Abstract
The onset of steady Bénard-Marangoni convection in two horizontal liquid layers of electrically conducting immiscible fluids subjected to a uniform vertical magnetic field and temperature gradient is analysed by means of a combination of analytical and numerical techniques. The free surface can be either deformable or nondeformable and the interface between the fluids is always assumed to be flat. The effect of the lower layer on the critical values of Rayleigh, Marangoni and wave numbers for the onset of steady convection is investigated. When the free surface is nondeformable, the critical parameters for the onset of pure Marangoni convection are increased, whereas for the onset of pure Bénard convection they are decreased compared to the single-layer model. The results for a single-layer and for two-layers are qualitatively similar for Bénard-Marangoni convection when the free surface is deformable. All disturbances can be stabilized with sufficiently strong magnetic field when the free surface is nondeformable. If the free surface is allowed to deform and gravity waves are excluded, then the layers are always unstable to disturbances with sufficiently small wave number with magnetic field. Inclusion of gravity waves has a stabilizing effect on certain disturbances of small wave number in the presence of weak or moderate magnetic field.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Lord Rayleigh, On the convection currents in a horizontal layer of fluid when the higher temperature is on the under side. Phil. Mag. 32 (1916) 529-546.
J. R. A. Pearson, On convection cells induced by surface tension. J. Fluid Mech. 4 (1958) 489-500.
D. A. Nield, Surface tension and buoyancy effects in cellular convection. J. Fluid Mech. 19 (1964) 341-532.
S. H. Davis and G. M. Homsy, Energy stability theory for free surface problems: buoyancy-thermocapillary. J. Fluid Mech. 98 (1980) 527-553.
M. Takashima, Surface tension driven instability in horizontal liquid layer with deformable free surface. I. Steady Convection. J. Phys. Soc. Japan 50 (1981) 2745-2750.
M. Takashima, Surface tension driven instability in horizontal liquid layer with deformable free surface. II. Overstability. J. Phys. Soc. Japan 50 (1981) 2751-2756.
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability. London: Oxford University Press (1961) 652 pp.
D. A. Nield, Surface tension and buoyancy effects in the cellular convection of an electrically conducting liquid in a magnetic field. Z. Angew. Math. Phys. 17 (1966) 131-139.
G. S. R. Sarma, Marangoni convection in a liquid layer under the action of a transverse magnetic field. Space Res. 19 (1979) 575-578.
G. S. R. Sarma, Interaction of surface-tension and buoyancy mechanisms in horizontal liquid layers. J. Thermophysics 1 (1987) 129-135.
S. K. Wilson, The effect of uniform magnetic field on the onset of Marangoni convection in a layer of conducting fluid. Quart. J. Mech. Appl. Math. 46 (1993) 211-248.
S. K. Wilson, The effect of a uniform magnetic field on the onset of steady Bénard-Marangoni convection in a layer of conducting fluid. J. Eng. Math. 27 (1993) 161-188.
A. Thess and K. Nitschke, On Bénard-Marangoni instability in the presence of a magnetic field. Phys. Fluids 7 (1995) 1176-1178.
K. A. Smith, On convective instability by surface-tension gradient. J. Fluid Mech. 24 (1966) 401-414.
R. W. Zeren and W. C. Reynolds, Thermal instability in two-fluid horizontal layers. J. Fluid Mech. 53 (1972) 305-327.
E. Crespo del Arco, G. P. Extremet and R. L. Sani, Steady thermocapillary flows in two-layer liquid system with flat interfaces. J. Crystal Growth 126 (1993) 335-346.
T. Doi and J. N. Koster, Thermocapillary convection in two immiscible liquid layers with free surface. Phys. Fluids A5 (1993) 1914-1927.
P. C. Biswal and A. Ramachandra Rao, Thermocapillary convection in two-layer liquid system with deformed interfaces. Microgravity Sci. Technol. 8 (1995) 240-248.
S. K. Wilson, University of Strathclyde, Glasgow, U.K. Private communication 7th June 1995.
A. Pellow and R. V. Southwell, On maintaining convective motion in a fluid heated from below. Proc. R. Soc. London A176 (1940) 312-343.
A. Vidal and A. Acrivos, Nature of the neutral state in surface tension driven convection. Phys. Fluids 9 (1966) 615-616.
S. K. Wilson, The effect of a uniform magnetic field on the onset of oscillatory Marangoni convection in a layer of molten silicon. Microgravity Sci. Technol. 7 (1994) 228-233.
C. V. Sternling and L. E. Scriven, Interfacial turbulence: hydrodynamic instability and the Marangoni effect. A. I. Ch. E. J. 5 (1959) 514-523.
A. Kaddame and G. Lebon, Overstability in Marangoni convection of an electrically conducting fluid in the presence of an external magnetic field. Microgravity Q. 3 (1993) 1-6.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chandra Biswal, P. The onset of steady Bénard-Marangoni convection in a two-layer system of conducting fluid in the presence of a uniform magnetic field. Journal of Engineering Mathematics 35, 385–404 (1999). https://doi.org/10.1023/A:1004439231890
Issue Date:
DOI: https://doi.org/10.1023/A:1004439231890