Evolution of the coupled Bénard-Marangoni convection.

Abstract

A weakly nonlinear theory that describes the onset of the combined B\'enard-Marangoni convection is presented. All relevant transport coefficients are taken to be temperature dependent. When the boundaries are, thermally, nearly insulating, the instability is weak and the perturbed fluid-air interface is found to be proportional to the temperature field F(\ensuremath{\zeta},\ensuremath{\tau}) which evolves according to ${F}_{\ensuremath{\tau}}$-${\ensuremath{\pi}}_{1}$(${F}_{\ensuremath{\zeta}}$${)}_{\ensuremath{\zeta}}^{3}$-${\ensuremath{\pi}}_{2}$ (${F}_{\ensuremath{\zeta}}$${)}_{\ensuremath{\zeta}\ensuremath{\zeta}}^{2}$+${\ensuremath{\pi}}_{3}$${F}_{\ensuremath{\zeta}\ensuremath{\zeta}\ensuremath{\zeta}\ensuremath{\zeta}}$ +${\ensuremath{\pi}}_{4}$${F}_{\ensuremath{\zeta}\ensuremath{\zeta}}$+${\ensuremath{\pi}}_{5}$ (${F}_{\ensuremath{\zeta}}$${)}^{2}$+${\ensuremath{\pi}}_{6}$(${F}^{2}$${)}_{\ensuremath{\zeta}\ensuremath{\zeta}}$ +${\ensuremath{\pi}}_{7}$(${F}^{2}$${)}_{\ensuremath{\zeta}}$+(${\ensuremath{\beta}}_{1}$ +${\ensuremath{\beta}}_{2}$)F=0 where ${\ensuremath{\beta}}_{i}$ are the Biot numbers, and ${\ensuremath{\pi}}_{i}$, i=1,...,7, are constants.