Dynamics of a simple model microswimmer in an anisotropic fluid: Implications for alignment behavior and active transport in a nematic liquid crystal

Several recent experiments investigate the orientational and transport behavior of self-driven bacteria and colloidal particles in nematic liquid crystals. Correspondingly, we study theoretically the dynamics of a minimal model microswimmer in a uniaxially anisotropic fluid. As a first step, the hydrodynamic Green's function providing the resulting fluid flow in response to a localized force acting on the anisotropic fluid is derived analytically. On this basis, the behavior of both puller- and pusher-type microswimmers in the anisotropic fluid is analyzed. Depending on the propulsion mechanism and the relative magnitude of different involved viscosities, we find alignment of the swimmers parallel or perpendicular to the anisotropy axis. Particularly, also an oblique alignment is identified under certain circumstances. The observed swimmer reorientation results from the hydrodynamic coupling between the self-induced fluid flow and the anisotropy of the surrounding fluid, which distorts the self-generated flow field. We support parts of our results by a simplified linear stability analysis. Our theoretical predictions are in qualitative agreement with recent experimental observations on swimming bacteria in nematic liquid crystals. They support the objective of utilizing the possibly switchable anisotropy of a host fluid to guide individual microswimmers and active particles along a requested path, enabling controlled active transport.

Figure 1

Illustration of the active model microswimmer. The swimmer is modeled as a sphere of hydrodynamic radius $a$ with no-slip surface condition, subject to hydrodynamic drag. Two point force centers exert active forces $+\mathit{f}$ and $-\mathit{f}$ separated a distance $L$ from each other. They are asymmetrically disposed with respect to the swimmer body. Then, the resulting self-generated flow field leads to propulsion by transporting the sphere. The positions of the force centers relative to the swimmer body are set by the parameter $\alpha$ that takes values between 0 and 1/2. The unit vector $\widehat{\mathit{t}}$ points toward the nearer force center, along the swimmer axis. A quiver plot of the self-induced velocity field in the surrounding fluid is schematically shown for the upper half of the fluid domain in the isotropic limit for ${\nu }_1={\nu }_2={\nu }_3$. The microswimmer displayed here is referred to as a pusher because it pushes out the fluid along the swimming axis. By directing the two point forces toward the spherical body, the swimmer is known as a puller because it pulls the fluid inward along the swimming path.

Figure 2

Swimming trajectories in the $(x,z)$ plane for $\left(\text{a}\right)$ pusher- and $\left(\text{b}\right)$ puller-type swimmers for a vertical director orientation along the $z$ axis and various values of the viscosity ratio $E={\nu }_2/{\nu }_3$. The swimmer is released from the origin with an initial inclination ${\psi }_0=\pi /6$. Arrows indicate the direction of time evolution. Here, $\alpha =2/5$ and $\overline{\nu }=0$.

Figure 3

Quiver plots of the velocity field induced by a pusher- [panels (a), (b), and (c)] and a puller-type [panels (d), (e), and (f)] microswimmer in a uniaxially anisotropic fluid, e.g., an aligned nematic LC, for various values of the viscosity ratio $E={\nu }_2/{\nu }_3$, while $\overline{\nu }=0$. The dash-dotted lines connecting the force centers are plotted to indicate the orientation of the swimmer. Here, we set an inclination $\psi =\pi /6$ and a swimmer asymmetry $\alpha =2/5$. The color bars show the magnitude of the flow velocity scaled by $f/\left({\nu }_{3}L\right)$. Spherical swimmer bodies are not shown here for clarity. In the isotropic case ($E=1$, center subfigures), the flow field is symmetric relatively to the swimmer axis. Therefore, no reorientation occurs. For $E>1$ (left subfigures), the flow is significantly more pronounced along the director axis (vertical). This results in asymmetric flow fields relatively to the swimmer axis and effective shear flows around the swimmer body. In panel (a), for a pusher, this leads to a counterclockwise and in panel (d), for a puller, to a clockwise rotation of the swimmer. Vice versa, for $E<1$ (right subfigures), the flow perpendicular to the director is more pronounced. This implies opposite asymmetry and then opposite sense of rotation.

Figure 4

Variation of the inclination angle versus the scaled horizontal distance for a pusher- [panels (a) and (b)] and a puller-type [panels (c) and (d)] swimmer released from the origin with an initial inclination of ${\psi }_0=\pi /3>{\psi }_A$ [panels (a) and (c)] and ${\psi }_0=\pi /6<{\psi }_A$ [panels (b) and (d)] for $\phi =0$ and various values of the viscosity ratio $A$, while $E=1$. Here, we set $\alpha =2/5$. Depending on the propulsion mechanism and on the values of $A$ and ${\psi }_0$, the swimmer tends to align parallel or perpendicular to the director, or tends to swim along the axis given by ${\psi }_A\simeq 0.24\pi$ indicated by the dashed lines. Arrows indicate the direction of time evolution. When the steady alignment parallel to the director is reached $(\psi =\pi /2)$, the swimmer moves along the $z$ axis.

Figure 5

State diagram of orientational swimming behavior in a uniaxially anisotropic fluid for a pusher [panels (a) and (b)] and a puller [panels (c) and (d)] as obtained by numerically integrating the full nonlinear equations. The swimmer is released from the origin with an initial inclination of ${\psi }_0=\pi /3$ [panels (a) and (c)] and ${\psi }_0=\pi /6$ [panels (b) and (d)] for $\phi =0$. The solid and dashed lines shown around the point $(A=0,E=1)$ are defined in Eq. (47). On the bottom right of each panel, the missing data points would fall into the unphysical regime of $A<A_{\mathrm{T}}=E-4$ [90].

Figure 6

Variation of the rescaled self-mobility function for the motion (a) parallel and (b) perpendicular to the director versus the viscosity ratios. Solid lines are the theoretical predictions given by Eqs. (C8) and (C13), respectively, while the dashed line displayed in panel (a) is the asymptotic result given by Eq. (C11) for $\eta ={\nu }_3$. Symbols give the boundary integral simulation results for two different mesh resolutions. Here we set $A=1/2$ in panel (b).

Figure 7

Variation of the rescaled pair-mobility function for two particles aligned with their connecting vector $\mathit{h}$ parallel or perpendicular to the director $\widehat{\mathit{n}}\parallel \widehat{\mathit{z}}$ for (a) the $zz$ and (b) the $xx$ components. Lines are the analytical predictions and symbols are the boundary integral simulation results obtained using 320 triangles (squares) and 1280 triangles (circles). Here we set $E=1/10$ and $A=1/2$. Insets: Illustration of typical configurations of a pair of particles disposed parallel or perpendicular to the director. The solid arrows indicate the force applied to the particle located at the origin, while the dotted arrows indicate the velocity of the particle located at distance $h$.