Bifurcations in annular electroconvection with an imposed shear

We report an experimental study of the primary bifurcation in electrically-driven convection in a freely suspended film. A weakly conducting, submicron thick smectic liquid crystal film was supported by concentric circular electrodes. It electroconvected when a sufficiently large voltage $V$ was applied between its inner and outer edges. The film could sustain rapid flows and yet remain strictly two-dimensional. By rotation of the inner electrode, a circular Couette shear could be independently imposed. The control parameters were a dimensionless number ${\cal R}$, analogous to the Rayleigh number, which is $\propto V^2$ and the Reynolds number ${\cal R}e$ of the azimuthal shear flow. The geometrical and material properties of the film were characterized by the radius ratio $\alpha$, and a Prandtl-like number ${\cal P}$. Using measurements of current-voltage characteristics of a large number of films, we examined the onset of electroconvection over a broad range of $\alpha$, ${\cal P}$ and ${\cal R}e$. We compared this data quantitatively to the results of linear stability theory. This could be done with essentially no adjustable parameters. The current-voltage data above onset were then used to infer the amplitude of electroconvection in the weakly nonlinear regime by fitting them to a steady-state amplitude equation of the Landau form. We show how the primary bifurcation can be tuned between supercritical and subcritical by changing $\alpha$ and ${\cal R}e$.