Equilibrium and stability of a free liquid film in a longitudinal gravitational field

Deformation of the free vertical liquid film by combined effect of gravity and thermocapillary forces is investigated. The system of equations connecting the flow rate, the film thickness and its temperature is found in the framework of thin-layer approximation. Using the shooting method, one-dimensional stationary problem is solved numerically for values of contact angle close to a right angle. The existence of a film having the constant thickness is found. It is shown that this solution is stable when the gravity is low. Using numerical continuation method, value of the gravity for which this solution becomes unstable is found.