Stability of a thin liquid film flowing on a vertical wall in the presence of mass transfer through the free surface

Abstract

We have studied the problem of the description of a thin liquid film flowing on a vertical wall in the presence of mass transfer through the free surface and have constructed a class of self-similar solutions in the case of a distributed nonstationary mass flux. The stability of the obtained self-similar flows with respect to small nonstationary harmonic perturbations of the liquid parameters has been studied and it is shown that the presence of a negative mass flux through the surface (evaporation) produces a destabilizing action on the flow. Similar to the case of a stationary flowing film of constant thickness, the self-similar flows are unstable with respect to perturbations of any frequency. In the case of evaporation, the development of instability has the character of an “explosion”: in the linear approximation, the amplitudes of perturbations exhibit infinite growth within a finite time (the time of evaporation of the liquid film). On the contrary, the presence of a positive mass flux through the surface (condensation, gas absorption, etc.) leads to the stabilization of the flow, whereby the amplitudes of perturbations exhibit limited growth in time over the entire film length and remain on the level of initial values. Moreover, in this case, there is a certain interval of frequencies in which small perturbations exhibit decay. It will be shown below that, using a feedback control, it is possible to stabilize the initially unstable self-similar film flows.