Weakly nonlinear analysis of Bénard–Marangoni instability in viscoelastic fluids

Abstract

The coupled thermogravitational and thermocapillary instability in a thin fluid layer of a viscoelastic medium heated from below is investigated. A linear and a weakly nonlinear analysis are successively presented. The viscoelastic medium is modelled by means of a general rheological model including the lower convected, the upper convected, and the co-rotational Jeffreys models as particular cases. in comparison with previous analyses, two new dimensionless numbers are introduced, namely the so-called gravitational and rate of heating numbers instead of the classical Rayleigh and Marangoni numbers. The critical values for the temperature gradient, wave number and oscillation frequencies corresponding to the onset of convection are determined from a linear approach. After motion has set in, particular patterns are predicted taking the form of either rolls, or hexagon, or squares. By means of a nonlinear technique, restricted to steady situations, it is determined under which specific conditions one pattern is preferred. The influence of the constitutive equation, the Prandtl and the Blot numbers is examined and discussed in details. (C) 2000 Elsevier Science B.V. All rights reserved