Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-06-16T15:41:14.720Z Has data issue: false hasContentIssue false
  • English
  • Français

Bénard convection in binary mixtures with Soret effects and solidification

Published online by Cambridge University Press:  26 April 2006

G. Zimmermann ,
U. Müller  and
S. H. Davis
G. Zimmermann
Affiliation:
Kernforschungszentrum Karlsruhe, Institut für Angewandte Thermo- und Fluiddynamik, Postfach 3640, W-7500, Karlsruhe, Germany Present address: Aachen Center for Solidification in Space, ACCESS e.V., Aachen, Germany.
U. Müller
Affiliation:
Kernforschungszentrum Karlsruhe, Institut für Angewandte Thermo- und Fluiddynamik, Postfach 3640, W-7500, Karlsruhe, Germany
S. H. Davis
Affiliation:
Department of Engineering Sciences and Applied Mathematics. Northwestern University, Evanston, IL 60208, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Bénard convection of a two-component liquid is considered. The liquid displays Soret effects and the boundary temperatures are fixed to span the solidification temperature of the mixture. Near the lower, heated plate the material is liquid and near the upper cooled plate there is a layer of pure solid solvent; all the solute is rejected during freezing. Linear stability theory is used to determine the effects on the critical conditions for Soret convection in the presence of the solidified layer and the interface between solid and liquid.

Experiments on mixtures of ethyl alcohol and water are performed using interferometry, photography and thermocouple measurements. The measured onset of instability to travelling waves at negative Soret coefficient compares well with those predicted by our linear theory. In the absence of ice the waves develop at finite amplitude to a fixed-amplitude state. However, when ice is present, these waves fail to persist but evolve to a state of steady finite-amplitude (overturning) convection. These differences are attributed to the presence of the ice and the nonlinear density profile of the basic state, both of which act as sources of non-Boussinesq effects.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 238 , May 1992 , pp. 657 - 682
Copyright
© 1992 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahlers, G., Cannell, D. S. & Heinrichs, R. 1987 Convection in a binary mixture. Nucl. Phys. B (Proc. Suppl.) 2, 77–86.Google Scholar
Ahlers, G. & Lücke, M. 1987 Some properties of an eight-mode Lorenz model for convection in binary fluids. Phys. Rev. A 35, 470–473.Google Scholar
Ahlers, G. & Rehberg, I. 1986 Convection in a binary mixture heated from below. Phys. Rev. Lett. 56, 1373–1376.Google Scholar
Antar, B. N. 1987 Penetrative double-diffusive convection. Phys. Fluids 30, 322–330.Google Scholar
Bensimon, D., Kolodner, P., Surko, C. M., Williams, H. & Croquette, V. 1990 Competing and coexisting dynamical states of travelling-wave convection in an annulus. J. Fluid Mech. 217, 441–467.Google Scholar
Brand, H. R., Lomdahl, P. S. & Newell, A. C. 1986 Benjamin-Feir turbulence in convective binary fluid mixtures. Physica 23D, 345–361.Google Scholar
Brand, H. R. & Steinberg, V. 1984 Analog of the Benjamin-Feir instability near the onset of convection in binary fluid mixtures. Phys. Rev. A 29, 2303–2304.Google Scholar
Bühler, K. 1979 Zellular Konvektion in rotierenden Behältern. Fortschritt-Berichte der VDI-Zeitschriften, Reihe 7, Nr. 54.Google Scholar
Bühler, K., Kirchartz, K. R. & Srulijes, J. 1978 Anwendung der Differentialinterferometrie bei thermischen konvektionsstromungen. In Applied Fluid Mechanics (ed. H. Oertel), pp. 54–69. Mitteil. Institüt für Strömungs lehre und Strömungsmaschinen, Universität Karlsruhe.
