Part 1
SUMMARY
Almost 150 years ago Weber [153], Thomson [145] and Marangoni [71] investigated the effect of spreading of a liquid with a lower surface tension over another liquid with a higher surface tension. So-called ”wine tears”, a phenomenon when in a wine glass, due to convection caused by variations of surface tension resulted from the evaporation of alcohol, the wine climbs up along the glass as a thin film, reaches a certain height and accumulates. Gravity causes the wine to flow down in the shape of ”tears” [145]. Surface tension - an important localized property - is represented as the magnitude of the force per unit length normal to a cut in the interface, or the free energy per unit area. It arises on the interface between two immiscible fluids or at a fluid-gas interface and can be attributed to unbalanced molecular attraction which tends to pull molecules into the interior of a liquid phase, and hence to minimize the surface area [35].
This results in a higher potential energy for the molecules at the interface. Consequently the interface tends to get its area reduced in order to minimize the potential energy, as if it were in a state of tension like a stretched membrane [129].
The surface tension is a function of many properties of liquid, e.g. temperature and con- centration. Any variations in temperature or composition result in local variations of surface tension along a liquid free surface and thus induce convective motion. This effect is called Marangoni convection. Thermodynamically the spreading of a liquid means that the interface will tend to assume a state of lower surface energy and does so by the spreading of areas of lower interfacial tension. Present thesis considers only temperature-induced convection, often called thermocapillary convection. In a terrestrial environment, Marangoni convection is usually overshadowed by buoyancy-driven flow. In the reduced gravity environment, however, buoyancy is greatly reduced and Marangoni convection could become very important. There are cases when the thermocapillary flows ruined experiments in fluid dynamics, e.g. due to presence of a bubble inside experimental cell [121], see detailed investigation of this problem in Appendix 10.
The ratio between the two factors, the so-called dynamic Bond number, defines whether gravity or thermocapillary forces are dominant.
In fluid convection with a free surface it is sometimes the case that surface tension effects become important even under normal gravity environment. For instance in such configurations as crystal growth by Bridgeman or floating-zone method the gravitational forces are negligible [17, 140]; Or, as in B´enard’s original experiments, the layer of fluid is very thin and the convection is surface-tension dominated [7, 88]. The criterion for the dominant thermocapillary convection over buoyancy is ratio between thermocapillary forces to buoyant forces written as:
Gr
Re4/3 = Bo Re1/3 <1,
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1.0 Summary
where Bo= Gr/Re =ρgβH2/σT is Bond number, definitions of the Grashof Gr and thermo- capillary Reynolds Re numbers and of the other terms are given below. Kamotani et al. [46]
measured critical temperature difference using containers of different sizes filled by P r = 27 liquid. They found an upper limit under which the flow is thermocapillary dominated, it is Bo≤0.24. Using liquids with different properties, the Bond number should be recalculated.
It has been recognized that thermocapillary Marangoni effects may be an important factor influencing convection instability in thin fluid layers [134, 26, 95, 109, 128]. Surface tension effects in heated fluid layers are becoming increasingly important in current research on micro- hydrodynamics where fluid layers are thin and surface tension effects are significant, e.g. they are of industrial interest in applications to microelectronic fabrication (such as dewetting). Fur- ther, these Marangoni flows arise in many diverse application areas such as coating, electronic cooling, welding processes, wafer drying, biofluidic chip fabrication and medicine delivery. This is a reason why much attention has been given in recent years to Marangoni flows.
Three different approaches can be found in the literature for a theoretical description of the influence of the Marangoni convection on mass transfer [39]. A detailed review about experi- mental studies concerning interfacial convections is given in [89, 105]. Here they are.
Analytical and numerical studies are based on linear or nonlinear differential equations which are solved for simplified boundary conditions. These theories predict the stability behavior of liquid-liquid and gas-liquid systems. The best known model of this kind has been proposed by Sternling and Scriven in [136] for quiescent and infinitely extended phases with a plane interface.
Mass transfer coefficients are calculated by semi-empirical correlations. Generally, these models modify the film theory of Lewis and Whitman in [67] or the surface renewal model of Higbie in [41]. A limited number of parameters is used in these models [96, 112].
Empirical correlations describe the relationship between mass transfer coefficient and inter- facial convection. These equations can only be applied to the corresponding test apparatus, systems and conditions studied.
Concerning the crystal growth from melt, much work has been done in the past, both exper- imentally and theoretically. Most of the experimental investigations were carried out in normal gravity but some results from microgravity experiments are now available. High Prandtl num- ber (Pr) fluids have been used in most experiments. Although Pr of crystal melts is generally smaller than unity, much less experimental information is available for low Pr fluids due to some experimental difficulties. The transport phenomena in the floating-zone melt have been simulated by many investigators in the so-called liquid bridge configuration, in which a liquid column is suspended vertically between two differentially heated rods.
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