## 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

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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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## References

## References

[1] D. Bourne, Hydrodynamic stability, the Chebyshev tau method and spurious eigenvalues, Continuum Mech. Thermodyn. 15 (2003) 571–579.

[2] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics, SpringerVerlag, 2007.

[3] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford University Press, 1961.

[4] L. Collatz, Numerical methods for free boundary problems, in: Proceedings of Free Boundary Problems: Theory and Applications, Montecatini, Italy, 1981.

[5] J.J. Dongarra, B. Straughan, D.W. Walker, Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problem, Appl. Numer. Math. 22 (1996) 399–434.

[6] F.I. Dragomirescu, Rayleigh number in a stability problem for a micropolar fluid, Turk. J. Math. 31 (2) (2007) 123–137.

[7] F.I. Dragomirescu, Bifurcation and numerical study in an EHD convection problem, An. St. Univ. ‘‘Ovidius” Constanta 16 (2) (2008) 47–57.

[8] D.R. Gardner, S.A. Trogdon, R.W. Douglas, A modified tau spectral method that eliminates spurious eigenvalues, J. Comput. Phys. 80 (1989) 137–167.

[9] A. Georgescu, Hydrodynamic Stability Theory, Kluwer, Dordrecht, 1985.

[10] A. Georgescu, D. Pasca, S. Gradinaru, M. Gavrilescu, Bifurcation manifolds in multiparametric linear stability of continua, ZAMM 73 (1993). 7/8 T767– T768.

[11] A. Georgescu, L. Palese, On a method in linear stability problems. Application to a natural convection in a porous medium, Rapp. Int. Dept. Math. Univ. Bari 9 (1996).

[12] A. Georgescu, M. Gavrilescu, L. Palese, Neutral thermal hydrodynamic and hydromagnetic stability hypersurface for a micropolar fluid layer, Indian J. Pure Appl. Math. 29 6 (1998) 575–582.

[13] C.I. Gheorghiu, I.S. Pop, A Modified Chebyshev–Tau Method for a Hydrodynamic Stability Problem, in: Proceedings of ICAOR 1996, vol. II, 1996, pp. 119–126.

[14] C.I. Gheorghiu, Spectral Methods for Differential Problems, Casa Cartii de Stiinta Publishing House, Cluj-Napoca, 2007.

[15] C.I. Gheorghiu, F.I. Dragomirescu, Spectral methods in linear stability. Applications to thermal convection with variable gravity field, Appl. Numer. Math. 59 (2009) 1290–1302.

[16] D. Gottlieb, S.A. Orszag, Numerical Analysis of Spectral Methods, SIAM, Philadelphia, PA., 1977.

[17] M.J. Gross, Mantles of the Earth and Terrestrial Planets, Wiley, 1967.

[18] P. Henrici, Bounds for iterates, inverses, spectral variation and fields of values of non-normal matrices, Numer. Math. 4 (1962) 24–40.

[19] A.A. Hill, B. Straughan, A legendre spectral element method for eigenvalues in hydromagnetic stability, J. Comput. Appl. Math. 193 (2003) 363–381.

[20] W. Huang, D.M. Sloan, The pseudospectral method for third-order differential equations, SIAM J. Numer. Anal. 29 (6) (1992) 1626–1647.

[21] W. Huang, D.M. Sloan, The pseudospectral method for solving differential eigenvalue problems, J. Comput. Phys. 111 (1994) 399–409.

[22] S.A. Orszag, Accurate solution of the Orr–Sommerfeld stability equation, J. Fluid Mech. 50 (1971) 689–703.

[23] P.H. Roberts, Electrohydrodynamic convection, Q.J. Mech. Appl. Math. 22 (1969) 211–220.

[24] R. Rosensweig, Ferrohydrodynamics, Cambridge Univ. Press, 1985.

[25] P.J. Schmid, D.S. Henningson, Stability and Transition in Shear Flows, Springer-Verlag, 2001.

[26] B. Straughan, The Energy Method, Stability, and Nonlinear Convection, second ed., Springer, Berlin, 2003.

[27] R. Tunbull, Electroconvective instability with a stabilizing temperature gradient. I. Theory, Phys. Fluids 11 (1968) 2588–2596.

[28] R. Turnbull, Electroconvective instability with a stabilizing temperature gradient. II. Experimental results, Phys. Fluids 11 (1968) 2597–2603.

[29] L.N. Trefethen, Computation of Pseudospectra, Acta Numer. (1999) 247–295.

[30] J.A.C. Weideman, S.C. Reddy, A MATLAB differentiation matrix suite, ACM Trans. Math. Softw. 26 (2000) 465–519.