LINEAR AND NONLINEAR LONGWAVE MARANGONI STABILITY OF TWO LIQUID LAYERS FLOWING DOWN BOTH SIDES OF A THICK VERTICAL WALL

The linear and nonlinear Marangoni instabilities of two thermally interacting liquid layers falling down both sides of a vertical solid wall were investigated. Here, the results differed completely from those of previous papers in that the wall was vertical and parallel to the acceleration of gravity and that the thin films had a main flow due to the Galilei number. In previous papers the wall was located perpendicular to gravity. Two coupled nonlinear evolution equations were calculated under the small wavenumber approximation. A temperature gradient was set across the system perpendicular to the wall. The linear growth rate of the instability was investigated with respect to a variety of parameters. For some magnitudes of the parameters, two different instability modes were found. A nonlinear two normal modes numerical solution was calculated to understand the thermal interaction of the two fluids when falling down both sides of the wall. It was found that only in one particular case it was possible to use the concepts of sinuous and varicose modes of instability. These calculations were performed using the parameters corresponding to the maximum growth rate of the linear theory. In the linear problem it was found that the thermocapillary terms disappear from the growth rate and phase velocity when the thicknesses of the two liquid layers were the same. However, it was interesting to find that, even in this case, the two liquid layers had nonlinear thermal interaction.