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Periodic and localized states in natural doubly diffusive convection. (English) Zbl 1138.76039
Summary: We employ a numerical continuation to follow branches of steady doubly diffusive convection in a vertical slot driven by imposed horizontal temperature and concentration gradients. No-slip boundary conditions are used on the lateral walls; periodic boundary conditions with large spatial period are used in the vertical direction. A variety of different states, both spatially periodic and spatially localized, are identified, and the profusion of the resulting solution branches is linked to a phenomenon known as homoclinic snaking.

MSC:
76E06 Convection in hydrodynamic stability
76R10 Free convection
76R50 Diffusion
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[1] Bardan, G.; Bergeon, A.; Knobloch, E.; Mojtabi, A., Nonlinear doubly diffusive convection in vertical enclosures, Physica D, 138, 91, (2000) · Zbl 0945.35072
[2] Batiste, O.; Knobloch, E.; Alonso, A.; Mercader, I., Spatially localized binary fluid convection, J. fluid mech., 560, 149, (2006) · Zbl 1122.76029
[3] Beghein, C.; Haghighat, F.; Allard, F., Numerical study of double diffusive natural convection in a square cavity, Int. J. heat mass transfer, 35, 833, (1992)
[4] Bennacer, R.; Gobin, D., Cooperating thermosolutal convection in enclosures: scale analysis and mass transfer, Int. J. heat mass transfer, 39, 2671, (1996) · Zbl 0964.76543
[5] Bensimon, D.; Shraiman, B.I.; Croquette, V., Nonadiabatic effects in convection, Phys. rev. A, 38, 5461, (1988)
[6] Bergeon, A.; Ghorayeb, K.; Mojtabi, A., Double diffusive instability in an inclined cavity, Phys. fluids, 11, 549, (1999) · Zbl 1147.76324
[7] Bergeon, A.; Knobloch, E., Natural doubly diffusive convection in three-dimensional enclosures, Phys. fluids, 14, 3233, (2002) · Zbl 1185.76049
[8] Burke, J.; Knobloch, E., Localized states in the generalized swift – hohenberg equation, Phys. rev. E, 73, 056211, (2006)
[9] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamics, (1991), Springer New York · Zbl 0717.76004
[10] Coullet, P.; Riera, C.; Tresser, C., Stable static localized structures in one dimension, Phys. rev. lett., 84, 3069, (2000)
[11] Crawford, J.D.; Knobloch, E., Symmetry and symmetry-breaking bifurcations in fluid mechanics, Annu. rev. fluid mech., 23, 341, (1991)
[12] Deville, M.; Fischer, P.F.; Mund, E.H., High-order methods for incompressible fluid flow, (2002), Cambridge University Press · Zbl 1007.76001
[13] Funaro, D., Polynomial approximation of differential equations, (1991), Springer New York
[14] K. Ghorayeb, Etude des écoulements de convection thermosolutale en cavité rectangulaire, Thèse de Doctorat de l’Université Paul Sabatier, Toulouse 3, France, 1997
[15] Ghorayeb, K.; Mojtabi, A., Double diffusive convection in a vertical rectangular cavity, Phys. fluids, 9, 2339, (1997)
[16] Gobin, D.; Bennacer, R., Double diffusion in a vertical fluid layer: onset of the convective regime, Phys. fluids, 6, 59, (1994) · Zbl 0822.76088
[17] Gobin, D.; Bennacer, R., Cooperating thermosolutal convection in enclosures: heat transfer and flow structure, Int. J. heat mass transfer, 39, 2683, (1996) · Zbl 0964.76543
[18] Han, H.; Kuehn, T.H., Double diffusive natural convection in a vertical rectangular enclosure, Int. J. heat mass transfer, 34, 449, (1991)
[19] Hiraoka, Y.; Ogawa, T., Rigorous numerics for localized patterns to the quintic swift – hohenberg equation, Japan J. indust. appl. math., 22, 57, (2005) · Zbl 1067.65146
[20] Iooss, G.; Pérouème, M.C., Perturbed homoclinic solutions in reversible 1:1 resonance vector fields, J. differential equations, 102, 62, (1993) · Zbl 0792.34044
[21] Jiang, H.D.; Ostrach, S.; Kamotani, Y., Thermosolutal convection with opposed buoyancy forces in shallow enclosures, (), 165
