_\theta + \frac{3}{2r^3} \sin 2\theta \sin 2\phi \ {\mathbf {e}}_\phi ,\end{aligned}$$

(127)

$$\begin{aligned}&{\mathbf {G}}({\mathbf {e}}_y, {\mathbf {e}}_x, {\mathbf {e}}_z)= {\mathbf {G}}({\mathbf {e}}_x,{\mathbf {e}}_y,{\mathbf {e}}_z) \nonumber \\&\qquad \qquad \qquad \quad = \frac{1}{r^3} \left[ \frac{3}{2}(\cos \theta -\cos 3\theta )\sin 2\phi \ {\mathbf {e}}_r - \frac{3}{8} (5\sin \theta +\sin 3\theta ) \sin 2\phi \ {\mathbf {e}}_\theta - \frac{3}{2} \sin 2\theta \cos 2\phi \ {\mathbf {e}}_\phi \right] , \end{aligned}$$

(128)

$$\begin{aligned}&{\mathbf {G}}({\mathbf {e}}_z,{\mathbf {e}}_x,{\mathbf {e}}_z) = {\mathbf {G}}({\mathbf {e}}_x,{\mathbf {e}}_z,{\mathbf {e}}_z) \nonumber \\&\qquad \qquad \qquad \quad = \frac{1}{r^3} \left[ (\sin \theta +3\sin 3\theta )\cos \phi \ {\mathbf {e}}_r- \frac{1}{4}(5\cos \theta +3\cos 3\theta )\cos \phi \ {\mathbf {e}}_\theta + \frac{1}{2}(1+3\cos 2\theta ) \ {\mathbf {e}}_\phi \right] ,\end{aligned}$$

(129)

$$\begin{aligned}&{\mathbf {G}}({\mathbf {e}}_y,{\mathbf {e}}_y,{\mathbf {e}}_z) = \ \frac{1}{2r^3} \left[ -(5\cos \theta +3\cos 3\theta ) - 3(\cos \theta -\cos 3\theta ) \cos 2\phi \right] {\mathbf {e}}_r \nonumber \\&\qquad \qquad \qquad \qquad \, + \frac{1}{8r^3} \left[ -(7\sin \theta +3\sin 3\theta ) + 3(5\sin \theta +\sin 3\theta )\cos 2\phi \right] {\mathbf {e}}_\theta - \frac{3}{2r^3} \sin 2\theta \sin 2\phi \ {\mathbf {e}}_\phi ,\end{aligned}$$

(130)

$$\begin{aligned}&{\mathbf {G}}({\mathbf {e}}_z, {\mathbf {e}}_y, {\mathbf {e}}_z) ={\mathbf {G}}({\mathbf {e}}_y, {\mathbf {e}}_z, {\mathbf {e}}_z) \nonumber \\&\qquad \qquad \qquad \quad = \!\frac{1}{r^3} \left[ (\sin \theta +3\sin 3\theta )\sin \phi \ {\mathbf {e}}_r \!-\! \frac{1}{4} (5\cos \theta \!+\!3\cos 3\theta )\!\sin \phi \!\ {\mathbf {e}}_\theta \!-\! \frac{1}{2} (1+3\cos 2\theta ) \!\cos \phi \ {\mathbf {e}}_\phi \right] \!,\quad \end{aligned}$$

(131)

$$\begin{aligned}&{\mathbf {G}}({\mathbf {e}}_z,{\mathbf {e}}_z,{\mathbf {e}}_z) = \frac{1}{r^3} \left[ (\cos \theta \!+\!3\cos 3\theta ) \ {\mathbf {e}}_r + \frac{1}{4}(3\sin 3\theta -\sin \theta ) \ {\mathbf {e}}_\theta \right] . \end{aligned}$$

(132)

1.6 Potential dipoles

The vectorial representation of a potential (source) dipole is given by Eq. (52) in the main text, where $\varvec{\alpha }$ denotes the direction of the dipole. The expressions of potential dipoles in different directions in spherical coordinates are given by

$$\begin{aligned} {\mathbf {P}}_\mathrm{D}({\mathbf {e}}_x)&= \frac{1}{r^3} \left[ 2 \sin \theta \cos \phi \ {\mathbf {e}}_r -\cos \theta \cos \phi \ {\mathbf {e}}_\theta + \sin \phi \ {\mathbf {e}}_\phi \right] , \end{aligned}$$

