Stochastic reorientations and the hydrodynamics of microswimmers near deformable interfaces

We study the hydrodynamic interaction between a microswimmer and a deformable interface when the swimmer can stochastically reorient itself. We consider a force- and torque-free swimmer, modeled as a slender body, that can execute random orientation tumbles or active Brownian rotations in the plane of the deformable interface. When the swimmer is in the more viscous fluid, our analysis shows that both tumbles and Brownian rotations acting on timescales comparable to that of interface deformations can lead to a pusher-type swimmer rotating away from the interface, while enhancing its attraction towards the interface. In turn, the intrinsic orientational stochasticity of the microswimmer favors a stronger migration of pushers towards the interface at short times, but migration away from the interface in the long-time limit. However, irrespective of the viscosity ratio of the two fluid medium, the tendency of a pusher to align parallel to the interface is suppressed; the results for puller-type swimmers are the opposite. Our study has potential consequences for the residence time of swimming microorganisms near deformable boundaries.

Figure 1

A schematic of the system for tumbling swimmers: A swimmer disturbs the fluid in fluid region 1. A deformable interface separates regions 1 and 2 with viscosities ${\eta }_1$ and ${\eta }_2$, respectively. The gray (dashed) plane represents the reference configuration of the undeformed interface and the black (solid) plane is the deformed configuration due to the disturbance flow field generated by the swimmer. On average, the swimmer spends a time $\tau$ between two orientation tumbles, and after tumbling it intrinsically adopts a new orientation. Here, ${\mathit{p}}_i$ represents the swimmer orientations before the $i\mathrm{th}$ tumble and is decomposed in the coordinate system described in the bottom left corner. The lighter swimmers are the in-plane projections on the interface of the corresponding swimmer located above them in fluid 1. The top left figure shows how the slender swimmer model relates to an actual swimmer of length $L$.

Figure 2

(a) The vertical component of the rotation rate ${\dot{p}}_z\left(t\right)$ plotted as a function of time for a tumbling ($\tau =0.15$) and straight ($\tau \to \infty$) pusher initially oriented at $\theta \left(0\right)={70}^{\circ }$, $\varphi \left(0\right)=0^{\circ }$. The black (open) circles denote tumbling events and the yellow (filled) circles denote those tumbles where $\mathrm{\Delta }\varphi <\pm {20}^{\circ }$ upon tumbling. (b) The relative orientation $\left[\theta \right(t)-\theta (0\left)\right]$ plotted as a function of time averaged over five different runs; the error bars indicate the extremities associated with the individual runs. In all plots, $r_{z_0}=1$, $\lambda =0.1$, $\kappa =10$, $\mathrm{\Gamma }=1$, and ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=1$.

Figure 3

Contour plots of the interface deformation (a) before the first tumble, (b) after two tumbles, and (c) after three tumbles, for swimmers initially oriented at $\theta \left(0\right)={70}^{\circ }$, $\varphi \left(0\right)=0^{\circ }$. The black arrow represents the in-plane projection of the swimmer orientation. In all plots, $r_{z_0}=1$, $\lambda =0.1$, $\tau =0.15$, $\kappa =10$, $\mathrm{\Gamma }=1$, and ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=1$.

Figure 4

(a) A single realization of the rotation rate ${\dot{p}}_z$ plotted as a function of time for $\theta \left(0\right)={70}^{\circ }$, $\varphi \left(0\right)=0^{\circ }$ with mean run durations $\tau =0.15,0.25,1$, and $\tau \to \infty$. (b) The relative rotation $\theta \left(t\right)-\theta \left(0\right)$ of a pusher plotted as a function of time for the same swimmer initial orientation for $\tau =0.15,0.25,0.5$, 0.75, 1, and $\tau \to \infty$, averaged over five runs; the error bars indicate the extremities associated with the individual runs. In all plots, $r_{z_0}=1$, $\lambda =0.1$, $\kappa =10$, $\mathrm{\Gamma }=1$, and ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=1$.

Figure 5

(a) The relative rotation $\theta \left(t\right)-\theta \left(0\right)$ of a pusher plotted as a function of time for different initial swimmer orientations at $\tau =0.15$ and $\lambda =0.1$. The curves are averages of five different runs and the error bars indicate the extremities associated with the individual runs. (b) The orientation phase portrait of a straight swimmer for $\lambda =0.1$. In all plots, $r_{z_0}=1$, $\kappa =10$, $\mathrm{\Gamma }=1$, and ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=1$.