Caldwell, D. R. 1970 Nonlinear effects in a Rayleigh-Bénard experiment. J. Fluid Mech. 42, 161–175.Google Scholar
Caldwell, D. R. 1973 Measurements of negative thermal diffusion coefficients by observing the onset of thermohaline convection. J. Phys. Chem. 77, 2004–2008.Google Scholar
Caldwell, D. R. 1974 Experimental studies on the onset of thermohaline convection. J. Fluid Mech. 64, 347–367.Google Scholar
Caldwell, D. R. 1975 Soret coefficient of 1N lithium iodide. J. Phys. Chem. 79, 1882–1884.Google Scholar
Caldwell, D. R. 1976 Thermosolutal convection in a solution with large negative Soret coefficient. J. Fluid Mech. 74, 129–142.Google Scholar
Chock, D. P. & Li, C.-H. 1975 Direct integration method applied to Sore-driven instability. Phys. Fluids 18, 1401–1406.Google Scholar
Cross, M. C. 1986a An eight-mode Lorenz-model of travelling waves in a binary fluid convection. Phys. Lett. A 119, 21–24.Google Scholar
Cross, M. C. 1986b Travelling and standing waves in binary-fluid convection in finite geometries. Phys. Rev. Lett. 57, 2935–2938.Google Scholar
Cross, M. C. 1988 Structure on nonlinear travelling-wave states in finite geometries. Phys. Rev. A 38, 3593–3600.Google Scholar
Cross, M. C. & Kim, K. 1988 Linear instability and the codimension-2 region in binary fluid convection between rigid impermeable boundaries. Phys. Rev. 137, 3909–3920.Google Scholar
D'an-Lax, E. 1967 Taschenbuch für Chemiker und Physiker, vol. 1 (Third Edn.) Springer.
Davis, S. H., Müller, U. & Dietsche, C. 1984 Pattern selection in single-component systems coupling Bénard convection and solidification. J. Fluid Mech. 144, 133–151.Google Scholar
Davis, S. H. & Segel, L. A. 1968 Effects of surface curvature and property variation on cellular convection. Phys. Fluids 11, 470–476.Google Scholar
Deane, A. E., Knobloch, E. & Toomre, J. 1988 Travelling waves and chaos in large-aspect-ratio thermosolutal convection. Phys. Rev. A 37, 1817–1820 and Erratum, Phys. Rev. A 38, 1661 (1988).Google Scholar
DeGroot, S. R. & Mazur, P. 1969 Non-equilibrium Thermodynamics (2nd edn). North-Holland.
Dietsche, C. 1984 Einfluss der Bénard-Konvektion auf Gefrierflachen. KfK-Bericht 3724.Google Scholar
Dietsche, C. & Müller, U. 1985 Influence of Bénard convection on solid-liquid interfaces. J. Fluid Mech. 161, 249–268.Google Scholar
Fineberg, J., Moses, E. & Steinberg, V. 1988a Spatially and temporally modulated travellingwave pattern in convecting binary mixtures. Phys. Rev. Lett. 61, 838–841.Google Scholar
Fineberg, J., Moses, E. & Steinberg, V. 1988b Nonlinear pattern and wave-number selection in convecting binary mixtures. Phys. Rev. A 38, 4939–4942.Google Scholar
Fineberg, J. & Moses, E. & Steinberg, V. 1989 Reply to: P. Kolodner (1989). Phys. Rev. Lett. 64, 579.Google Scholar
Grauer, T. & Haken, H. 1988 Generalized Ginzberg-Landau equations applied to instabilities in systems coupling convection and solidification. In Physicochemical Hydrodynamics, Interfacial Phenomena (ed. M. G. Velarde). NATO ASI Series, Vol. 174, pp. 571–582. Plenum.
Hadji, L. & Schell, M. 1990 Soret-driven convection coupled to the morphology of a solid-liquid interface. Phys. Fluids A 2, 1597–1606.Google Scholar
Heinrichs, R., Ahlers, G. & Cannell, D. S. 1987 Travelling waves and spatial variations in the convection of a binary mixture. Phys. Rev. A 35, 2761–2764.Google Scholar
Hurle, D. T. J. & Jakeman, E. 1969 Significance of the Soret effect in the Rayleigh-Jeffreys problem. Phys. Fluids 12, 2704–2705.Google Scholar
Hurle, D. T. J. & Jakeman, E. 1971 -Soret-driven thermosolutal convection. J. Fluid Mech. 47, 667–687.Google Scholar
Hurle, D. T. J. & Jakeman, E. 1973a Thermal oscillations in convecting fluids. Phys. Fluids 16, 2056–2059.Google Scholar
Hurle, D. T. J. & Jakeman, E. 1973b Natural oscillations in heated fluid layers. Phys. Lett. 43A, 127–129.Google Scholar
Kirchartz, K.-R. 1980 Zeitabhangige zellularkonvektion in horizontalen und geneigten behaltern. Dissertation an der Universitat Karlsruhe.
