Weakly nonlinear analysis of electroconvection in a suspended fluid film

Abstract

It has been experimentally observed that weakly conducting suspended films of smectic liquid crystals undergo electroconvection when subjected to a large enough potential difference. The resulting counter-rotating vortices form a very simple convection pattern and exhibit a variety of interesting nonlinear effects. The linear stability problem for this system has recently been solved. The convection mechanism, which involves charge separation at the free surfaces of the film, is applicable to any sufficiently two-dimensional fluid.In this paper, we derive an amplitude equation which describes the weakly nonlinear regime, by starting from the basic electrohydrodynamic equations. This regime has been the subject of several recent experimental studies. The lowest order amplitude equation we derive is of the Ginzburg-Landau form, and describes a forward bifurcation as is observed experimentally. The coefficients of the amplitude equation are calculated and compared with the values independently deduced from the linear stability calculation.

Bifurcations in annular electroconvection with an imposed shear

“…We find that g can become slightly negative for small α, so that the bifurcation is weakly backward. The result for g in the limit Re = 0, α → 1 is compared to the theoretical result for rectangular films [25]. Finally, in Section IV D we describe the results for g in sheared films, where Re > 0.…”

“…The reduced Nusselt number n is related to the electric Nusselt number N by n = N − 1. One can show [25,5] that the amplitude can be scaled so that n = |A m | 2 . To solve for the real and imaginary parts of the amplitude equation, let A m (t) = A(t)e iΦ(t) , where A(t) and Φ(t) are real.…”

Electrically driven convection in a thin annular film undergoing circular Couette flow

“…Surprizingly, this remarkably ideal situation is experimentally realizeable in the electroconvection of thin, freely suspended liquid crystal films. [1] This system, which can be accurately described by electrohydrodynamic theory [2,3], presents a unique opportunity to quantitatively study convection with a novel combination of forcing and shear. In this paper, our treatment is both theoretical and experimental; we present a complete linear stability analysis and use it to make the first quantitative comparisons with experiment.…”

“…For example, the standard case of buoyancy-driven Rayleigh-Bénard convection (RBC) has been studied in the presence of rotation [7,8,9,10] and shearing due to an open throughflow [11,12], as well as in the geophysically interesting cases of radial gravitation with rotation [5,6]. The phenomenology of two-dimensional (2D) electroconvection in a rectangular geometry was the subject of our previous experimental [13,14,15,16,17] and theoretical [2,3] work. These studies made precise the degree of analogy with RBC; the electrical nature of the forcing introduces some crucial differences in detail, even at the level of linear stability.…”

“…If a voltage is applied across a conducting fluid film, the free surfaces acquire a potentially unstable distribution of surface charge. [2,3] This charge must be present simply to satisfy the standard matching conditions on electric fields across an interface. We do not consider charge generation by electrode injection or other bulk effects.…”