Swimming in an anisotropic fluid: How speed depends on alignment angle

Orientational order in a fluid affects the swimming behavior of flagellated microorganisms. For example, bacteria tend to swim along the director in lyotropic nematic liquid crystals. To better understand how anisotropy affects propulsion, we study the problem of a sheet supporting small-amplitude traveling waves, also known as the Taylor swimmer, in a nematic liquid crystal. For the case of weak anchoring of the nematic director at the swimmer surface and in the limit of a minimally anisotropic model, we calculate the swimming speed as a function of the angle between the swimmer and the nematic director. The effect of the anisotropy can be to increase or decrease the swimming speed, depending on the angle of alignment. We also show that elastic torque dominates the viscous torque for small-amplitude waves and that the torque tends to align the swimmer along the local director.

Figure 1

Taylor swimmer with wave vector $q$, amplitude $b$, and wave speed $c$ in a two-dimensional liquid crystal. The local tangent vector of the swimmer makes an angle ${\varphi }_{\mathrm{s}}$ with the $x$ axis and $\widehat{\mathbf{N}}$ is the outward pointing normal for the region on the side of the swimmer with $y>0$. The director $\mathbf{n}$ makes an angle $\theta$ with the $x$ axis and the director has a fixed angle $\varphi$ for large $\left|y\right|$.

Figure 2

Dimensionless change in swimming speed as a function of the angle $\varphi$ between the swimmer and the nematic direction, for $\lambda =0$ and various values of the dimensionless penetration depth $q{\ell }_{\xi }$: $q{\ell }_{\xi }=0.5$ (blue, bottom curve at $\varphi =\pi /2),q{\ell }_{\xi }=1$ (red, middle curve), and $q{\ell }_{\xi }=10$ (green, top curve at $\varphi =\pi /2)$. The $q{\ell }_{\xi }=10$ curve is indistinguishable from the $q{\ell }_{\xi }=\infty$ curve.

Figure 3

Dimensionless change in swimming speed as a function of the angle $\varphi$ between the swimmer and the nematic direction, for $\lambda =1$ and various values of the dimensionless penetration depth $q{\ell }_{\xi }$: $q{\ell }_{\xi }=0.5$ (blue, bottom curve at $\varphi =\pi /2),q{\ell }_{\xi }=1$ (red, middle curve), and $q{\ell }_{\xi }=2$ (green, top curve at $\varphi =\pi /2)$.

Figure 4

Dimensionless correction to swimming speed as a function of alignment angle for weak anisotropic viscosity in the transversely isotropic fluid model.