(133)

$$\begin{aligned} {\mathbf {P}}_\mathrm{D}({\mathbf {e}}_y)&= \frac{1}{r^3} \left[ 2 \sin \theta \sin \phi \ {\mathbf {e}}_r - \cos \theta \sin \phi \ {\mathbf {e}}_\theta - \cos \phi \ {\mathbf {e}}_\phi \right] ,\end{aligned}$$

(134)

$$\begin{aligned} {\mathbf {P}}_\mathrm{D}({\mathbf {e}}_z)&= \frac{1}{r^3} \left[ 2 \cos \theta \ {\mathbf {e}}_r + \sin \theta \ {\mathbf {e}}_\theta \right] . \end{aligned}$$

(135)

1.7 Rotlet dipoles

One can take a derivative of a rotlet to obtain a rotlet dipole, which is given by Eq. (56) in the main text, where $\varvec{\zeta }$ and $\varvec{\gamma }$ denote respectively the direction of the rotlet and the direction along which the derivative is taken. The expressions of rotlet dipoles of different configurations in spherical coordinates are given by

$$\begin{aligned}&{\mathbf {R}}_D({\mathbf {e}}_x, {\mathbf {e}}_x) = \frac{1}{r^3} \left[ -\frac{3}{2} \sin \theta \sin 2 \phi \ {\mathbf {e}}_\theta - \frac{3}{4} \sin 2\theta \left( 1+\cos 2\phi \right) {\mathbf {e}}_\phi \right] , \end{aligned}$$

(136)

$$\begin{aligned}&{\mathbf {R}}_\mathrm{D}({\mathbf {e}}_y,{\mathbf {e}}_y) = \frac{1}{r^3} \left[ \frac{3}{2} \sin \theta \sin 2 \phi \ {\mathbf {e}}_\theta - \frac{3}{4} \sin 2 \theta \left( 1- \cos 2\phi \right) {\mathbf {e}}_\phi \right] ,\end{aligned}$$

(137)

$$\begin{aligned}&{\mathbf {R}}_\mathrm{D}({\mathbf {e}}_z,{\mathbf {e}}_z) = \frac{3 \sin 2\theta }{2r^3}{\mathbf {e}}_\phi , \end{aligned}$$

(138)

$$\begin{aligned}&{\mathbf {R}}_\mathrm{D}({\mathbf {e}}_y,{\mathbf {e}}_x) = \frac{1}{r^3} \left[ -\cos \theta \ {\mathbf {e}}_r -\frac{1}{2}\sin \theta \left( 1-3 \cos 2\phi \right) \ {\mathbf {e}}_\theta - \frac{3}{4} \sin 2\theta \sin 2 \phi \ {\mathbf {e}}_\phi \right] ,\end{aligned}$$

(139)

$$\begin{aligned}&{\mathbf {R}}_\mathrm{D}({\mathbf {e}}_z,{\mathbf {e}}_x)= \frac{1}{r^3} \left[ \sin \theta \sin \phi \ {\mathbf {e}}_r - 2 \cos \theta \sin \phi \ {\mathbf {e}}_\theta - \frac{1}{2} \left( 1+3 \cos 2\theta \right) \cos \phi \ {\mathbf {e}}_\phi \right] ,\end{aligned}$$

(140)

$$\begin{aligned}&{\mathbf {R}}_\mathrm{D}({\mathbf {e}}_x,{\mathbf {e}}_y)= \frac{1}{r^3} \left[ \cos \theta \ {\mathbf {e}}_r + \frac{1}{2}\sin \theta \left( 1 +3 \cos 2\phi \right) \ {\mathbf {e}}_\theta - \frac{3}{4} \sin 2 \theta \sin 2 \phi \ {\mathbf {e}}_\phi \right] , \end{aligned}$$

(141)

$$\begin{aligned}&{\mathbf {R}}_\mathrm{D}({\mathbf {e}}_z,{\mathbf {e}}_y)= \frac{1}{r^3} \left[ -\sin \theta \cos \phi \ {\mathbf {e}}_r + 2 \cos \theta \cos \phi \ {\mathbf {e}}_\theta - \frac{1}{2}\left( 1+3 \cos 2\theta \right) \sin \phi \ {\mathbf {e}}_\phi \right] , \end{aligned}$$

