Thermocapillary diffusion

Thermocapillary Diffusion. Temperature induced Marangoni flow. The movement of suspended drops or bubbles when subjected to a temperature gradient, due to the resulting surface/interfacial tension gradient. 

Here, we consider only the simpler situation in which the surfactant is assumed to be relatively dilute so that it is mobile on the interface and contributes a change only in the interfacial tension, without any more complex dynamical or rheological effects. In this case, the boundary conditions derived for a fluid interface still apply. Specifically, the dynamic and kinematic boundary conditions, in the form (2 122) and (2-129), respectively, and the stress balance, in the form (2 134), can still be used. However, the interfacial tension, which appears in the stress balance, now depends on the local concentration of surfactant. We shall discuss how this concentration is defined shortly. First, however, we note that flows involving an interface with surfactant are qualitatively similar to thermocapillary flows. The primary difference is that the concentration distribution of surfactant on the interface is almost always dominated by convection and diffusion within the interface, whereas the... 

Further, following [419], we consider in detail another mechanism of the surface tension variability produced in the process of motion. We consider a drop moving at a constant velocity, on whose surface an exothermic or endothermal reaction is involved. It is assumed that a surfactants takes part in the reaction being dissolved in the surrounding liquid. Let the liquid temperature and the concentration of surfactants be constant remote from the drop, while on the interface, the concentration of the surfactants (the reactant) is zero (the diffusion regime of reaction). In this symmetric situation, the temperature variability, and, consequently, the thermocapillary stresses can be produced only in the process of motion of liquids. 

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. 

Several elementary aspects of mass diffusion, heat transfer and fluid flow are considered in the context of the separation and control of mixtures of liquid metals and semiconductors by crystallization and float-zone refining. First, the effect of convection on mass transfer in several configurations is considered from the viewpoint of film theory. Then a nonlinear, simplified, model of a low Prandtl number floating zone in microgravity is discussed. It is shown that the nonlinear inertia terms of the momentum equations play an important role in determining surface deflection in thermocapillary flow, and that the deflection is small in the case considered, but it is intimately related to the pressure distribution which may exist in the zone. However, thermocapillary flows may be vigorous and can affect temperature and solute distributions profoundly in zone refining, and thus they affect the quality of the crystals produced. 

A typical order of magnimde for the number of molecules of interest comes from the number in a mole, or Avogadro s numb, that is, on the order of 10 That many differential equations cannot be solved, van Remoortere et al. (1999) consider no more than 10 molecules (hquid and sohd combined) and solve the equations of motion (for liquids) for 5 x 10 seconds in real time. Because vray few molecules are used, the drop is two dimensional, almost like a cardboard cutout, because it is very thin in the perpendicular direction. Yet the results are very novel. The very tiny drop spreads, showing diffusion-like behavior. Waves are seen on the surface which are reminiscent of thermocapillary waves, suggeshng that we may be observing a fundamental mechanism of drop movement at this scale. 

The strength of the thermocapillary flow is determined by the non-dimensional Marangoni number (Ma) defined in Equation 11 where (dy/dx) is the temperature gradient, Tj is the viscosity, a is the thermal diffusivity and L is the characteristic... 

When the particle relative motion is driven by a body force or by the thermocapillary migration (rather than by self-diffusion). Equation 4.337 is no longer valid. Instead, in Equation 4.336 we have to formally substitute the following expression for (see Rogers and Davis [943]) ... 

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