Knobloch, E. & Moore, D. R. 1988 Linear stability of experimental Soret convection. Phys. Rev. A 37, 860–870.Google Scholar
Kolodner, P., Bensimon, D. & Surko, C. M. 1988a Travelling-wave convection in an annulus. Phys. Rev. Lett. 60, 1723–1726.Google Scholar
Kolodner, P., Passner, A., Surko, C. M. & Walden, R. W. 1986 Onset of oscillatory convection in a binary fluid mixture. Phys. Rev. Lett. 56, 2621–2624.Google Scholar
Kolodner, P., Passner, A., Williams, H. L. & Surko, C. M. 1987a The transition to finiteamplitude travelling-wave convection in binary fluid mixtures. Nucl. Phys. B (Proc. Suppl.) 2, 97–108.Google Scholar
Kolodner, P. & Surko, C. M. 1988 Weakly nonlinear travelling-wave convection. Phys. Rev. Lett. 61, 842–845.Google Scholar
Kolodner, P., Surko, C. M., Passner, A. & Williams, H. L. 1987b Pulses of oscillatory convection. Phys. Rev. A 36, 2499–2502.Google Scholar
Kolodner, P., Surko, C. M., Williams, H. L. & Passner, A. 1988b Two-frequency states at the onset of convection in binary fluid mixtures. In Propagation in Systems Far From Equilibrium (ed. J. E. Wesfreid, H. R. Brand, P. Manneville, A. Albinet, and N. Boccara), pp. 282–291. Springer.
Kolodner, P., Williams, H. & Moe, C. 1988c Optical measurement of Soret coefficient of ethanol/water solutions. J. Chem. Phys. 88, 6512–6524.Google Scholar
Krishnamurti, R. 1968 Finite amplitude convection with changing mean temperatures. J. Fluid Mech. 33, 445–455.Google Scholar
Kurz, W. & Fisher, D. J. 1989 Fundamentals of Solidification. Switzerland: Trans. Tech. Publications.
Legros, J. C., Platten, J. K. & Poty, P. G. 1972 Stability of a two-component fluid layer heated from below. Phys. Fluids 15, 1383–1390.Google Scholar
Legros, J. C. & Platten, K. 1977 Two-component Bénard problem with Poiseuille flow. J. Non-Equilib. Thermodynam. 2, 211–232.Google Scholar
Lhost, O., Linz, S. J. & Müller, H. W. 1991 Onset of convection in binary liquid mixtures: Improved Galerkin approximations. J. Phys. II 1, 279–287.Google Scholar
Lhost, O. & Platten, J. K. 1988 Transitions between steady states, travelling waves and modulated travelling waves in the system water-isopropanol heated from below. Phys. Rev. A 38, 3147–3150.Google Scholar
Lhost, O. & Flatten, J. K. 1989 Experimental study of the transition from nonlinear travelling waves to steady overturning convection in binary mixtures. Phys. Rev. A 40, 4552–4557.Google Scholar
Linz, S. J. & Lücke, M. 1987 Convection in binary mixtures: A Galerkin model with impermeable boundary conditions. Phys. Rev. A 35, 3397–4000.Google Scholar
Linz, S. J., Lücke, M., Müller, H. W. & Niederlander, J. 1988 Convection in binary fluid mixtures: Travelling waves and lateral currents. Phys. Rev. A 38, 5727–5741.Google Scholar
Moses, E., Fineberg, J. & Steinberg, V. 1987 Multistability and confined travelling-wave patterns in a convecting binary mixture. Phys. Rev. A 35, 2757–2760.Google Scholar
Moses, E. & Steinberg, V. 1986 Flow patterns and nonlinear behavior of travelling waves in a convective binary fluid. Phys. Rev. A 34, 693–696 and Erratum. Phys. Rev. A 35, 1444–1445 (1987).Google Scholar
Moses, E. & Steinberg, V. 1988 Mass transfer in propagating patterns of convection. Phys. Rev. Lett. 60, 2030–2033.Google Scholar
Ott, J. B., Goates, J. R. & Waite, B. A. 1979 (Solid-liquid) phase equilibria and solid-hydrate formation in water+methyl+ethyl+isopropyl, and +terciary butyl alcohols. J. Chem. Thermodyn. 11, 739–746.Google Scholar
Platten, J. K. 1971 Le probleme de Bénard dans les Melanges: cas de surface libres. Bull. Classe Sci. Acad. Roy. Belg. 57, 669–683.Google Scholar
Platten, J. K. & Chavepeyer, G. 1972a Oscillations in a water-ethanol liquid layer heated from below. Phys. Lett. 40A, 287–288.Google Scholar
Platten, J. K. & Chavepeyer, G. 1972b Soret driven instability. Phys. Fluids 15, 1555–1557.Google Scholar
Platten, J. K. & Legros, J. C. 1984 Convection in Liquids. Springer.
