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 $\mathcal{R},$ analogous to the Rayleigh number, which is $\propto V^2$ and the Reynolds number Re of the azimuthal shear flow. The geometrical and material properties of the film were characterized by the radius ratio $\alpha ,$ and a dimensionless number P, analogous to the Prandtl number. Using measurements of current-voltage characteristics of a large number of films, we examined the onset of electroconvection over a broad range of $\alpha ,$ P, and Re. 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 Re.