Weakly Nonlinear Analysis of Electroconvection in a Suspended Fluid Film

It has been experimentally observed that weakly conducting suspended lms of smectic liquid crystals undergo electroconvection when subjected to a large enough potential di erence. The resulting counter-rotating vortices form a very simple convection pattern and exhibit a variety of interesting nonlinear e ects. The linear stability problem for this system has recently been solved. The convection mechanism, which involves charge separation at the free surfaces of the lm, is applicable to any su ciently two-dimensional uid. 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 coe cients of the amplitude equation are calculated and compared with the values independently deduced from the linear stability calculation. Typeset using REVTEX 1