Wall-curvature driven dynamics of a microswimmer

Microorganisms navigate through fluid, often confined by complex environments, to survive and sustain life. Inspired by this fact, we consider a model system and seek to understand the wall curvature driven dynamics of a squirmer, a mathematical model for a microswimmer, using (i) lattice Boltzmann simulations and (ii) analytical theory by D. Papavassiliou and G. P. Alexander [Eur. Phys. Lett. 110, 44001 (2015)]. The instantaneous dynamics of the system is presented in terms of fluid velocity fields, and the translational and angular velocities of the microswimmer, whereas the long time dynamics is presented by plotting the squirmer trajectories near curved boundaries in physical and dynamical space, as well as characterizing them in terms of fixed points and experimentally relevant measures, namely, (i) proximity parameter, (ii) retention time, (iii) swimmer orientation and (iv) tangential velocity near the boundary, and (v) scattering angle during the collision. Our detailed analysis shows that irrespective of the type and strength, microswimmers exhibit a greater affinity towards a concave boundary due to hydrodynamic interactions compared to a convex boundary. In the presence of additional repulsive interactions with the boundary, we find that pullers (propel by forward thrust) have a slightly greater affinity towards the convex-curved walls compared to pushers (propel by backward thrust). Our study provides a comprehensive understanding of the consequence of hydrodynamic interactions in a unified framework that encompasses the dynamics of pullers, pushers, and neutral swimmers in the neighborhood of flat, concave, and convex walls. In addition, the combined effect of oppositely curved surfaces is studied by confining the squirmer in an annulus. The results presented in a unified framework and insights obtained are expected to be useful to design geometrical confinements to control and guide the motion of microswimmers in microfluidic applications.

Figure 1

Velocity field generated by an unconfined squirmer: (a) neutral swimmer ($B_1>0$, $B_2=0$) and (b) shaker ($B_1=0$, $B_2>0$). The black circle in the center represents the squirmer, and the thick horizontal arrow shows its orientation. Continuous lines are streamlines, and the color field shows the magnitude of velocity (normalized by $B_n/2$).

Figure 2

Schematic illustration of the system considered. (a) A squirmer of radius $r_s$ is located near a circular post of radius $r_b$ at a distance of $d=d_s+r_s+r_b$, where $d_s$ is the shortest distance between the surface of the swimmer and the surface of the boundary. The orientation vector of the squirmer makes an angle ${\theta }_s$ with the separation vector ($\widehat{\mathit{s}}$). (b) Change in the curvature $\alpha =r_s/r_b$ corresponds to different configurations (i) $\alpha \to -\infty$, squirmer near an infinitesimally small post, (ii) $\alpha =-1$, post has the same radius as the squirmer, (iii) $\alpha =-0.1$, squirmer near a convex boundary, (iv) $\alpha =0$, squirmer near a flat wall, (v) $\alpha =0.1$, squirmer near a concave boundary, and (vi) $\alpha =0.65$, squirmer in a strong circular confinement.

Figure 3

The curved boundary of the solid particle (continuous black line) is approximated with a staircase construction (dotted line) in the simulations. There are four types of nodes: (i) ${\mathbf{x}}_p$, solid nodes that interact with the fluid nodes (solid black circles), (ii) ${\mathbf{x}}_f$, the fluid nodes that interact with the solid nodes (blue filled circles), (iii) solid nodes that don't interact with the fluid nodes (grey filled circles), and (iv) fluid nodes that don't interact with the solid nodes (hollow circles). Lattice Boltzmann populations moving along the links (brown lines) from ${\mathbf{x}}_f$ to ${\mathbf{x}}_p$ (or from ${\mathbf{x}}_p$ to ${\mathbf{x}}_f$) are bounced back at boundary nodes ${\mathbf{x}}_b$ (green squares).

Figure 4

Instantaneous velocity fields of the fluid around the squirmer are obtained from the lattice Boltzmann simulations for ${\theta }_s={45}^{\circ }$ and $d_s/r_s=0.5$. The first row is for a neutral swimmer ($B_1>0$, $B_2=0$): (a) $\alpha =-0.5$, (b) $\alpha =-0.2$, (c) $\alpha =0$, (d) $\alpha =0.2$, and (e) $\alpha =0.5$. The continuous lines are streamlines, and the background color corresponds to the magnitude of the normalized velocity field (red, highest; blue, lowest). The second row is for a shaker ($B_1=0$, $B_2>0$), and for the same $\alpha$ as in first row.

Figure 5

Effect of curvature of a neighboring boundary ($\alpha$) on the instantaneous dynamics of a squirmer. (a) $V_{\parallel }$, (b) $V_{\perp }$ (the components of instantaneous translational velocity parallel and perpendicular to the separation vector), and (c) ${\mathrm{\Omega }}_z$ (the angular velocity) of a neutral swimmer ($B_1>0$, $B_2=0$) are plotted as a function of squirmer orientation. Panels (d)–(f) are similar plots for a shaker ($B_1=0$, $B_2\ne 0$). In all cases, the surface to surface separation is maintained as $d_s=0.2r_s$. Legends of all plots are same as that given in (a). $V_{\parallel }$ and $V_{\perp }$ are normalized by the $U_0$, and ${\mathrm{\Omega }}_z$ is normalized by $r_s{U}_0$, where $U_0=B_1/2$ for the top row and $B_2/2$ for the bottom row.