Rehberg, I. & Ahlers, G. 1986 Codimension-two bifurcation in a convection experiment. Contemp. Maths 56, 277–282.Google Scholar
Rosenberger, F. 1979 Fundamentals of Crystal Growth I. Springer.
Schechter, R. S., Prigogine, I. & Hamm, J. R. 1972 Thermal diffusion and convective stability. Phys. Fluids 15, 379–386.Google Scholar
Scott, M. R. & Watts, H. A. 1975 Subroutine SUPORT. Rep. SAND75–0198. Saudia Labs Albuquerque.Google Scholar
Soret, M. Ch. 1879 Une dissolution saline primitivement homogene. Arch. Sci. Phys. Nat. Geneve 2, 48–61.Google Scholar
Steinberg, V. & Moses, E. 1987 Experiments on convection in binary mixtures. In Pattern, Defects and Microstructures in Nonequilibrium Systems, Proc. NATO Advanced Research Workshop (ed. D. Walgraef). NATO ASI Series, pp. 309–335. Martinus Nijhoff.
Steinberg, V., Moses, E. & Fineberg, J. 1987 Spatio-temporal complexity at the onset of convection in a binary fluid. Nucl. Phys. B (Proc. Suppl.) 2, 109–124.Google Scholar
Sullivan, T. S. & Ahlers, G. 1988a Hopf bifurcation to convection near the codimension-two-point in a 3He-4He mixture. Phys. Rev. Lett. 61, 78–81.Google Scholar
Sullivan, T. S. & Ahlers, G. 1988b Nonperiodic time dependence at the onset of convection in a binary liquid mixture. Phys. Rev. A 38, 3143–3146.Google Scholar
Surko, C. M., Kolodner, P., Passner, A. & Walden, R. W. 1986 Finite-amplitude travelling-wave convection in binary fluid mixtures. Physica 23D, 220–229.Google Scholar
Villers, D. & Platten, J. K. 1984 Heating curves in the two-component Bénard problem. J. Non-Equilib. Thermodyn. 9, 131–146.Google Scholar
Walden, R. W., Kolodner, P., Passner, A. & Surko, C. M. 1985 Travelling waves and chaos in convection in binary fluid mixtures. Phys. Rev. Lett. 55, 496–499.Google Scholar
Zielinska, B. J. A. & Brand, H. R. 1987 Exact solution of the linear-stability problem for the onset of convection in binary fluid mixtures. Phys. Rev. A 35, 4349–4353 and Erratum. Phys. Rev. A 37, 1786 (1988).Google Scholar
Zimmermann, G. 1990 Bénard-Konvektion in binaren Flüssigkeitsmischungen mit thermodiffusion. KfK-Bericht 4683.Google Scholar
Zimmermann, G. & Müller, U. 1992 Bénard convection in binary mixtures with Soret effects. Intl J. Heat Mass Transfer 35 (in press).Google Scholar
Zimmermann, G., Müller, U. & Davis, S. H. 1986 Bénard convection in a partly solidified two-component system. KfK-Bericht 4122.Google Scholar