Flows Driven by Surface Tension Gradients

The final example of a shallow-cavity problem that we consider is the flow driven by surface-tension gradients at the upper surface, which is again assumed to be an interface as sketched in Fig. 7(c).13 The surface-tension gradient is produced when a fixed temperature differential is maintained between the two walls at x = 0 and L. 

Arguably the most important application of sol-gel technology is the preparation of films to exploit their optical or electronic properties or their chemical or mechanical resistance. Hurd (Sandia National Laboratories) examines the physics of film deposition by dip coating, elucidating the factors that control the thickness and structure of the film. Direct observation of the deposition process reveals the kinetics of the drying process and the existence of flows driven by surface tension gradients when binary solvents are used. Extensive discussion of the preparation of, and applications for, such films are presented in references 4 and 5. 

A drop of surfactant solution will, under certain conditions, undergo a fingering instability as it spreads on a surface [27, 28]. This instability is attributed to the Marongoni effect (Section IV-2D) where the process is driven by surface tension gradients. Pesach and Marmur have shown that Marongoni flow is also responsible for enhanced spreading... 

The flow can be forced, e.g., by a pump or an impeller, it can be driven by gravity or even by surface tension gradients. In most cases, the flow is too complex to be analysed theoretically, and empirical correlations are used for predicting values of the mass transfer coefficient for given conditions. These correlations are usually presented as functions of dimensionless groups, in order to generalize the applicability. 

Convection in Melt Growth. Convection in the melt is pervasive in all terrestrial melt growth systems. Sources for flows include buoyancy-driven convection caused by the solute and temperature dependence of the density surface tension gradients along melt-fluid menisci forced convection introduced by the motion of solid surfaces, such as crucible and crystal rotation in the CZ and FZ systems and the motion of the melt induced by the solidification of material. These flows are important causes of the convection of heat and species and can have a dominant influence on the temperature field in the system and on solute incorporation into the crystal. Moreover, flow transitions from steady laminar, to time-periodic, chaotic, and turbulent motions cause temporal nonuniformities at the growth interface. These fluctuations in temperature and concentration can cause the melt-crystal interface to melt and resolidify and can lead to solute striations (25) and to the formation of microdefects, which will be described later. 

Finally, we consider the problem of Marangoni instability, namely convection in a thin-fluid layer driven by gradients of interfacial tension at the upper free surface. This is another problem that was discussed qualitatively in Chap. 2, and is a good example of a flow driven by Marangoni stresses. 

This parameter is termed the Marangoni number. As discussed beJow, if Ma exceeds a critical value, an unstable convective flow will develop. The Marangoni number can also be interpreted as a thermal Peclet number (Eq. 3.5.16) if the characteristic velocity for the surface tension driven viscous flow is taken to be that of Eq. (10.5.5). We emphasize that this velocity is not a given parameter but rather a derived quantity. Expressing this velocity in terms of the imposed uniform temperature gradient p, with the aid of continuity, we arrive at Eq. (10.6.10). Interpreted as a Peclet number, the Marangoni number is a measure of the heat transport by convection due to surface tension gradients to the bulk heat transport by conduction. 

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