Jose Pontes, Daniel Walgraef and Christo I. Christov

We study pattern formation and selection in Rayleigh-B'enard systems confined between well conducting horizontal boundaries and subjected to a weak horizontal gradient of the Rayleigh number. The study is based on the numerical integration of the Swift-Hohenberg equation and addresses the questions of the preferred orientation of the patterns with respect to the gradient of the Rayleigh number, boundary effects observed at subcritial sidewalls, the characteristics of long-term evolution of the patterns with emphasis on the wavelength selection, and the effect of non-potential modifications of the Swift-Hohenberg equation. It is shown that, contrary to common belief, the rolls do not align with the direction of the horizontal temperature gradient (Dewel, 1989; Malomed, 1993), due to the influence of the walls. Rather, rolls approach a sidewall perpendicularly when the local bifurcation parameter is sufficiently beyond the threshold but tend to be parallel to a subcritical or critical sidewall. Simulations performed with non-potential modifications of the Swift-Hohenberg equation lead in most cases, to asymptotic time-dependent behaviours.

Rayleigh-B'enard convection, Pattern Formation, Nonlinear Systems, Temperature gradients.

ARMERO F & SIMO JC. 1992. A new unconditionaly stable fractional step method for nonlinear coupled thermo-mechanical problems. Int. J. Num. Meth Engn., 45: 737. doi: 10.1016/0749-6419(93)90036-P

BRAGARD J, PONTES J & VELARDE MG. 1996. Patterns, defects and evolution of B'enard-Marangoni cells. Int. J. Bifurcation and Chaos, 6(9): 1665-1671.

BROWN SN & STEWARSON K. 1977. On thermal convection in a large box. SIAM, 57: 187-204.

CHANDRASEKHAR S. 1961. Hydrodynamic and Hydromagnetic Stability. Dover, New York.

CHRISTOV CI. 1994. Gaussian elimination with pivoting for multidiagonal systems. University of Reading.

CHRISTOV CI, PONTES J, WALGRAEF D & VELARDE MG. 1997. Implicit time splitting for fourth-order parabolic equations. Comput. Methods Apll. Mech. Engrg., 148: 209-224. doi: 10.1016/S0045-7825(96)01176-0

CROSS MC. 1982. Boundary conditions on the envelope functions of convective rolls close to the onset. Phys. Fluids, 25: 936-941. doi: 10.1063/1.863835

CROSS MC. 1982. Ingredients of a theory of convective textures close to onset. Phys. Rev. A, 25: 1065-1076. doi: 10.1103/PhysRevA.25.1065

CROSS MC, TESAURO G & GREENSIDE HS. 1986. Waveneumber selection and persistent dynamics in models of convection. Physica D, 10: 12-18. doi: 10.1016/0167-2789(86)90105-3

DE WIT A. 1993. Brisure de sym'etrie spatiale et dynamique spatiotemporalle dans les syst`emes r'eaction-diffusion. PhD thesis, Universit'e Libre de Bruxelles, Brussels.

DEWEL G & BORCKMANS P. 1989. Effects of slow spatial variations on dissipative structures. Phys. Lett. A, 138: 189-192. doi: 10.1016/0375-9601(89)90025-X

DOUGLASS J & RACHFORD HH. 1956. On the numerical solution of heat conduction problems in two and three space variables. Trans. Amer. Math. Soc., 82: 421-439.

EKEBJAERG L & JUSTTESEN P. 1991. An explicit scheme for advection-diffusion modelling in two-dimensions. Comp. Meth. Appl. Mech. ngineering., 88: 287-297. doi: 10.1016/0045-7825(91)90091-J

FRATI A, PASQUARELLY & QUARTERONI A. 1992. Spectral Approximation to advection-diffusion problems by fictious interfce method. J. Comput. Physics., 107: 201-212.

