Electroconvection in a dielectric liquid between two concentric half-cylinders with rigid walls: Linear and nonlinear analysis

Abstract

We study the linear stability and nonlinear behavior of the electroconvection between two concentric halfcylinders with no-slip conditions on all boundaries. The no-slip condition makes impossible to apply the standard modal approach. Hence, we apply a finite element technique similar to the one we have used in a previous paper about the electroconvection in a rectangular enclosed domain. When compared to the classical problem of electroconvection between two full concentric cylinders, the linear criterion is higher, due to the viscous shear introduced by the lateral walls. As a consequence, the structure of the eigenmodes is very different. There is a repulsion between modeswith the same symmetry, forcing pairs of modes to cross each other repeatedly. For inner injection and small value of the ratio between the inner and outer radii the no-slip condition changes the nature of the bifurcation from subcritical to supercritical, while it is always subcritical for outer injection. To understand this behavior, we perform a modal analysis using the eigenfunctions obtained from the linear stability analysis as modal basis. We show that the supercritical branch is originated by the nonorthogonality of the modes when no-slip boundary conditions are imposed. These mechanism explains the previously unexplained appearance of the supercritical branch in the enclosed rectangular configuration.