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Thermocapillary flow in a liquid layer at minimum in surface tension. (English) Zbl 0927.76024

The problem of thermocapillary (Marangoni) flow in a liquid layer has been investigated by many researchers and under various starting assumptions. In this paper, the authors obtain exact similarity solutions to the steady two-dimensional Navier-Stokes equations and the temperature equation for thermocapillary flow in a thin viscous layer on a horizontal rigid plane. The upper free surface of the layer, that is exposed to the constant property motionless ambient gas, is assumed flat and non-deformable, which is a crucial assumption for the existence of similarity solutions. Thermally the liquid layer is subjected to temperature gradients in three distinguished ways, both in transverse as well as in the streamwise direction. The surface tension is assumed to have quadratic dependence on temperature, whereas all other physical characteristics of the liquid are supposed to be constant. The gravitational force is retained in the Navier-Stokes equations, while buoyancy forces and the dissipation term in the energy equation are disregarded. Contrary to the usual convention, in this paper \(M\) stands for the Reynolds number \(Re\) in the dimensionless Navier-Stokes equations, and it is (for this reviewer misleadingly) named the Marangoni number. Flow patterns and temperature fields are presented for different \(M\) and for Prandtl number \(Pr=1\). From the numerical investigation the authors have concluded that for \(M\) larger than a certain value, solutions of the problem do not exist.

MSC:

76D45 Capillarity (surface tension) for incompressible viscous fluids
76R05 Forced convection
80A20 Heat and mass transfer, heat flow (MSC2010)
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[1] Vochten, R., P?tr?, G. J.: Study of the heat of reversible adsorption at the air-solution interface. J. Colloid Interface Sci.42, 320-327 (1973). · doi:10.1016/0021-9797(73)90295-6
[2] Legros, J. C.: Problems related to non-linear variations of surface tension. Acta Astron.13, 697-703 (1986). · doi:10.1016/0094-5765(86)90020-2
[3] Limbourg-Fontaine, M. C., P?tr?, G., Legros, J. C.: Effects of a surface tension minimum on thermocapillary convection. PCH Physico Chem. Hydrodyn.6, 301-310 (1985).
[4] Legros, J. C., Limbourg-Fontaine, M. C., P?tr?, G.: Surface tension minimum and Marangoni convection. In: Fluid dynamics and space, Proceedings of ESA Symp., Rhode-Saint-Genese, 25-26 June (ESA SP-265) pp. 137-143 (1986).
[5] Platten, J. K., Villers, D.: On thermocapillary flows in containers with differentially heated side walls. In: Physicochemical hydrodynamics. Interfacial phenomena (Velarde, M. G., ed.), pp. 311-336. New York: Plenum Press 1988.
[6] P?tr?, G., Azouni, M. A., Tshinyama, K.: Marangoni convection at alcohol aqueous solution-air interfaces. Appl. Sci. Res.50, 97-106 (1993). · doi:10.1007/BF00849547
[7] Napolitano, L. G., Golia, C., Viviani, A.: Effects of non-linear tension on combined free convection in cavities. L’Aerotechnica63, 29-36 (1984). · Zbl 0559.76083
[8] Villers, D., Platen, J. K.: Marangoni convection in systems presenting a minimum in surface tension. PCH Physico Chem. Hydrodyn.6, 435-451 (1985).
[9] Dubovik, K. G., Slavtchev, S. G.: Numerical modeling of thermocapillary convection in a liquid layer at a non-linear dependence of the surface tension on temperature. Mech. Zhidkosti Gaza,1, 138-143 (1991) (in Russian).
[10] Slavtchev, S. G., Dubovik, K. G.: Thermocapillary convection in a rectangular cavity at minimum of surface tension. Theor. Appl. Mech. (Sofia)23, 85-90 (1992).
[11] Pukhnatchov, V. V.: Display of anomalous thermocapillary effect in a thin liquid layer. In: Hydrodynamics and heat mass transfer in liquids with free surfaces, (Schreiber, I. R., ed.), pp. 119-127. Novosibirsk: Inst. Teplofiziki, 1985 (in Russian).
[12] Pukhnatchov, V. V.: Thermocapillary convection in weak force fields. Preprint No. 178-88, Inst. Teplofiziki, Novosibirsk, 1988 (in Russian).
[13] Cloot, A., Lebon, G.: Marangoni convection induced by a nonlinear temperature-dependent surface tension. J. Phys.47, 23-29 (1986).
[14] Gupalo, Yu. P., Ryazantsev, Yu. S.: Thermocapillary motion of a liquid with free surface with non-linear dependence of the surface tension on the temperature. Fluid Dyn.23, 752-757 (1989). · Zbl 0677.76034 · doi:10.1007/BF02614155
[15] Oron, A., Rosenau, Ph.: On a nonlinear thermocapillary effect in thin liquid layers. J. Fluid Mech.273, 361-374 (1994). · Zbl 0825.76240 · doi:10.1017/S0022112094001977
[16] Gupalo, Yu. P., Ryazantsev, Yu. S., Skvortsova A. V.: Effect of thermocapillary forces on free surface fluid motion. Fluid Dyn.24, 657-661 (1990). · Zbl 0705.76034 · doi:10.1007/BF01051714
[17] Brady, J. F., Acrivos, A.: Steady flow in a channel or tube with an accelerating surface velocity. An exact solution to the Navier-Stokes equations with reverse flow. J. Fluid Mech.112, 127-150 (1981). · Zbl 0491.76037 · doi:10.1017/S0022112081000323
[18] Davis, S. H.: Thermocapillary instabilities. Annu. Rev. Fluid Mech.19, 403-435 (1987). · Zbl 0679.76052 · doi:10.1146/annurev.fl.19.010187.002155
[19] Schlichting, H.: Boundary layer theory. New York: McGraw-Hill, 1968. · Zbl 0096.20105
[20] Terril, R. M.: Laminar flow in a uniformly porous channel. Aeron. Q.15, 299-310 (1964).
[21] Terril, R. M., Thomas, P. W.: On laminar flow through a uniformly porous pipe. Appl. Sci. Res.21, 37-67 (1969). · Zbl 0179.56904 · doi:10.1007/BF00411596
[22] Robinson, W. A.: The existence of multiple solutions for laminar flow in a uniformly porous channel with suction at both walls. J. Eng. Math.10, 23-40 (1976). · Zbl 0325.76035 · doi:10.1007/BF01535424
[23] Szaniawski, A.: Quasi-isobaric solutions of the Hiemenz equation. Arch. Mech.45, 689-725 (1993). · Zbl 0809.76018
[24] Samarsky, A., Nikolaev, E.: Methods for solving finite-difference equations. Moscow: Nauka 1978 (in Russian).
[25] Riabouchinsky, D.: Quelques considerations sur les mouvements plans rotationnels d’un liquide. C. R. Acad. Sci.179, 1133-1136 (1924). · JFM 50.0683.03
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