Flow properties and hydrodynamic interactions of rigid spherical microswimmers

We analyze a minimal model for a rigid spherical microswimmer and explore the consequences of its extended surface on the interplay between its self-propulsion and flow properties. The model is the first order representation of microswimmers, such as bacteria and algae, with rigid bodies and flexible propelling appendages. The flow field of such a microswimmer at finite distances significantly differs from that of a point-force (Stokeslet) dipole. For a suspension of microswimmers, we derive the grand mobility matrix that connects the motion of an individual swimmer to the active and passive forces and torques acting on all the swimmers. Our investigation of the mobility tensors reveals that hydrodynamic interactions among rigid-bodied microswimmers differ considerably from those among the corresponding point-force dipoles. Our results are relevant for the study of collective behavior of hydrodynamically interacting microswimmers by means of Stokesian dynamics simulations at moderate concentrations.

Figure 1

(a) Schematics of our minimal model for a spherical swimmer of radius $a$. The swimmer is driven by an active force ${\mathbf{f}}^{\mathrm{sp}}=f^{\mathrm{sp}}\widehat{\mathbf{n}}$ directed along its intrinsic orientation $\widehat{\mathbf{n}}$. Force neutrality of the system is incorporated via a conjugate force ${\mathbf{f}}^{\mathrm{spc}}=-{\mathbf{f}}^{\mathrm{sp}}$ that acts on the fluid at the position ${\mathbf{r}}_{\mathrm{c}}={\mathbf{r}}_{\mathrm{cm}}\pm \ell \widehat{\mathbf{n}}$ where ${\mathbf{r}}_{\mathrm{cm}}$ is the center of mass of the spherical body. For a pusher (top), ${\mathbf{r}}_{\mathrm{c}}={\mathbf{r}}_{\mathrm{cm}}-\ell \widehat{\mathbf{n}}$ and the forces ${\mathbf{f}}^{\mathrm{sp}}$ and ${\mathbf{f}}^{\mathrm{spc}}$ point away from each other, whereas for a puller (bottom), ${\mathbf{r}}_{\mathrm{c}}={\mathbf{r}}_{\mathrm{cm}}+\ell \widehat{\mathbf{n}}$, the forces point towards each other. (b) The model swimmer in a coordinate system whose origin $\mathbf{O}$ coincides with the swimmer's center of mass ${\mathbf{r}}_{\mathrm{cm}}$. $\mathbf{r}$ denotes any point in the fluid and ${\mathbf{r}}_{\mathrm{c}}^{\mathrm{eff}}$ the center of the effective point-force dipole that produces the same flow in the far field as the swimmer. Any point in the $x\text{−}y$ plane containing the vector $\widehat{\mathbf{n}}=-\widehat{\mathbf{x}}$ can also be denoted by the vector $\mathbf{r}-{\mathbf{r}}_{\mathrm{c}}^{\mathrm{eff}}$, which originates from ${\mathbf{r}}_{\mathrm{c}}^{\mathrm{eff}}$ and makes an angle $\theta$ with $\widehat{\mathbf{x}}$. The point ${\mathbf{r}}^{*}=-(a^2/\ell )\widehat{\mathbf{n}}$ denotes the position of the hydrodynamic image systems of the point force ${\mathbf{f}}^{\mathrm{spc}}$ generated by the sphere.

Figure 2

The normalized swimming speed $v_n^{\mathrm{sp}}=v^{\mathrm{sp}}/(f^{\mathrm{sp}}/6\pi \eta a)$ of a spherical swimmer of radius $a$, plotted against $a/\ell$, where $\ell$ is the separation between the active forces ${\mathbf{f}}^{\mathrm{sp}}$ and ${\mathbf{f}}^{\mathrm{spc}}=-{\mathbf{f}}^{\mathrm{sp}}$. The plots of $v_n^{\mathrm{sp}}$ for both pushers and pullers are identical. Inset: The ratio $S^{\mathrm{eff}}/S^0$ of the dipole strengths of the effective point-force dipole producing the correct far field and the point-force dipole obtained in the limit of zero swimmer size.

