We rigorously derive from first principles the generic Landau amplitude equation that describes the primary bifurcation in electrically driven convection. Our model accurately represents the experimental system: a weakly conducting, submicron thick liquid crystal film suspended between concentric circular electrodes and driven by an applied voltage between its inner and outer edges. We explicitly calculate the coefficient $g$ of the leading cubic nonlinearity and systematically study its dependence on the system's geometrical and material parameters. The radius ratio $\alpha$ quantifies the film's geometry while a dimensionless group ${ \cal P}$, similar to the Prandtl number, fixes the ratio of the fluid's electrical and viscous relaxation times. Our calculations show that for fixed $\alpha$, $g$ is a decreasing function of ${ \cal P}$, as ${ \cal P}$ becomes smaller, and is nearly constant for ${ \cal P} \gtrsim 1$. As ${ \cal P} \to 0$, $g \to \infty$. We find that $g$ is a nontrivial and discontinuous function of $\alpha$. We show that the discontinuities occur at codimension-two points that are accessed by varying $\alpha$.
Categories: nlin.PS cond-mat.soft physics.flu-dyn