Spontaneous helical flows in active nematics lying on a cylindrical surface

Within the framework of the two-dimensional Ericksen-Leslie model, we explore the effect of geometric confinement on the spontaneous flow of active nematic gels. The nematic particles are assumed to flow on a cylindrical surface, while a degenerate tangential anchoring is enforced. Using the linear approximation of the motion equations, we show that there is a close interplay among extrinsic curvature, flow, director alignment, and activity. We find that the extrinsic curvature promotes the director alignment parallel to the cylindrical axis and is responsible for raising the critical threshold with respect to the flat case. Our analysis reveals a very rich scenario where the key quantities are the activity coefficient, the tumbling parameter, and the anisotropic viscosity ratio. Thus, solutions can exhibit a double periodicity in both the azimuthal and axial variables. As a consequence, the velocity field can make a finite angle with the cylinder axis and the active flow winds on the surface with a helical pattern, while the director oscillates around the cylinder generators. Our results can be validated on thin layers of nematic gels placed between two concentric cylinders and suggest which material properties are most suited for the design of active microfluidic devices.

Figure 1

Schematic representation of the system geometry, where $\mathbf{n}$ represents the nematic director and $\mathbf{v}$ the tangential flow. The figure shows also the Darboux frame $\{\mathbf{n},\mathit{\tau },\mathit{\nu }\}$ and the principal frame $\{{\mathbf{e}}_{\theta },{\mathbf{e}}_z,\mathit{\nu }\equiv {\mathbf{e}}_r\}$.

Figure 2

The control plane $(\lambda ,\gamma )$ divided into five fundamental regions $(n=1)$. The gray area represents the accessible part of the control space, where the (passive) dissipation function is positively defined. Each subregion shows analogous qualitative features. The insets show the critical activity as a function of the wave number: ${\zeta }_c={\zeta }_c\left(q\right)$. The minimum value of $|{\zeta }_c|$, when corresponding to $q\ne 0$, determines the pitch and the angle of the helical pattern at the transition. Both first- and second-order transitions can be observed, depending on the values of $\lambda$ and $\gamma$. The blue boundary lines are given in Eqs. (13) and (14).

Figure 3

Typical helical flow with $n=1, \lambda =0.02, \gamma =-1, \rho k/{\alpha }_4^2=0.2$. The amplitudes of velocity and director fields are arbitrary since they remain undetermined in the linear approximation (11). The minimal critical activity and the corresponding helical pitch, as deduced from the critical points of Eq. (10), are $R^2{\zeta }_c/k\approx -8.87, \xi \approx 0.468$.