Figure 3

The flow streamlines (blue arrows) created by a spherical swimmer of radius $a$, with orientation $\widehat{\mathbf{n}}=-\widehat{\mathbf{x}}$ and $\ell =5a$, plotted in the $x\text{−}y$ plane. Panels (a) and (b) correspond to a pusher and a puller type swimmer, respectively. The red arrows represent the active forces ${\mathbf{f}}^{\mathrm{sp}}$ and ${\mathbf{f}}^{\mathrm{spc}}=-{\mathbf{f}}^{\mathrm{sp}}$, and the red dot indicates the effective center ${\mathbf{r}}_{\mathrm{c}}^{\mathrm{eff}}$ of the equivalent far-field point-force dipole in each case.

Figure 4

Angular dependence of the flow fields of a swimmer with $\ell =2.5a$ (pusher) and its associated point-force dipole and that of the bare point-force dipole obtained in the limit of zero swimmer size. For each case, the normalized flow field $u_{\mathrm{n}}^{\alpha }\left(\right|\mathbf{r}-{\mathbf{r}}_{\mathrm{c}}^{\mathrm{eff}}|,\theta )=u^{\alpha }\left(\right|\mathbf{r}-{\mathbf{r}}_{\mathrm{c}}^{\mathrm{eff}}|,\theta )/(f^{\mathrm{sp}}/6\pi \eta a)$ with $\alpha =x,y$, is shown as a function of the angle $\theta$ about the effective center ${\mathbf{r}}_{\mathrm{c}}^{\mathrm{eff}}$ measured at distances $|\mathbf{r}-{\mathbf{r}}_{\mathrm{c}}^{\mathrm{eff}}|=5a$ [(a), (c)] and $15a$ [(b), (d)].

Figure 5

Comparison of the flow fields of the swimmer with $\ell =2.5a$ (pusher) and its associated point-force dipole. The figure shows the contour plot of the magnitude of the difference of the flows of the swimmer and its associated point-force dipole, $\left|{\mathbf{u}}^{\mathrm{sp}}\left(\mathbf{r}\right)-{\mathbf{u}}^{\mathrm{dip}}\left(\mathbf{r}\right)\right|$, normalized by the angular averaged flow speed of the dipole, ${\langle u^{\mathrm{dip}}\left(\left|\mathbf{r}-{\mathbf{r}}_{\mathrm{c}}^{\mathrm{eff}}\right|\right)\rangle }_{\theta }$, at the corresponding distance from the center ${\mathbf{r}}_{\mathrm{c}}^{\mathrm{eff}}$. The body of the swimmer is shown in black; the active forces ${\mathbf{f}}^{\mathrm{sp}}$ and ${\mathbf{f}}^{\mathrm{spc}}$ and the effective center ${\mathbf{r}}_{\mathrm{c}}^{\mathrm{eff}}$ of the associated point-force dipole are shown in red. The white region (lighter color) represents values in the range 0.2–0.8 not shown in the color bar.

Figure 6

The effect of the finite size of the swimmers on hydrodynamic interactions. (a) Schematics of six configurations for two swimmers $i,j$ with respective orientations ${\widehat{\mathbf{n}}}_i,{\widehat{\mathbf{n}}}_j$ and a separation vector ${\mathbf{r}}_{ij}$. (b)–(e) The ratios $\delta v_{\left|\right|}/\delta v_{\left|\right|}^{\mathrm{dip}}$ and $\delta v_{\perp }/\delta v_{\perp }^{\mathrm{dip}}$ plotted as functions of the separation $r_{ij}/a$ for $\ell =2.5a$. Here $\delta v_{\left|\right|}$ ($\delta v_{\perp }$) is the velocity increment in the direction parallel (perpendicular) to the orientation of a swimmer resulting from its hydrodynamic interactions with another swimmer. Similarly, $\delta v_{\left|\right|}^{\mathrm{dip}}$ ($\delta v_{\perp }^{\mathrm{dip}}$) is the value for the corresponding associated point-force dipoles.