Thermocapillary motion

The N-S equation for incompressible flow with constant viscosity is 

Assuming steady, 2-D, negligible inertial terms and with lateral velocity gradients small compared to vertical gradients, the momentum equation in the x-direction becomes 

The z-momentum equation assuming vanishing z-component velocity due to absence of any driving force in this direction becomes 

It should be recognized that the liquid surface layer is set in motion by the surface tension force which is balanced by the motion of the fluid in the opposite direction below the surface. Thus, there is no net flux across any cross-section, and the continuity equation gives 

Note that the air viscosity being negligibly small compared to that of the liquid, the viscous force from air is neglected in the above equation. 

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Similar analyses are available for surface-tension-driven flows in a slender cavity with the additional assumption that the meniscus at the top of the cavity is also flat (36). Smith and Davis (37-39) have used this configuration to study the stability of the flow with respect to wavelike instabilities (see also reference 40). Homsy and co-workers (41, 42) have analyzed the effect of a surface-active agent on the thermocapillary motion in a slender cavity. 

Figure 7-17. Streamlines for thermocapillary motion of a gas bubble for a = 0.8 (a2pg//3), where the bubble velocity is reduced to 20% of its value in the absence of thermocapillary effects. The stream-function values are calculated from Eq. (7-244) with coefficients C and D from Eqs. (7-248). Contour values are plotted in increments of 0.7681.

The thermogravitational motion is described in the Boussinesq approximation in which the variable density in the equations of motion (5.9.1)—(5.9.3) and in the convective heat conduction equation (5.9.4) is taken into account only in the Archimedes term (the last term in (5.9.2)). This term is proportional to the temperature deviation T from the mean value. The thermocapillary motion... 

Statement of the problem. Let us consider the problem of a steady-state thermocapillary motion in a liquid layer of thickness h. The motion is assumed to be two-dimensional. The dependence of the surface tension on temperature is assumed to be quadratic according to (5.9.19). The thermogravitational effect is not taken into account. It is assumed that the linear temperature distribution is maintained on the hard lower surface, and the plane surface of the layer is thermally insulated. The origin of the Cartesian coordinates X, Y is placed on the solid surface at the point with temperature To. The velocity and temperature fields are described by Eqs. (5.9.1)-(5.9.4) with jg = 0. 

Some additional effects were considered for the thermocapillary motion of drops and bubbles in an external temperature gradient interaction of a drop with a plane wall [285], and interaction of drops with bubbles or of bubbles with each other [12, 146], In particular, it was shown in [12] that the interaction of drops of radius a decreases with increasing distance l between them as (a/i f for thermocapillary drift, compared with a/l for the motion in the gravitational field. 

Radiation-induced thermocapillary motion of a drop. The temperature gradient is the simplest but not the unique method for bringing about the thermocapillary drift of a drop. If the drop is opaque and the fluid is transparent, one can move the drop by a light beam in a uniformly heated fluid. The radiation absorbed by the drop will heat it nonuniformly, thus producing thermocapillary stresses. For dcr/dT < 0, the drop will drift towards the warmer part, that is, towards the beam. 

Remark. The problem of mass transfer to a drop for the diffusion regime of reaction on its surface under the conditions of thermocapillary motion is stated in the same way as in its absence (see Section 4.4) taking into account the corresponding changes in the fluid velocity field. In [144], a more complicated problem is considered for the chemocapillary effect with the heat production, which was described in [147-149,419], It was assumed that a chemical reaction of finite rate occurs on the drop surface. 

Gupalo, Yu. P. and Ryazantsev, Yu. S., Thermocapillary motion of a liquid with a free surface with nonlinear dependence of the surface tension on the temperature, Fluid Dynamics, Vol. 23, No. 5, pp. 752-757, 1988. 

Rednikov, A. E. and Ryazantsev, Yu. S., On thermocapillary motion of a drop under the action of radiation, J. Appl. Mech. Techn. Phys., No. 2,1989. 

Szymczyk, J. and Siekmann, J., Numerical calculation of the thermocapillary motion of a bubble under microgravity, Chem. Eng. Comm., Vol. 69, pp. 129-147, 1988. 

Cartesian coordinate normal to direction of wave motion on free surface or thermocapillary motion in pan Cartesian coordinate transverse to direction of motion of spherical particle and translating with it Charge number... 

Figure 44 Schematic diagram showing the thermocapillary motion and circulation flows in the molten salt layer on the turbine blade [68].

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