# Analytical and numerical solutions to an electrohydrodynamic stability problem

## Abstract

A linear hydrodynamic stability problem corresponding to an electrohydrodynamic convection between two parallel walls is considered. The problem is an eighth order eigenvalue 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 polynomials. 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.

## Authors

Călin-Ioan Gheorghiu
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

F.I. Dragomirescu
Dept. of Math., Univ. ‘‘Politehnica” of Timisoara

## Keywords

Linear hydrodynamic stability; bifurcation manifolds; high order eigenvalue problems; hinged boundary conditions; direct analytical methods; Fourier type methods; spectral methods.

### References

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## Paper coordinates

I.F. Dragomirescu, C.I. Gheorghiu, Analytical and numerical solutions to an electrohydrodynamic stability problem. Appl. Math. Comput., 216 (2010) 3718-3727.
doi: 10.1016/j.amc.2015.10.078

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##### Journal

Applied Mathematics and Computation

Elsevier

0096-3003