Charge transport scaling in turbulent electroconvection

We describe a local-power-law scaling theory for the mean dimensionless electric current Nu in turbulent electroconvection. The experimental system consists of a weakly conducting, submicron-thick liquid-crystal film supported in the annulus between concentric circular electrodes. It is driven into electroconvection by an applied voltage between its inner and outer edges. At sufficiently large voltage differences, the flow is unsteady and electric charge is turbulently transported between the electrodes. Our theoretical development, which closely parallels the Grossmann-Lohse model for turbulent thermal convection, predicts the local-power law $\mathrm{Nu}\sim F\left(\Gamma \right){\mathcal{R}}^{\gamma }{\mathcal{P}}^{\delta }$. $\mathcal{R}$ and $\mathcal{P}$ are dimensionless numbers that are similar to the Rayleigh and Prandtl numbers of thermal convection, respectively. The dimensionless function $F\left(\Gamma \right)$, which is specified by the model, describes the dependence of Nu on the aspect ratio $\Gamma$. We find that measurements of Nu are consistent with the theoretical model.

Figure 1

Schematics of the experiment: (a) top and (b) side.

Figure 2

(Color online) Qualitative visualization of electroconvective flow, characterized by a film of uneven thickness: at (a) $V=100\mathrm{V}$ and (b) $V=250\mathrm{V}$. Data, however, were acquired only from uniformly thick films in which no flow is visible.

Figure 3

A representative current-voltage curve for an annular film showing the onset of electroconvection. Data obtained for increasing (decreasing) voltages are shown in ▵(▿). The inset shows the rms fluctuations of the current vs the applied voltage. Here $\Gamma =3.74\pm 0.02$ and $\mathcal{P}=23\pm 2$.

Figure 4

(Color online) (a) shows representative plots of Nu vs $\mathcal{R}$ for $\Gamma =0.33\pm 0.01$ (▵), $3.74\pm 0.02$ (◻), and $6.60\pm 0.02$ (엯), analyzed by method A. For these, $\mathcal{P}=8.8\pm 0.5$, $21\pm 1$, and $36\pm 1$, respectively. Least-square fits to the power law $\mathrm{Nu}\sim {\mathcal{R}}^{\gamma }$ give best-fit values $\gamma =0.19\pm 0.01$, $0.21\pm 0.02$, and $0.26\pm 0.01$, respectively. The solid reference lines in the figure have slopes of $1∕5$ (lower one) and $1∕4$ (upper one). In (b) we plot the compensated scaling $\mathrm{Nu}{\mathcal{R}}^{-\gamma }$ vs $\mathcal{R}$ for the same data as in (a). The inset in (b) shows a more expanded scale for $\Gamma =0.33$ (▵) and 3.74 (◻).

Figure 5

(Color online) Plots of averaged $\mathrm{Nu}∕{\mathcal{R}}^{\gamma }$ vs $\mathcal{P}$ for $\Gamma =6.60\pm 0.05$ and $11.1\pm 0.1$, where $\gamma$ is taken from the best fit to $\mathrm{Nu}\sim {\mathcal{R}}^{\gamma }$. Circular symbols (엯, ●) show results obtained by analysis method A, while (∗) symbols are obtained by method B. Solid (●) and open (엯) symbols indicate when the scaling exponent $\gamma \sim 1∕4(1∕5)$. Note that the crossover between $1∕4$ and $1∕5$ exponents occurs at different $\mathcal{P}$ for different aspect ratios $\Gamma$, as indicated by the shaded rectangles.

Figure 6

(Color online) The average value of fully compensated $\mathrm{Nu}∕\left[F\left(\Gamma \right){\mathcal{R}}^{\gamma }\right]$ vs $\mathcal{P}$ on a log-log scale for various $\Gamma$. Circular symbols (엯) show results obtained by analysis method A, while (∗) symbols are obtained by method B. All of these data had a Nu vs $\mathcal{R}$ scaling exponent of $\gamma \sim 1∕5$. The solid (dashed) line is the best fit to $\sim {\mathcal{P}}^{\beta }$ with $\beta =0.20\left(0.26\right)$, using analysis method A (B).

Figure 7

Plot of $kF\left(\Gamma \right)$ vs $\Gamma$ from the scaling theory (solid line), with $k$ chosen so that $kF(\Gamma =1)$ is unity. Experimental data for turbulent electroconvection are shown by solid symbols (●) for method A and by (∗) for method B. The data span $5⩽\mathcal{P}⩽250$. The inset shows the same data on a logarithmic scale.