(142)

$$\begin{aligned}&{\mathbf {R}}_\mathrm{D}({\mathbf {e}}_x,{\mathbf {e}}_z)=\frac{1}{r^3} \left[ -\sin \theta \sin \phi \ {\mathbf {e}}_r -\cos \theta \sin \phi \ {\mathbf {e}}_\theta + \frac{1}{2} \left( 1-3\cos 2\theta \right) \cos \phi \ {\mathbf {e}}_\phi \right] , \end{aligned}$$

(143)

$$\begin{aligned}&{\mathbf {R}}_\mathrm{D}({\mathbf {e}}_y,{\mathbf {e}}_z)= \frac{1}{r^3} \left[ \sin \theta \cos \phi \ {\mathbf {e}}_r + \cos \theta \cos \phi \ {\mathbf {e}}_\theta + \frac{1}{2}\left( 1-3\cos 2\theta \right) \sin \phi \ {\mathbf {e}}_\phi \right] . \end{aligned}$$

(144)

Appendix B: Swimming of a squirmer with radial deformation

In the main text, we considered squirmers with a purely tangential deformation. In this appendix, we complement these results by addressing the case of squirmers also undergoing radial deformation. The results here might also be useful for modeling jet-driven microscopic swimmers, for instance, the locomotion of bacteria expelling slime.

For the axisymmetric case without the restriction to purely tangential deformations, Eq. (19), the solution to the pumping problem reads

$$\begin{aligned} u_r&= \sum _{n=1}^\infty \frac{(n+1) P_n}{2 (2n-1) \eta r^{n+2}} \left[ A_{0n} r^2 - 2 B_{0n} (2n-1) \eta \right] ,\end{aligned}$$

(145)

$$\begin{aligned} u_\theta&= \sum _{n=1}^\infty \frac{\sin \theta P_n^{\prime }}{2 r^{n}} \left[ \frac{ n-2}{n(2n-1) \eta } A_{0n} - \frac{2}{r^2} B_{0n} \right] , \end{aligned}$$

(146)

$$\begin{aligned} u_\phi&= \sum _{n=1}^\infty \frac{\sin \theta P_n^{\prime }}{r^{n+1}} C_{0n}, \end{aligned}$$

(147)

with the surface velocities

$$\begin{aligned} u_r(r=a)&= \sum _{n=1}^\infty \left[ \frac{(n+1)A_{0n} }{2 (2n-1) \eta a^{n}} - \frac{B_{0n}}{a^{n+2}} \right] P_n,\end{aligned}$$

(148)

$$\begin{aligned} u_\theta (r=a)&= \sum _{n=1}^\infty \left[ \frac{ n-2}{2n(2n-1)a^n \eta } A_{0n} - \frac{1}{a^{n+2}} B_{0n} \right] \sin \theta P_n^{\prime } , \end{aligned}$$

(149)

$$\begin{aligned} u_\phi (r=a)&= \sum _{n=1}^\infty \frac{\sin \theta P_n^{\prime }}{a^{n+1}} C_{0n} . \end{aligned}$$

(150)

Without the azimuthal modes $C_{0n}$, the preceding surface velocities reduce to the form in Lighthill [40] and Blake [41]. However, notice that the coefficients $A_{0n}$ and $B_{0n}$ do not correspond directly to the coefficients $A_n$ and $B_n$ used in Lighthill [40] and Blake [41], which represent directly the radial and polar modes, respectively. With Lamb’s general solution, the radial and polar modes are represented by a combination of the $A_{0n}$ and $B_{0n}$ modes. The relation between the two sets of coefficients is given by

$$\begin{aligned} A_{0n}&= \frac{a^n n (2n-1)\eta }{n+1} A_n - \frac{2a^n (2n-1)\eta }{n+1} B_n, \end{aligned}$$

(151)

$$\begin{aligned} B_{0n}&= \frac{a^{n+2}(n-2)}{2(n+1)}A_n - \frac{a^{n+2}}{n+1}B_n . \end{aligned}$$

(152)

The translational and rotational velocities are computed similarly to Sect. 3.1. Without the restriction to tangential deformation, Eq. (19), the propulsion velocity becomes

$$\begin{aligned} {\mathbf {U}}&= \frac{2}{3 \eta a} \nabla \left[ r \left( P_1 A_{01} \right) \right] = -\frac{2}{3a \eta } A_{01} {\mathbf {e}}_z, \end{aligned}$$

(153)

while the computation of the rotational velocity is unaffected (Eq. 32).

