Stability of a vertical liquid film with consideration of the Marangoni effect and heat exchange with the environment

The stability of a free vertical liquid film under the combined action of gravity and thermocapillary forces has been studied. An exact solution of the Navier–Stokes and thermal conductivity equations is obtained for the case of plane steady flow with constant film thickness. It is shown that if the free surfaces of the film are perfectly heat insulated, the liquid flow rate through the cross section of the layer is zero. It is found that to close the model with consideration of the heat exchange with the environment, it is necessary to specify the liquid flow rate and the derivative of the temperature with respect to the longitudinal coordinate or the flow rate and the film thickness. The stability of the solution with constant film thickness at small wave numbers is studied. A solution of the spectral problem for perturbations in the form of damped oscillations is obtained.