Eur. Phys. J. B 8, 125-135 (1999)
https://doi.org/10.1007/s100510050674

Influence of the surface deformability and variable viscosity on buoyant -thermocapillary instability in a liquid layer

¹ Institute of Mechanics, 4, Acad. G. Bonchev Str., 1113 Sofia, Bulgaria
² LESP (UMR 6614) , CNRS-INSA et Université de Rouen, B.P. 8, 76131 Mont-Saint-Aignan Cedex, France

Received: 15 May 1997
Published online: 15 March 1999

Abstract

The primary stationary and oscillatory Bénard-Marangoni instability is investigated in a fluid layer of infinite horizontal extent, bounded below by a rigid plane and above by a deformable upper surface, subjected to a vertical temperature gradient. Since the viscosity is temperature-dependent the consequences of relaxing Oberbeck-Boussinesq approximation and free surface deformability are theoretically examined by means of small disturbance analysis. The problem has been solved numerically by the Taylor series expansion method. The results obtained confirm that when the free surface is undeformable, stationary convection develops in the form of polygonal cells, and oscillatory motion cannot be detected. When the surface deformability is considered, stationary convection sets in, either as a short-wavelength hexagonal instability or as a long-wavelengh mode or as both, and oscillatory convection is also possible. The stability threshold for the short-wavelength mode depends mainly on the viscosity variation while the long-wavelength mode is determined by the surface deformation. Numerically, it is found that the neutral oscillatory Marangoni numbers are only negative. When a variable-viscosity model is used the theoretical and experimental results are in better agreement.