Codimension-two points in annular electroconvection as a function of aspect ratio

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 number $\mathcal{P}$, 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 $\mathcal{P}$, as $\mathcal{P}$ becomes smaller, and is nearly constant for $\mathcal{P}\gtrsim 1$. As $\mathcal{P}\to 0$, $g\to \infty$. 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 $\alpha$.

Figure 1

The coordinate system and film geometry.

Figure 2

(Color online) (a) A representative current-voltage characteristic. The dashed line is a linear fit to the current-voltage data for $V<V_c^0$, indicating the region of ohmic response. At the onset of electroconvection, the data depart from the ohmic response. Part (b) shows the amplitude-$ϵ$ curve (◻) for the same data as part (a), as well as the fit to the Landau amplitude equation (solid line). In this case, we have $\alpha =0.47$, $\mathcal{P}=20.4$, and best-fit parameters $g=1.9\pm 0.1$, $h=5.1\pm 0.2$, and $f=0.008\pm 0.002$.

Figure 3

$g$ versus $\alpha$ for $\mathcal{P}=123$.

Figure 4

The fractional discontinuity in $g$, $2\Delta g∕(g_{m+1}+g_m)$, versus the fractional change in the critical mode number $m=m_c^0$, $2\Delta m∕(2m+1)$, at CoD2 points with $\mathcal{P}=123$.

Figure 5

(Color online) The coefficient $g$ versus $\mathcal{P}$ at $\alpha =0.674$ (solid line) and $\alpha =0.676$ (dashed line). The inset shows $\mid dg∕d\mathcal{P}\mid$ versus $\mathcal{P}$ for the same $\alpha$.