Hydrodynamic theory for nematic shells: The interplay among curvature, flow, and alignment

We derive the hydrodynamic equations for nematic liquid crystals lying on curved substrates. We invoke the Lagrange-Rayleigh variational principle to adapt the Ericksen-Leslie theory to two-dimensional nematics in which a degenerate anchoring of the molecules on the substrate is enforced. The only constitutive assumptions in this scheme concern the free-energy density, given by the two-dimensional Frank potential, and the density of dissipation which is required to satisfy appropriate invariance requirements. The resulting equations of motion couple the velocity field, the director alignment, and the curvature of the shell. To illustrate our findings, we consider the effect of a simple shear flow on the alignment of a nematic lying on a cylindrical shell.

Figure 1

Darboux frame $\{\mathbf{n},\mathbf{\tau },\mathbf{\nu }\}$ on the nematic shell $\mathcal{S}$. The vector $\mathbf{v}$ is the velocity of the center of mass of the molecule and $\theta$ is the angle contained by the director and the dashed line of curvature.

Figure 2

Alignment angles as functions of the Leslie viscosity parameters ${\alpha }_2$ and ${\alpha }_3$ for cylindrical and planar geometries. (a) refers to the case ${\alpha }_3>0$, (b) to the case ${\alpha }_3<0$, and (c) to ${\alpha }_3=0$. The solid lines represent linearly stable steady solutions, and the dashed lines unstable alignments.