GREENSIDE HS and COUGHRAM Jr WM. 1984. Non-linear pattern formation near the onset of Rayleigh-B'enard convection. Phys. Rev. A, 30: 398-428. doi: 10.1103/PhysRevA.30.398

HEWETT DW, LARSON DJ & DOSS S. 1992. Solution of simultaneous partial differential equations using dynamic ADI: Solution of streamlined darwin field equation. J. Comp. Phys., 101: 11-24. doi: 10.1016/0021-9991(92)90039-2

HOYLE RB. 1995. Steady squares and hexagons on a subcritical ramp. Phys. Rev. E, 51: 310-315.

KELLY RE & PALL D. 1976. Thermal convection induced between nonuniformly heated horizontal surfaces. In: Proceedings of the 1976 Heat Transfer and Fluid Mechanics Institute, pages 1-17. Stanford U.P.

KIRCHARTZ KR, SRULIJES JA & OERTEL Jr H. 1983. Steady and timedependent Rayleigh-B'enard convection under influence of shear flows. Adv. Space Res., 3(5): 19-22.

LANDAU EM. 1944. On the problem of turbulence. C.R. Acad. Sci. URSS, 44: 311.

MALOMED BA and NEPOMNYASHCHY AA. 1993. Two-dimensional stability of convection rolls in the presence of a ramp. Europhys. Lett., 21(2): 195-200.

MALOMED BA, NEPOMNYASHCHY AA & TRIBELSKY MI. 1990. Domain boundaries in convection patterns. Phys. Rev. A, 41(12): 7244-7263. doi: 10.1103/PhysRevA.42.7244

MAMPAEY F. 1990. A stable alternating direction method for simulating multi-dimensional solidification problems. Int. J. Num. Meth. Engn., 30: 711-728. doi: 10.1002/nme.1620300411

MANNEVILLE P. 1990. Dissipative Structures and Weak Turbulence. Academic Press, San Diego.

MANNEVILLE P & POMEAU Y. 1983. A grain boundary in cellular structures near the onset of convection. Phil. Mag. A, 48: 607. doi: 10.1080/01418618308234915

NORMAND C, POMEAU Y & VELARDE MG. 1977. Convective instability: A physicist's approach. Rev. Mod. Phys., 49: 581-624. doi: 10.1103/RevModPhys.49.581

PEACEMAN DW & RACHFORD Jr HH. 1955. The numerical solution of paraboloic and eliptic differential equations. SIAM, 3: 28-41.

PONTES J. 1994. Pattern Formation in Spatially Ramped Rayleigh- B'enard Systems. PhD thesis, Universit'e Libre de Bruxelles, Brussels.

PONTES J, CHRISTOV CI & VELARDE MG. 1996. Numerical study of patterns and their evolution in finite geometries. Int. J. Bifurcation and Chaos, 6(10): 1883-1890. doi: 10.1142/S0218127496001211

ROSENFELD M & YASSOUR Y. 1994. The alternating direction multizone implicit method. J. comp. Phys., 110: 212-220.

SRULIJES JA. 1979. Zellularkonvektion in Behšaltern mit Horizontalen Temperaturgradienten. PhD thesis, Fakultšat fšur Maschinenbau, Univ. Karlsruhe, Karlsruhe.

SWIFT J & HOHENBERG PC. 1977. Hydrodynamic fluctuations at the convective instability. Phys. Rev. A, 15: 319-328. doi: 10.1103/PhysRevA.15.319

VELARDE MG & NORMAND C. 1980. Convection. Sci. Am., 243: 92-108.

WALGRAEF D. 1997. Spatio-Temporal Pattern Formation. Springer, New York.

WALTON IC. 1982. On the onset of Rayleigh-B'enard convection in a fluid layer of slowly increasing depth. Stud. Appl. Maths., 67: 199-216.

WALTON IC. 1983. The onset of cellular convection in a shallow twodimensional container of fluid heated non-uniformelly from below. J. Fluid Mech., 131: 455-470.

WESFREID J, POMEAU Y, DUBOIS M, NORMAND C & BERG'E P. 1978. Critcal effects in Rayleigh-B'enard convection. J. Phys. Lett., 39 (7): 725-731.

YANENKO NN. 1971. The Method of Fractional Steps. Springer, New York.