Figure 6

Effect of curvature of a neighboring boundary ($\alpha$) on the instantaneous dynamics of a squirmer. The components of instantaneous velocity (a) $V_{\parallel }$ and (b) $V_{\perp }$, and (c) the angular velocity ${\mathrm{\Omega }}_z$ of a neutral swimmer are plotted as a function of surface to surface separation distance $d_s$. Panels (d)–(f) are similar plots for a shaker. In all cases, the orientation of the squirmer is chosen as ${\theta }_s={60}^{\circ }$. Legends of all plots are the same and are given in (a). Both $V_{\parallel }$ and $V_{\perp }$ are normalized by $U_0$, and ${\mathrm{\Omega }}_z$ is normalized by $r_s{U}_0$, where $U_0=B_1/2$ for top row and $B_2/2$ for bottom row. The markers in each plot are obtained from lattice Boltzmann simulations for the case of $\alpha =0.33$.

Figure 7

Trajectories of a squirmer near a curved boundary ($\left|\alpha \right|=0.1$). (a) A neutral swimmer and (b) a shaker near a convex boundary with an initial orientation ${\theta }_s^i={45}^{\circ }$ (shown by the black arrow). (c) A neutral swimmer and (d) a shaker near a concave boundary with ${\theta }_s^i={135}^{\circ }$. Here ($x,y$) are normalized by $r_s$. Color bar indicates the orientation of the squirmer with the surface tangent, ${90}^{\circ }-{\theta }_s$. Panels (e)–(f) are altered trajectories corresponding to cases (a)–(d) but when the boundary has a repulsive potential.

Figure 8

The trajectories of a squirmer in a dynamical space ($d_s$, ${\theta }_s$) for various values of size ratio and activity: (a) $\alpha =-0.5$, $\beta =0.1$, (b) $\alpha =-0.5$, $\beta =4$, (c) $\alpha =-0.2$, $\beta =4$, (d) $\alpha =0.2$, $\beta =0.1$, and (e) $\alpha =0.2$, $\beta =4$. Fixed points corresponding to crawling trajectories, marked as thick black dots, with ${\theta }_s<{90}^{\circ }$ and ${\theta }_s>{90}^{\circ }$ are referred to as Type I and Type II, respectively. The regions of existence of Type I and Type II fixed points in $\alpha -\beta$ space is illustrated in (f). Labels A–E indicate different regions which are also colored differently. Note that plots (a)–(e) are enclosed in boxes colored according to the color coding of the regions in (f).

Figure 9

The contour plot of average proximity parameter, $\langle \varphi \rangle$, as a function of curvature ($\alpha$) and activity ($\beta$). (a) In the absence of repulsion ($\delta =0$) and (b) in the presence of repulsion ($\delta$ = 0.05). In these calculations, the swimmer is considered to be in the proximity of the boundary, if the surface to surface separation distance is less than $d_c=0.2r_s$. Note that $d_c$ is an arbitrary choice, and the behavior depicted here is insensitive to the value of $d_c$. (c) The contour plot of average retention time, $\langle t_r\rangle$, as a function of the curvature ($\alpha$) and $\beta$, and in the absence of repulsion ($\delta =0$). Color bar of (a) and (b) is the same as that of (c). For the analysis, we approximate the integral in Eqs. (20) and (21) with summation, and divide the orientational space into 18 equal size intervals.

Figure 10

The contour plot of (a) $\langle {90}^{\circ }-{\theta }_s\rangle$, the average orientation with respect to the surface tangent, and (b) $\langle V_{\perp }\rangle$, the average tangential velocity near the surface are plotted as a function of $\alpha$ and $\beta$. (c) Scattering angle ($S_{\theta }$) of a neutral swimmer ($\beta =0$) as a function of angle made by the orientation vector with the separation vector (${\theta }_s^i$) at time $t=0$ for various size ratios. The inset in (c) shows the definition of the scattering angle $S_{\theta }$.

Figure 11

Instantaneous velocity fields of a squirmer in annular confinement obtained from the lattice Boltzmann simulations for ${\alpha }_o=0.125$, ${\alpha }_i=-1$, and ${\theta }_s={45}^{\circ }$. For $d_n=0.1$, (a) a neutral swimmer ($B_1>0$, $B_2=0$), (b) a shaker ($B_1=0$, $B_2>0$). For $d_n=0.9$, (c) a neutral swimmer, (d) a shaker. Here the continuous lines are streamlines, and the color field corresponds to the magnitude of the velocity. Note that ${\theta }_s$ is defined as an angle made by the orientation vector with the separation vector joining the center of the squirmer and the surface of the concave boundary.

Figure 12

Trajectories of neutral swimmers and shakers in an annular confinement. Effect of (a) ${\alpha }_i$ for ${\alpha }_o=0.1$, ${\theta }_s^i={45}^{\circ }$, (b) initial orientation for ${\alpha }_i=-0.5$, ${\alpha }_o=0.1$, and (c) initial location for ${\theta }_s^i={45}^{\circ }$, ${\alpha }_i=-0.5$, ${\alpha }_o=0.1$ for a neutral swimmer. Panels (d), (e), and (f) are corresponding plots for a shaker. In (a) and (b) and (d) and (e) the squirmer is initialized at an equal distance from both the confining surfaces. Hard repulsion is imposed at a distance $0.05r_s$ on the confining surfaces.

Figure 13

Instantaneous velocities of a squirmer in an annular confinement as a function of eccentricity for an orientation of ${\theta }_s={30}^{\circ }$. (a) Neutral swimmer ($B_1>0$, $B_2=0$) and (b) shaker ($B_1=0$, $B_2\ne 0$). Continuous lines are expressions based on simple superposition [Eqs. (23) and (24)], and markers indicate the results obtained from the numerical simulations.