The flow field around an axisymmetric swimming squirmer is thus given by

$$\begin{aligned} v_r&= \left( \frac{a^2}{3 \eta }A_{01}-2B_{01} \right) \frac{\cos \theta }{r^3}+ \sum _{n=2}^\infty \frac{(n+1) P_n}{r^{n+2}} \left[ \frac{A_{0n} r^2}{2(2n-1) \eta } - B_{0n} \right] , \end{aligned}$$

(154)

$$\begin{aligned} v_\theta&= \left( \frac{a^2}{6\eta } A_{01}-B_{01}\right) \frac{\sin \theta }{r^3} + \sum _{n=2}^\infty \frac{\sin \theta P_n^{\prime }}{r^{n+2}} \left[ \frac{(n-2)r^2}{2n (2n-1) \eta } A_{0n} - B_{0n}\right] , \end{aligned}$$

(155)

$$\begin{aligned} v_\phi&= \sum _{n=2}^\infty \frac{\sin \theta P_n^{\prime }}{r^{n+1}} C_{0n} . \end{aligned}$$

(156)

The computation of the propulsion speed of a nonaxisymmetric squirmer with radial deformation follows the same procedures as in Sect. 4.1, but without the restriction to tangential deformation, Eq. (19). The propulsion speed is given by

$$\begin{aligned} {\mathbf {U}}&= \frac{2}{3 \eta a} \nabla \left[ r \left( P_1 A_{01}+ P_1^1 \cos \phi A_{11}+ P_1^1 \sin \phi \tilde{A}_{11} \right) \right] \nonumber \\&= \left. \frac{2}{3a \eta } \left( A_{11} {\mathbf {e}}_x + \tilde{A}_{11} {\mathbf {e}}_y - A_{01} {\mathbf {e}}_z \right) \right. , \end{aligned}$$

(157)

while the expression for the rotational speed remains the same, Eq. (61).

To obtain the overall swimming flow field, we follow the same procedures as in the axisymmetric case. Superimposing Lamb’s general solution in the pumping problem, Eqs. (16)–(17), with the flow fields due to the induced translation and rotation at the velocities determined earlier, we arrive at the flow field surrounding a general swimming squirmer without the assumption of purely tangential deformation:

$$\begin{aligned} v_r&= \frac{1}{r^3} \left[ \left( 2 B_{11} - \frac{a^2}{3 \eta }A_{11} \right) \sin \theta \cos \phi + \left( 2 \tilde{B}_{11}- \frac{a^2}{3 \eta } \tilde{A}_{11} \right) \sin \theta \sin \phi - \left( 2B_{01}- \frac{a^2}{3 \eta }A_{01} \right) \cos \theta \right] \nonumber \\&+ \sum _{n=2}^\infty \sum _{m=0}^n \frac{(n+1) P^m_n}{r^{n+2}} \left\{ \left[ \frac{A_{mn} r^2}{2(2n-1) \eta } - B_{mn} \right] \cos m\phi + \left[ \frac{\tilde{A}_{mn} r^2}{2(2n-1) \eta } - \tilde{B}_{mn} \right] \sin m\phi \right\} ,\end{aligned}$$

(158)

$$\begin{aligned} v_\theta&= -\frac{1}{r^3} \left[ \left( B_{11}- \frac{a^2}{6 \eta } A_{11}\right) \cos \theta \cos \phi + \left( \tilde{B}_{11}-\frac{a^2}{6\eta } \tilde{A}_{11} \right) \cos \theta \sin \phi + \left( B_{01}-\frac{a^2}{6\eta } A_{01}\right) \sin \theta \right] \nonumber \\&+ \sum _{n=2}^\infty \sum _{m=0}^n \frac{\sin \theta P_n^{m^{\prime }}}{r^{n+2}} \left\{ \left[ \frac{(n-2)r^2}{2n (2n-1) \eta } A_{mn} - B_{mn}\right] \cos m \phi + \left[ \frac{(n-2)r^2}{2n(2n-1) \eta } \tilde{A}_{mn} - \tilde{B}_{mn} \right] \sin m\phi \right\} \nonumber \\&+ \sum _{n=2}^\infty \sum _{m=0}^n \frac{mP^m_n}{r^{n+1}\sin \theta } \left( \tilde{C}_{mn} \cos m\phi - C_{mn} \sin m\phi \right) ,\end{aligned}$$

