Modelling the Influence of Temperature Gradients on the State of a Free Liquid Surface

Various heat and mass exchange processes (condensation, evaporation, and others) in film devices occur in the conditions of interfacial instability. We study the interfacial instability, associated with the Marangoni effect, of a liquid film flowing due to gravity at moderate Reynolds numbers. We derive an equation for its free surface in the framework of a mathematical model of the wave flow of a non-isothermal liquid film, namely, the system of Navier - Stokes and continuity equations with boundary conditions accounting for heat and mass transfer. The coefficients in the equation depend on the temperature gradients, which causes variations in the surface tension creating thermocapillary forces on the interphase surface. The model equation of state for the free surface of a liquid film is a nonlinear fourth-order PDE, which we solve by the finite difference method. The results of computer simulations of the influence of temperature gradients on the nonlinear development of perturbations on the free surface of a liquid film showed that the Marangoni effect both strengthens perturbations and the possibility of rupture and suppresses perturbations. When the grow rate of perturbations increases, liquid films with small Reynolds numbers are the most resistant to temperature gradients.

Keywords

liquid film; temperature gradients; interfacial instability; Marangoni instability.

References

1. Kholpanov L.P., Chkadov V.Ya. Gidrodinamika i teplomassoobmen s poverkhnost'yu razdela [Hydrodynamics and Heat-Mass Exchange with the Surface of Section]. Moscow, Nauka, 1990. (in Russian)
2. Prokudina L.A., Vyatkin G.P. [Instability of non-Isothermal Liquid Films]. Doklady Akademii Nauk, 1998, vol. 362, no. 6, pp. 770-772. (in Russian)
3. Linde H., Schwarz P., Wilke H. Dissipative Structures and Nonlinear Kinetics of Marangoni-Instability. Lecture Notes in Physics, 1979, no. 105, pp. 75-120.
4. Schwarz P., Bielcki J., Linde H. Origin and Behaviour of a Dissipative Structure of Marangoni Instability. Z. Phys. Chem., 1985, vol. 266, pp. 731-739.
5. Ivanilov U.P. [Rolling Waves in an Inclined Channel]. Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki [Computational Mathematics and Mathematical Physics], 1961, vol. 1, no. 6, pp. 1061-1076. (in Russian)