Nanofluidics of nematic liquid crystals in hollow capillaries

The aim of this paper is to investigate the response of a homogeneously aligned nematic nanosized hollow cavity (HANNHC) confined between two charged horizontal coaxial cylinders and subjected to both a radially applied electrostatic field $\mathbf{\text{E}}$, arising from the surface charge density $\kappa$ and the temperature gradient $\mathbf{\nabla }T$ set between these cylinders. This was done within the framework of an extension of the classical Ericksen-Leslie theory, supplemented by thermomechanical correction of the shear stress and Rayleigh dissipation function, as well as taking into account the entropy balance equation. The physical mechanism responsible for the excitation of the hydrodynamic flow in the HANNHC is based on the interaction of the director and temperature gradients and the static electric field. Calculations show that under the influence of both the $\mathbf{\nabla }T$ and $\mathbf{\text{E}}$, a stationary flow $u^{\mathrm{st}}$ is excited in the HANNHC in the horizontal direction. It is shown that the electric force enforced by the flexoelectric polarization plays a crucial role in the excitation of $u^{\mathrm{st}}$ between these cylinders.

Figure 1

The coordinate system specifying the orientation of director field $\widehat{\mathbf{\text{n}}}$.

Figure 2

Distance dependence $r-r_1$ of the electrostatic field $E(r-r_1)$, calculated using Eq. (3). As explained in the text, four values of the dimensionless cavity thickness $D=d/{\lambda }_D=2.0$ (curve 1), 4.0 (curve 2), 8.0 (curve 3), and 16.0 (curve 4), have been used throughout.

Figure 3

Plot of the stationary radial distribution of the polar angle ${\theta }^{\mathrm{st}}(r-r_1)$ across the dimensionless HAN cavity $r_1\le r\le r_2$, under the effect of the electric field $E_0$, caused by the surface density $\kappa =4\times {10}^{-3}$ C/${\mathrm{m}}^2$, and the temperature difference $\mathrm{\Delta }\chi =0.0162$, both for the cases of $\mathbf{\text{P}}=0$ (a) and $\mathbf{\text{P}}\ne 0$ (b), respectively. As explained in the text, four values of the dimensionless cavity thickness $D=d/{\lambda }_D=2.0$ (curve 1), 4.0 (curve 2), 8.0 (curve 3), and 16.0 (curve 4), have been used throughout.

Figure 4

Plot of the stationary radial distribution of the polar angle ${\theta }^{\mathrm{st}}(r-r_1)$ across the dimensionless HAN cavity $r_1\le r\le r_2$, under the effect of the electric field $E_0$, caused by the surface density $\kappa =4\times {10}^{-3}$ C/${\mathrm{m}}^2$, both for the cases of $\mathbf{\text{P}}=0$ (a) and $\mathbf{\text{P}}\ne 0$ (b), respectively. Here the temperature difference $\mathrm{\Delta }\chi$ is equal to 0. [(c) and (d)] Same as described in the cases (a) and (b), but with accounting the temperature difference $\mathrm{\Delta }\chi =0.0162$. As explained in the text, four values of the dimensionless cavity thickness $D=d/{\lambda }_D=2.0$ (curve 1), 4.0 (curve 2), 8.0 (curve 3), and 16.0 (curve 4), have been used throughout.

Figure 5

Same as described in the caption of Fig. 3, but a plot of the stationary radial distribution of the velocity component $u^{\mathrm{st}}(r-r_1)$ across the dimensionless HAN cavity $r_1\le r\le r_2$.

Figure 6

Plot of the stationary radial distribution of the velocity component $u^{\mathrm{st}}(r-r_1)$ across the dimensionless HAN cavity $r_1\le r\le r_2$, under the effect of the electric field $E_0$, caused by the surface density $\kappa =4\times {10}^{-3}$ C/${\mathrm{m}}^2$, both for the cases of $\mathbf{\text{P}}=0$ (a) and $\mathbf{\text{P}}\ne 0$ (b), respectively. Here the temperature difference $\mathrm{\Delta }\chi =0.0$. As explained in the text, four values of the dimensionless cavity thickness $D=d/{\lambda }_D=2.0$ (curve 1), 4.0 (curve 2), 8.0 (curve 3), and 16.0 (curve 4), have been used throughout.