(159)

$$\begin{aligned} v_\phi&= \frac{1}{r^3} \left[ \left( B_{11}-\frac{a^2}{6 \eta }A_{11} \right) \sin \phi - \left( \tilde{B}_{11}-\frac{a^2}{6 \eta }\tilde{A}_{11} \right) \cos \phi \right] + \sum _{n=2}^\infty \sum _{m=0}^n \frac{\sin \theta P_n^{m^{\prime }}}{r^{n+1}} \left( C_{mn} \cos m\phi + \tilde{C}_{mn} \sin m\phi \right) \nonumber \\&- \sum _{n=2}^\infty \sum _{m=0}^n \frac{mP_n^m}{r^{n+2}\sin \theta } \left\{ \left[ \frac{(n-2)r^2}{2n(2n-1)\eta } \tilde{A}_{mn} - \tilde{B}_{mn} \right] \cos m\phi - \left[ \frac{(n-2)r^2}{2n(2n-1)\eta } A_{mn} - B_{mn} \right] \sin m\phi \right\} . \end{aligned}$$

(160)

Appendix C: Rate of work with radial deformation

We now compute the rate of work of a swimmer with a radial deformation. Lengthy calculations allow us to compute the integral from Eq. (76) as

$$\begin{aligned} \mathcal {P}&= \frac{48 \pi \eta }{a^5} \left( B_{01}^2 + B_{11}^2 + \tilde{B}_{11}^2 \right) + \frac{4 \pi }{3 a \eta } \left( A_{01}^2 + A_{11}^2 + \tilde{A}_{11}^2 \right) - \frac{16 \pi }{a^3} \left( A_{01}B_{01}+A_{11} B_{11}+\tilde{A}_{11} \tilde{B}_{11} \right) \nonumber \\&\!+ \sum _{n=2}^\infty \frac{4 \pi n (n\!+\!1)}{2n\!+\!1} \left[ \frac{2n^3 \!+\! n^2 \!-\!2n\!+\!2}{2n^2 (2n\!-\!1)^2 \eta a^{2n-1}} A_{0n}^2 \!-\! \frac{2(n+2)}{n a^{2n+1}} A_{0n}B_{0n} \!+\! \frac{(10n+4 \!+\!4n^2) \eta }{n a^{2n+3}} B_{0n}^2 \!+\! \frac{(n+2) \eta }{a^{2n+1}} C_{0n}^2 \right] \nonumber \\&+ \sum _{n=2}^\infty \sum _{m=1}^n \frac{2\pi n (n+1)(n+m)!}{(2n+1)(n-m)!} \left[ \frac{2n^3+n^2-2n+2}{2n^2 (2n-1)^2 \eta a^{2n-1}} (A_{mn}^2+ \tilde{A}_{mn}^2)- \frac{2(n+2)}{n} (A_{mn} B_{mn}+\tilde{A}_{mn} \tilde{B}_{mn}) \right. \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad +\frac{(10n +4+4n^2) \eta }{n a^{2n+3}} (B_{mn}^2+ \tilde{B}_{mn}^2)+ \left. \frac{(n+2) \eta }{a^{2n+1}} (C_{mn}^2 + \tilde{C}_{mn}^2) \right] . \end{aligned}$$

(161)

We have again employed Eq. (78) to obtain the preceding result. Despite the presence of cross-terms, it can be shown by algebraic manipulations that the rate of work is positive definite. Removing all the nonaxisymmetric modes and transforming the coefficients with Eqs. (151) and (152), the preceding expression agrees with the results in Blake [41]. For the case of purely tangential deformation, with Eq. (19), the preceding expression reduces to Eq. (77) in the main text.