[22] Jiang, H.D.; Ostrach, S.; Kamotani, Y., Unsteady thermosolutal transport phenomena due to opposed buoyancy forces in shallow enclosures, J. heat transfer, 113, 135, (1991)
[23] Kamotani, Y.; Wang, L.W.; Ostrach, S.; Jiang, H.D., Experimental study of natural convection in shallow enclosures with horizontal temperature and concentration gradients, Int. J. heat mass transfer, 28, 165, (1985)
[24] Karniadakis, G.Em.; Israeli, M.; Orszag, S.A., High-order splitting method for the incompressible navier – stokes equations, J. comput. phys., 97, 414, (1991) · Zbl 0738.76050
[25] Knobloch, E., Convection in binary fluids, Phys. fluids, 23, 1918, (1980)
[26] Lee, J.; Hyun, M.T.; Kim, K.W., Natural convection in confined fluids with combined horizontal temperature and concentration gradients, Int. J. heat mass transfer, 31, 1969, (1988)
[27] Lee, J.W.; Hyun, J.M., Double-diffusive convection in a rectangle with opposing horizontal temperature and concentration gradients, Int. J. heat mass transfer, 33, 1619, (1990)
[28] Lynch, R.E.; Rice, J.R.; Thomas, D.H., Tensor product analysis of partial difference equations, Bull. amer. math. soc., 70, 378, (1964) · Zbl 0126.12704
[29] Mamun, C.K.; Tuckerman, L.S., Asymmetry and Hopf bifurcation in spherical Couette flow, Phys. fluids, 7, 80, (1995) · Zbl 0836.76033
[30] Meca, E.; Mercader, I.; Batiste, O.; Ramírez Piscina, L., Complex dynamics in double-diffusive convection, Theoret. comput. fluid dynamics, 18, 231, (2004) · Zbl 1178.76152
[31] Peyret, R., ()
[32] Prat, J.; Mercader, I.; Knobloch, E., Resonant mode interactions in rayleigh-Bénard convection, Phys. rev. E, 58, 3145, (1998)
[33] Quarteroni, A.; Vallis, A., Domain decomposition methods for partial differential equations, (1999), Oxford University Press
[34] Schmidt, R.W., Double diffusion in oceanography, Annu. rev. fluid. mech., 26, 255, (1994)
[35] Sezai, I.; Mohamad, A.A., Double diffusive convection in a cubic enclosure with opposing temperature and concentration gradients, Phys. fluids, 12, 2210, (2000) · Zbl 1184.76494
[36] Shyy, W.; Chen, M.-H., Double-diffusive flow in enclosures, Phys. fluids, A3, 2592, (1991) · Zbl 0825.76793
[37] Sorokin, L.E., Subcritical motions of a binary mixture with anomalous thermodiffusion in a vertical layer, Fluid dynam., 36, 11, (2001) · Zbl 1018.76507
[38] Tsitverblit, N., Bifurcation phenomena in confined thermosolutal convection with lateral heating: commencement of the double-diffusive region, Phys. fluids, 7, 718, (1995) · Zbl 1039.76506
[39] Tsitverblit, N., Multiplicity of equilibrium states in laterally heated thermosolutal systems with equal diffusivity coefficients, Phys. fluids, 11, 2516, (1999) · Zbl 1149.76573
[40] Tsitverblit, N.; Kit, E., The multiplicity of steady flows in confined double-diffusive convection with lateral heating, Phys. fluids, A5, 1062, (1993)
[41] Tuckerman, L.S., Steady-state solving via Stokes preconditioning: recursion relations for elliptic operators, (), 573
[42] Turner, J.S., Double diffusive phenomena, Annu. rev. fluid. mech., 6, 37, (1974) · Zbl 0312.76028
[43] Turner, J.S., Multicomponent convection, Annu. rev. fluid. mech., 17, 11, (1985)
[44] Wilcox, W.R., Transport phenomena in crystal growth from solution, Prog. crystal growth charact., 26, 153, (1993)
[45] Woods, P.D.; Champneys, A.R., Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate hamiltonian – hopf bifurcation, Physica D, 129, 147, (1999) · Zbl 0952.37009
[46] Xin, S.; Le Quéré, P.; Tuckerman, L.S., Bifurcation analysis of double-diffusive convection with opposing horizontal thermal and solutal gradients, Phys. fluids, 10, 850, (1998)
[47] Yang, T.-S.; Akylas, T.R., On asymmetric gravity – capillary solitary waves, J. fluid mech., 330, 215, (1997) · Zbl 0913.76011
[48] Yochelis, A.; Burke, J.; Knobloch, E., Reciprocal oscillons and nonmonotonic fronts in forced nonequilibrium systems, Phys. rev. lett., 97, 254501, (2006)
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