Analytical and numerical solutions to an electrohydrodynamic stability problem

Keywords: Linear hydrodynamic stability Bifurcation manifolds High order eigenvalue problems Hinged boundary conditions Direct analytical methods Fourier type methods Spectral methods a b s t r a c t A linear hydrodynamic stability problem corresponding to an electrohydrodynamic con-vection between two parallel walls is considered. The problem is an eighth order eigen-value one supplied with hinged boundary conditions for the even derivatives up to sixth order. It is first solved by a direct analytical method. By variational arguments it is shown that its smallest eigenvalue is real and positive. The problem is cast into a second order differential system supplied only with Dirichlet boundary conditions. Then, two classes of methods are used to solve this formulation of the problem, namely, analytical methods (based on series of Chandrasekar–Galerkin type and of Budiansky–DiPrima type) and spectral methods (tau, Galerkin and collocation) based on Chebyshev and Legendre polynomi-als. For certain values of the physical parameters the numerically computed eigenvalues from the low part of the spectrum are displayed in a table. The Galerkin and collocation results are fairly closed and confirm the analytical results. Nowadays the industry and ecology need more and more results from hydrodynamic stability theory for sophisticated fluid motions occurring in complicated circumstances. In certain of these situations, a direct application of numerical methods can lead one to false results due to the bifurcation problems of the stationary solutions set of the Navier–Stokes equations (or of some more general models) and to the dependence of the eigenvalue on physical parameters. That is why, an analytic along with a numerical study of the eigenvalue problems from hydrodynamic stability theory is highly requested. This is in fact one of the most difficult topic in hydrodynamic stability. The occurrence of false secular points in the linear stability of continua was first pointed out by Collatz in 1981 in his paper [4]. A considerable number of theoretical and numerical studies have been devoted to the interaction of electromagnetic fields with fluids. Rosenswieg [24] pointed out that there are three main categories on this subject, i.e. electrohydrodynamics (EHD), magnetohydrodynamics (MHD) and ferrohydrodynamics (FHD). Here we are interested in an eigenvalue problem in EHD which implies the presence of electric forces. The problem at hand consists into an eighth order differential equation, containing only even order derivatives, supplied with homogeneous boundary conditions for the even order derivatives up to sixth order, i.e. the so called hinged …