Appendix D: Squirming with arbitrary surface velocities and radial deformation

Here we allow in the general squirming profile additional radial velocity components of the form $D_{m} (\theta ), \tilde{D}_m (\theta )$,

$$\begin{aligned} \left. u_r\right| _{r=a}= \sum _{m=0}^\infty D_m (\theta ) \cos m\phi + \tilde{D}_m (\theta ) \sin m \phi , \end{aligned}$$

(162)

$$\begin{aligned} u_\theta \big |_{t=a} =\sum _{m=0}^\infty E_m (\theta ) \cos m \phi + \tilde{E}_m (\theta ) \sin m \phi , \end{aligned}$$

(163)

$$\begin{aligned} u_\phi \big |_{r=a}=\sum _{m=0}^\infty F_m (\theta ) \cos m \phi + \tilde{F}_m (\theta ) \sin m \phi , \end{aligned}$$

(164)

and follow the same matching conditions, Eqs. (83) and (84), in order to determine the coefficients $A_{mn}$, $\tilde{A}_{mn}$, $B_{mn}$, $\tilde{B}_{mn}$, $C_{mn}$, and $\tilde{C}_{mn}$ in Lamb’s general solution. The only difference is that we no longer have the purely tangential deformation condition (Eq. 19) and therefore are required to determine $A_{mn}, \tilde{A}_{mn},B_{mn}, \tilde{B}_{mn}$ such that the radial (Eq. 83) and polar (Eq. 84) matching conditions are satisfied simultaneously. Using the orthogonality of the associated Legendre polynomials and simultaneously solving the equations, we obtain

$$\begin{aligned} A_{mn}&= \frac{(2n-1) a^n \eta }{2} \frac{(2n+1)(n-m)!}{(n+1)(n+m)!} \mathop \int \limits _{-1}^1 \left[ n D_m + \frac{\partial }{\partial \mu } \left( E_m \sin \theta \right) - \frac{m \tilde{F}_m}{\sin \theta } \right] P_n^m \mathrm{d}\mu ,\end{aligned}$$

(165)

$$\begin{aligned} \tilde{A}_{mn}&= \frac{(2n-1) a^n \eta }{2} \frac{(2n+1)(n-m)!}{(n+1)(n+m)!} \mathop \int \limits _{-1}^1 \left[ n \tilde{D}_m + \frac{\partial }{\partial \mu } \left( \tilde{E}_m \sin \theta \right) + \frac{m F_m}{\sin \theta } \right] P_n^m \mathrm{d}\mu , \end{aligned}$$

(166)

$$\begin{aligned} B_{mn}&= \frac{a^{n+2}}{4} \frac{(2n+1)(n-m)!}{(n+1)(n+m)!} \mathop \int \limits _{-1}^1 \left[ (n-2) D_m + \frac{\partial }{\partial \mu } \left( E_m \sin \theta \right) -\frac{m \tilde{F}_m}{\sin \theta } \right] P_n^m \mathrm{d}\mu , \end{aligned}$$

(167)

$$\begin{aligned} \tilde{B}_{mn}&= \frac{a^{n+2}}{4} \frac{(2n+1)(n-m)!}{(n+1)(n+m)!} \mathop \int \limits _{-1}^1 \left[ (n-2) \tilde{D}_m + \frac{\partial }{\partial \mu } \left( \tilde{E}_m \sin \theta \right) + \frac{m F_m}{\sin \theta } \right] P_n^m \mathrm{d}\mu ,\end{aligned}$$

(168)

$$\begin{aligned} C_{mn}&= \frac{a^{n+1}}{2n}\frac{(2n+1)(n-m)!}{(n+1)(n+m)!} \mathop \int \limits _{-1}^1 \left[ -\frac{\partial }{\partial \mu } \left( F_m \sin \theta \right) -\frac{m \tilde{E}_m}{\sin \theta } \right] P_n^m \mathrm{d}\mu , \end{aligned}$$

(169)

$$\begin{aligned} \tilde{C}_{mn}&= \frac{a^{n+1}}{2n}\frac{(2n+1)(n-m)!}{(n+1)(n+m)!} \mathop \int \limits _{-1}^1 \left[ -\frac{\partial }{\partial \mu } \left( \tilde{F}_m \sin \theta \right) +\frac{m E_m}{\sin \theta } \right] P_n^m \mathrm{d}\mu , \end{aligned}$$

(170)

for $0\le m \le \infty$ and $m \le n \ne 0$. When there is no radial deformation, i.e., $D_m = \tilde{D}_m= 0$, we recover the results without radial deformation (Eqs. 90–93 and 19).