Thermocapillary flow

The complexity of detailed calculations of thermocapillary flows is the coupling between the velocity and temperature fields, primarily by means of the boundary condition, (2-146). To determine gradvy (and thus u by means of the equation of motion), we need to solve the thermal energy equation (2-110) to determine the temperature distribution. However, to do that, we generally need u because it appears as a coefficient in (2-110). We shall later consider some simple problems of this type. 

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... 

The last of the three problems in Fig. 6-7 is qualitatively related to the thermocapillary flows thatare important in the processing of single crystals for microelectronics applications. A typical configuration is sketched in Fig. 6 8, in which a cylindrical solid passes through a heating coil (a furnace), is melted, and then resolidifies into a single crystal of high quality. 

Figure 5.4. Streamlines and the profile of the longitudinal velocity component for thermocapillary flow in a liquid layer...

To study special features of the thermocapillary flow, we consider an approximate analytical solution of the problem at small Marangoni numbers under the assumption that the Prandtl number is of the order of 1. 

In Figure 5.4, the streamlines of the thermocapillary flow (5.9.24) are shown together with the profile of the longitudinal dimensionless component u = Vx/U of the flow velocity. The arrow directions correspond to the case Ma > 0. 

Spatial gradients in surface tension may arise from a variety of causes, including spatial variations at the interface in temperature (Eq. 10.1.3), in surface concentrations of an impurity or additive (Eq. 10.1.4), or in electric charge or surface potential. The resulting flows are termed, respectively, thermocapillary flows, diffusocapillary flows, and electrocapillary flows. We shall limit our discussion of electrocapillary phenomena because of space restrictions but instead refer the reader to Levich (1962) and Newman (1991). 

PIMPUTKAR, S.M. OSTRACH, S. 1980. Transient thermocapillary flow in thin liquid layers. Phys. Fluids 23, 1281—1285. 

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. 

Gill et al. (14) have shown by numerical computation that Equation 29e is a good approximation to a constant heat flux for fluids with finite values of Pr which are typical of liquid metals. The following discussion applies to all Pr fluids, but low Pr is the category that includes essentially all fluids of interest in semiconductor technology as well as all liquid metals. On the other hand, the most complete data on thermocapillary flows in molten zones has been reported by Preisser, Scharmann and Schwabe (15) and Schwabe and... 

Mashayek and Ashgriz [98] considered effects of the heat transfer from the liquid to the surrounding ambient, the liquid thermal conductivity, and the temperature-dependent surface tension coefficient on the jet instability and the formation of satellite drops. Two different disturbances were imposed on the jet. In the first case, the jet is exposed to a spatially periodic ambient temperature. In addition to the thermal boundary condition, an initial surface disturbance with the same wave number as the thermal disturbance is also imposed on the jet. Both in-phase and out-of-phase thermal disturbances with respect to surface disturbances are considered. For the in-phase thermal disturbances, a parameter set is obtained at which capillary and thermocapiUary effects can cancel each other and the jet attains a stable configuration. No such parameter set can be obtained when the thermocapillary flows are in the same direction as the capillary flows, as in the out-of-phase thermal disturbances. In the second case, only an initial thermal disturbance is imposed on the surface of the liquid while the ambient temperature is kept spatially and temporally uniform (Fig. 1.20). 

Thermocapillary Flows and Interface Deformations Produced by Localized Laser Heating... 

Chraibi H, Delville JP (2012) Thermocapillary flows and interface deformations produced by localized laser heating in conflned environment. Phys Fluids 24 032102... 

Fig.1 Dimensionless coordinates for the description of thermocapillary flow around a liquid-vapor meniscus...

Thermocapillarity, Fig. 2 Thermocapillary flow induced by a vertical temperature gradient has been shown to counter the bubble rise due to gravity... 

These results are complicated by the fact that most experiments with drops involve the presence of surfactants which serve to stabilize droplets and aid in their formation. However, surfactant dynamics and their solubility also vary with temperature. Since the effect of surfactant adsorption can be more dramatic than that of temperature, the surfactant dynamics can greatly modify the effect of temperature gradients. Indeed anomalous thermocapillary flow was... 

Since thermocapillary flow occurs from regions of low to high surface tension, the direction of flows will occur from high temperature regions to low temperature when (dy/dT) is positive and from high to low temperature when (dy/dT) is negative. Thus the direction of flow is dependent upon the sign of (dy/dT). 

These are, for the most part, thermocapillary forces but diffusocapillary forces can arise when welding steels with different sulphur contents (see Section 4.1.7). The direction of the thermocapillary flow is determined by the concentration of O or S in the alloy. 

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 an electric field is applied to a system consisting of droplets of liquid phase B present in liquid A, electrocapillary forces can bring about the movement of these droplets. These forces can be used to recover trace metals and metal mattes from waste pyrometallurgical slags [47]. The driving force in thermocapillary flows is (dy/dT) i.e. the temperature dependence of the surface tension. Whereas in electrocapillarity the driving force is (dy/dE) where E is the electrical potential at constant chemical potential X, and (dy/dE) is equal [47] to the surface excess charge density (qi) at the droplet interface (Equation 18). 

---