Hydrodynamic interactions in polar liquid crystals on evolving surfaces

We consider the derivation and numerical solution of the flow of passive and active polar liquid crystals, whose molecular orientation is subjected to a tangential anchoring on an evolving curved surface. The underlying passive model is a simplified surface Ericksen-Leslie model, which is derived as a thin-film limit of the corresponding three-dimensional equations with appropriate boundary conditions. A finite element discretization is considered and the effect of hydrodynamics on the interplay of topology, geometric properties, and defect dynamics is studied for this model on various stationary and evolving surfaces. Additionally, we consider an active model. We propose a surface formulation for an active polar viscous gel and exemplarily demonstrate the effect of the underlying curvature on the location of topological defects on a torus.

Figure 1

Top: Evolution of the director field $\widehat{\mathbf{p}}$ on a stationary ellipsoid of the dry case (top row) and the wet case (bottom row) for $t=1,2,3,4,6$ (left to right). Bottom: Surface Frank-Oseen energy ${\mathcal{F}}^{\mathbf{P}}$ and surface kinetic energy ${\mathcal{F}}^{\mathrm{kin}}$ vs time $t$.

Figure 2

Top: Evolution of the director field $\widehat{\mathbf{p}}$ on a torus of the dry case (top row) and the wet case (bottom row) for $t=0.3,4.5,6.6$ (left to right). Middle: Surface Frank-Oseen energy ${\mathcal{F}}^{\mathbf{P}}$ and surface kinetic energy ${\mathcal{F}}^{\mathrm{kin}}$ vs time $t$. Bottom: Velocity field $\widehat{\mathbf{v}}$ for the annihilation of a source ($+1$, left) and a saddle ($-1$, right) defect in the director field $\widehat{\mathbf{p}}$ (red dots) for $t=3,3.75,3.81,4.05,4.5,5.7$ (left to right and top to bottom).

Figure 3

Top: Schematic defect positions of the initial condition (left) and the final configuration (right) on a torus with the analytical initial condition for the director field $\widehat{\mathbf{p}}$ and zero initial condition for the velocity field $\widehat{\mathbf{v}}$. Red dots indicate $+1$ defects (sources or sinks) and blue dots indicate $-1$ defects (saddle). Bottom: Evolution of the director field $\widehat{\mathbf{p}}$ for $t=1,5,25$ (left to right).

Figure 4

Surface Frank-Oseen energy ${\mathcal{F}}^{\mathbf{P}}$ and surface kinetic energy ${\mathcal{F}}^{\mathrm{kin}}$ vs time $t$ for the analytical initial condition for the director field $\widehat{\mathbf{p}}$ and the killing vector field as initial condition for the velocity $\widehat{\mathbf{v}}$.

Figure 5

Left: Schematic description of the ellipsoid evolution for a half period of oscillation. Descending gray scale indicates increasing time. The motion in the second half of the oscillation is reversed, respectively. Right: Major axes parameters for the ellipsoid over a full period of oscillation. The time of one oscillation period is considered to be $T=160$. The major axes parameters are chosen such that the surface area of the ellipsoid is conserved over time.

Figure 6

Top: Evolution of the director field $\widehat{\mathbf{p}}$ on an evolving ellipsoid of the dry case (top row) and the wet case (bottom row) for $t=3,40,95,145,160$ (left to right). Middle: Surface Frank-Oseen energy ${\mathcal{F}}^{\mathbf{P}}$ and surface kinetic energy ${\mathcal{F}}^{\mathrm{kin}}$ vs time $t$ for the first period of oscillation. Bottom: Surface Frank-Oseen energy ${\mathcal{F}}^{\mathbf{P}}$ and surface kinetic energy ${\mathcal{F}}^{\mathrm{kin}}$ vs time $t$ over five periods of oscillation.

Figure 7

Surface Frank-Oseen energy ${\mathcal{F}}^{\mathbf{P}}$ and surface kinetic energy ${\mathcal{F}}^{\mathrm{kin}}$ vs time $t$ for the simulation with the damped oscillation of the defects around the minimal defect configuration.

Figure 8

Top: Snapshot of the director field $\widehat{\mathbf{p}}$ of the surface active polar viscous gel model on the torus for $t=14.3$. Bottom: Number of defects per area for the inner ($\kappa <0$) and the outer ($\kappa >0$) region of the torus vs time $t$.