  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 ${\cal R}e$.
