Mathematics & Physics 2015, 8(3), 253–259 УДК 532.61.096 Equilibrium and Stability of a Free Liquid Film in a Longitudinal Gravitational Field Oksana A.Burmistrova∗ Lavrentyev Institute of Hydrodynamics of SB RAS Lavrentyev prospect, 15, Novosibirsk, 630090 Russia Received 24.03.2015, received in revised form 30.04.2015, accepted 06.05.2015 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. <...> Using numerical continuation method, value of the gravity for which this solution becomes unstable is found. <...> Keywords: liquid film, free surface, thin-layer approximation, thermocapillary effect, small disturbances, stability of solution. <...> DOI: 10.17516/1997-1397-2015-8-3-253-259 Introduction This paper motivated to interpret results of experiments that were carried out by professors O.Kabov, L.Tadrist and there colleagues from Marseille [1]. <...> The most part of papers on this subject considers films trickling down along the solid wall. <...> Thermocapillary effect plays a significant part in liquid motions [3–5]. <...> In [6,7] deformation of a free liquid filmby thermocapillary forces is considered when the gravity is absent. <...> In [8] a deformation and a rupture of a thin liquid film which is hanging between two solid flat walls under the action of concentrated thermal load action are considered under micro-gravity conditions. <...> The paper [9] is devoted to the problem for infinite non-isothermal liquid film with thermoisolated free surfaces under gravitation. <...> Spectral problem for perturbations gives the solution in the form of damped oscillations. ∗oksanabur@mail.ru Siberian Federal University. <...> When x2 = 0 and x2 = l, the liquid is bounded by solid walls. the two-dimension gradient and the Laplacian. <...> Reasoning by analogy with [6] we obtain two equations connecting the flow rate vector q through the cross-section of the layer, the film thickness h and the mean value T∗(x1,x2, t) of its temperature using thin-layer approximation <...>