The onset of convection in a horizontal layer of fluid heated from below in the presence of a gravity field varying across the layer is numerically investigated. The eigenvalue problem governing the linear stability of the mechanical equilibria of the fluid, in the case of free boundaries, is a sixth order differential equation with Dirichlet and hinged boundary conditions. It is transformed into a system of second order differential equations supplied only with Dirichlet boundary conditions. Then it is solved using two distinct classes of spectral methods namely, weighted residuals (Galerkin type) methods and a collocation (pseudospectral) method, both based on Chebyshev polynomials. The methods provide a fairly accurate approximation of the lower part of the spectrum without any scale resolution restriction. The Viola’s eigenvalue problem is considered as a benchmark one. A conjecture is stated for the first eigenvalue of this problem.
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)
See the expanding block below.
C.I. Gheorghiu, F.-I. Dragomirescu, Spectral methods in linear stability. Applications in thermal convection with variable gravity field, Appl. Numer. Math., 59 (2009) 1290-1302
Applied Numerical Mathematics
Google Scholar Profile
google scholar link
 M.G. Blyth, C. Pozrikidis, Effect of surfactant on the stability of film flow down an inclined plane, J. Fluid Mech. 521 (2004) 241–250.
 J.P. Boyd, Chebyshev and Fourier Spectral Methods, second ed., Dover, New York, 2000.
 C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, 1987.
 S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford Univ. Press, 1961.
 J.J. Dongarra, B. Straughan, D.W. Walker, Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems, Appl. Numer. Math. 22 (1996) 399–434.
 I. Dragomirescu, Approximate neutral surface of a convection problem for variable gravity field, Rend. Sem. Mat. Univ. Politec Torino 64 (2006) 331–342.
 I. Dragomirescu, A. Georgescu, Linear stability bounds in a convection problem for variable gravity field, Bul. Acad. Stiinte Repub. Mold. Mat. 52 (2006) 51–66.
 I. Dragomirescu, A SLP-based method for a convection problem for a variable gravity field, in: Proc. of Aplimat 2007, Bratislava, pp. 149–154.
 I. Dragomirescu, Shifted polynomials in a convection problem, arXiv:0709.2240v1 [math-ph], 2007.
 P.G. Drazin, W.H. Reid, Hydrodynamic Stability, Cambridge University Press, London, 1981.
 C.I. Gheorghiu, I.S. Pop, A modified Chebyshev-tau method for a hydrodynamic stability problem, in: Proc. ICAOR ’97, vol. II, Transylvania Press, Cluj-Napoca, 1997, pp. 119–126.
 C.I. Gheorghiu, Spectral Methods for Differential Problems, Casa Cartii de Stiinta Publishing House, Cluj-Napoca, 2007.
 L. Greenberg, M. Marleta, Numerical solution of non-self-adjoint Sturm–Liouville problems and related systems, SIAM J. Numer. Anal. 38 (2001) 1800–1845.
 W. Heinrichs, Improved condition number for spectral methods, Math. Comput. 53 (1998) 103–119.
 I. Herron, On the principle of exchange of stabilities in Rayleigh–Bénard convection, SIAM J. Appl. Math. 61 (2000) 1362–1368.
 A.A. Hill, B. Straughan, A Legendre spectral element method for eigenvalues in hydromagnetic stability, J. Comput. Appl. Math. 193 (2003) 363–381.
 W. Huang, D.M. Sloan, The pseudospectral methods for third-order differential equations, SIAM J. Numer. Anal. 29 (1992) 1626–1647.
 J.C. Mason, D.C. Handscomb, Chebyshev Polynomials, Chapman & Hall, 2003.
 J.M. Melenk, N.P. Kirkner, V. Schwab, Spectral Galerkin discretization for hydrodynamic stability problems, Computing 65 (2000) 97–118.
 S. Orszag, Accurate solutions of Orr–Sommerfeld stability equation, J. Fluid Mech. 50 (1971) 689–703.
 K. Prakash, H. Raj, Effect of variable gravitational field on thermal instability of a rotating fluid layer with magnetic field in porous medium, Czech. J. Phys. 47 (1997) 793–800.
 S. Pruess, C.T. Fulton, Mathematical software for Sturm–Liouville problems, ACM Trans. Softw. 19 (1998) 360–376.
 A. Quarteroni, F. Saleri, Scientific Computing with MATLAB and Octave, second ed., Springer-Verlag, 2006.
 S.C. Reddy, P.J. Schmid, D. Henningson, Pseudospectra of Orr–Sommerfeld operator, SIAM J. Appl. Math. 53 (1993) 15–47.
 J. Shen, Efficient spectral-Galerkin method II, Direct solvers of second and fourth order equations by using Chebyshev polynomials, SIAM J. Sci. Comput. 16 (1995) 74–87.
 G.W. Stewart, J.-G. Sun, Matrix Perturbation Theory, Academic Press, New York, 1990.
 B. Straughan, The Energy Method, Stability, and Nonlinear Convection, Springer, Berlin, 2003.
 L.N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, PA, 2000.
 P.G. Vekilov, Protein crystal growth—microgravity aspects, Advances in Space Research 24 (1999) 1231–1240.
 J.A.C. Weideman, S.C. Reddy, A MATLAB differentiation matrix suite, ACM Trans. on Math. Software 26 (2000) 465–519.