Direct numerical simulation of supercritical annular electroconvection

We use direct numerical simulation to study electrically driven convection in an annular thin film. The simulation models a laboratory experiment that consists of a weakly conducting, submicron thick liquid crystal film suspended between two concentric electrodes. The film is driven to convect by imposing a sufficiently large voltage across it. The flow is driven by a surface charge density inversion which is unstable to the imposed electrical force. This mechanism is closely analogous to the mass density inversion which is unstable to the buoyancy force in conventional, thermally driven Rayleigh-Bénard convection. The simulation uses a pseudospectral method with Chebyshev polynomials in the radial direction and Fourier modes in the azimuthal direction. The numerical results, which are in good agreement with previous experimental data and theoretical predictions, reveal several insights. The mode competition near a codimension-two point exhibits hysteresis. The primary bifurcation is supercritical for a broad range of fluid and geometrical parameters.

Figure 1

Schematics of the annular electroconvection experiment: (a) Top view and (b) side view.

Figure 2

(Color online) Representative numerical data for the dimensionless convective current, $\text{Nu}-1$, as the Rayleigh-like number $\mathcal{R}$ changes. Here the other parameters are $\alpha =0.56$ and $\mathcal{P}=10$. Data obtained for increasing (decreasing) $\mathcal{R}$ are shown as $▵$ $(▿)$. (b) The corresponding amplitude of convection $A=\sqrt{\text{Nu}-1}$ as a function of $\mathcal{R}$. The solid line is a nonlinear least-squares fit of the data to the Landau amplitude equation given by Eq. (32).

Figure 3

(Color) The basic fields for steady convection at $\mathcal{R}=199.8$, $\alpha =0.56$, and $\mathcal{P}=10$: The total electric charge density $q$ (color) and the velocity field (black). Only the upper half of the annular cell is shown. Positive charge moves away from the high potential electrode at the inner radius, while negative charge moves away from the grounded outer electrode.

Figure 4

(Color) (a) The streamfunction and (b) the perturbed 2D electric potential ${\psi }_2^{\left(1\right)}$ for the same control parameters as in Fig. 3. For this specific $\alpha$, the state is dominated by the $m=7$ Fourier mode and there are seven counter-rotating vortex pairs.

Figure 5

(Color online) (a) The radius ratio dependence of the critical Rayleigh number ${\mathcal{R}}_c$. The solid circles (●) are the results of the numerical simulation. Open and filled boxes $(◻,∎)$ are the theoretical predictions of nonlocal linear stability analysis, using third order and sixth order expansions in the radial direction, respectively. (b) The critical number of counter-rotating vortex pairs $m_c$. All the simulation data (●) are for a fixed $\mathcal{P}=10$. The solid lines are the predictions of fully nonlocal linear stability theory and $◇$ are experimental results from Ref. [6].

Figure 6

(Color online) The radius ratio dependence of the coefficient of the cubic nonlinearity $g$. Numerical data (●) for $\mathcal{P}=10$ with various radius ratios $\alpha$ show a continuous and supercritical bifurcation $\left(g>0\right)$. They agree well with a nonlocal theory prediction for $\mathcal{P}=123$ shown by red filled squares $(∎)$. Black filled diamonds $(◆)$ are experimental results, from Ref. [6], for various $\mathcal{P}>1$.

Figure 7

(Color online) Simulation data of the cubic nonlinearity coefficient $g$ over a wide range of $\mathcal{P}$ for a fixed $\alpha =0.33$. The numerical data show supercritical bifurcations $g>0$ for various $\mathcal{P}$ and the $\mathcal{P}$ independence of $g$ for $\mathcal{P}>0.1$.

Figure 8

(Color online) Mode competition close to the CoD2 point at $\alpha =0.452$ and $\mathcal{R}=87$. (a) The streamfunction of a transient mixed-mode state with $m=5$ and $m=6$ components. This state corresponds to the $Z_2$ symmetric solution predicted in Ref. [17]. (b) The early time growth of the Fourier amplitudes of modes $m=5$ and $m=6$, starting from an initial condition of equally large amplitudes for both modes. The growth rates of ${\widehat{\psi }}_{2m}$ are both $\approx 0.05$, for early times $\lesssim 90{\tau }_c$. (c) The full time evolution of the amplitudes of the two modes, showing that mode competition eventually results in one mode suppressing the other.

Figure 9

(Color online) Perturbed 2D electric potential ${\psi }_2$ near a codimension-two point (CoD2) at $\alpha =0.452$. The Rayleigh-like numbers were $\mathcal{R}=87.006$ for $m=5$ (left) and $\mathcal{R}=87.859$ for $m=6$ (right). Both patterns used different random white noise as the initial condition.