Figure 6

(a) A single instance of the vertical translation velocity $V_z^T$ of a pusher plotted as a function of time for initial swimmer orientations of $\theta \left(0\right)={45}^{\circ }$, ${55}^{\circ }$, and ${70}^{\circ }$ at $\tau =0.15$ and $\lambda =0.1$. The black (open) markers denote tumbling events and the yellow (filled) markers denote those tumbles where $\mathrm{\Delta }\varphi <\pm {20}^{\circ }$ upon tumbling. (b) The relative vertical translation $r_z\left(t\right)-r_z\left(0\right)$ for initial orientations $\theta \left(0\right)={45}^{\circ }$, ${50}^{\circ }$, ${55}^{\circ }$, and ${70}^{\circ }$. The solid curves are averages of five different runs and the error bars indicate the extremities associated with the individual runs. In both (a) and (b), the dashed curves correspond to straight swimmers ($\tau \to \infty$). In all plots, $r_{z_0}=1$, $\kappa =10$, $\mathrm{\Gamma }=1$, and ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=1$.

Figure 7

(a) The relative rotation $\theta \left(t\right)-\theta \left(0\right)$ of a pusher plotted as a function of time for $\theta \left(0\right)={56}^{\circ }$. (b) The corresponding relative vertical translation $r_z\left(t\right)-r_z\left(0\right)$ plotted as a function of time. In both (a) and (b), the dashed curve corresponds to straight swimmers ($\tau \to \infty$) and the solid curve is for a tumbling swimmer with $\tau =0.15$. In all plots, $r_{z_0}=1$, $\lambda =0.1$, $\kappa =10$, $\mathrm{\Gamma }=1$, and ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=1$.

Figure 8

(a) The relative rotation $\theta \left(t\right)-\theta \left(0\right)$ of a pusher plotted as a function of time for $\theta \left(0\right)={70}^{\circ }$ for viscosity ratio $\lambda =0.1,0.5,1,10$. (b) The corresponding relative vertical translational $r_z\left(t\right)-r_z\left(0\right)$ plotted as a function of time. In both (a) and (b), the dashed curves correspond to straight swimmers $(\tau \to \infty )$ and the solid curves are averages of five different runs for $\tau =0.15$; the error bars indicate the extremities associated with the individual runs. In all plots, $r_{z_0}=1$, $\lambda =0.1$, $\kappa =10$, $\mathrm{\Gamma }=1$, and ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=1$.

Figure 9

(a) The vertical component of the rotation rate ${\dot{p}}_z\left(t\right)$ plotted as a function of time for a tumbling ($\tau =0.15$), rotationally diffusing ($D_r=6.67$) and straight ($\tau \to \infty$ or $D_r\to 0$) pusher initially oriented at $\theta \left(0\right)={60}^{\circ }$, $\varphi \left(0\right)=0^{\circ }$; the black (open) circles mark tumbling events. (b) The relative orientation $\theta \left(t\right)-\theta \left(0\right)$ for the same initial orientation averaged over five different runs plotted as a function of time; the error bars indicate the extremities associated with the individual runs. The inset of (b) shows the orientation change of a single realization over long times. In all plots, $r_{z_0}=1$, $\lambda =0.1$, $\kappa =10$, $\mathrm{\Gamma }=1$, and ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=1$.

Figure 10

The in-plane component of the swimmer orientation $\varphi \left(t\right)$ plotted as a function of time for a (a) tumbling and (b) rotationally diffusing pusher initially oriented at $\theta \left(0\right)={60}^{\circ }$, $\varphi \left(0\right)=0^{\circ }$. The black (open) circles in (a) denote 27 tumbling events and the yellow (filled) circles denote those four tumbles where change in $\varphi <{20}^{\circ }$ upon tumbling. The yellow circles in (b) show the 616/800 rotational diffusion events in which $\mathrm{\Delta }\varphi \le {20}^{\circ }$. In all plots, $r_{z_0}=1$, $\lambda =0.1$, $\kappa =10$, $\mathrm{\Gamma }=1$, and ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=1$.

Figure 11

The in-plane orientation autocorrelation functions (a) $\langle {\mathit{p}}_{\parallel }\left(t\right)·{\mathit{p}}_{\parallel }\left(0\right)\rangle$ and (b) $\langle {[{\mathit{p}}_{\parallel }\left(t\right)·{\mathit{p}}_{\parallel }\left(0\right)]}^2\rangle$ plotted for both a tumbling ($\tau =0.15$) and a rotationally diffusing ($D_r=6.67$) swimmer averaged over five different runs; the error bars indicate the extremities associated with the individual runs. In both plots, $\theta \left(0\right)={70}^{\circ }$, $r_{z_0}=1$, $\lambda =0.1$, $\kappa =10$, $\mathrm{\Gamma }=1$, and ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=1$.