Number Marangoni

The effect can be important in mass-transfer problems (see Ref. 57 and citations therein). The Marangoni instability is often associated with a temperature gradient characterized by the Marangoni number Ma ... 

Should there be Marangoni instability in a layer of water 1.5 mm deep, with the surface 0.3°C cooler than the bottom What dimensions does the Marangoni number (Ma) have (Assume T 25°C.)... 

Circulation of fluid is promoted by surface tension gradients but inhibited by viscosity, which slows the flow, and by molecular diffusion, which tends to even out the concentration differences. The onset of instabibty is described by a critical Marangoni number (Mo), an analogue of the Rayleigh... 

Linear stability analysis has been successfully applied to derive the critical Marangoni number for several situations. 

When surface tension differences appear or are produced between some points or some small regions of an interface, the flow produced is called the Marangoni flow or flow with Marangoni effect. The Marangoni number, used to characterize the flow shown on Fig. 6.9, is a combination of the Reynolds number, the Weber number and the Schmidt number ... 

To quantify the impact of Marangonl effects, as compared to those of the interfacial tension only, sometimes the Marangoni number Ma is used. It is defined as... 

Kj / y, where K is the surface dilationcil modulus, defined in 13.6.19). An alter-native Marangoni number was introduced by Edwards et al., who considered creep flow around an emulsion droplet. Their definition is Ma s K° /kar/, where a is the radius of the droplet, rj the bulk viscosity and fc (in s ) a rate constant, characteristic of the rate of supply of surfactants to the interface by transport from the bulk. The second definition rather applies to Gibbs monolayers it is a measure of the extent to which surface tension gradients can develop against the counteracting replenishment of the surface. 

This defines the Marangoni number Ma. The expression for the slenderness ratio of our gravity-driven film is again used in Equation 9. We call this Criterion B. Having now introduced the reference velocity U., we note that Criterion A as expressed by Equation 7. is the ratio U j./Uj.. 

Croell, A. Miieller-Sebert, W. Nitsche, R. Critical Marangoni number for the onset of time-dependent convection in silicon. Mater. Res. Bull. 1989, 24, 995-1004. 

Stemling and Scriven wrote the interfacial boundary conditions on nonsteady flows with free boundary and they analyzed the conditions for hydrodynamic instability when some surface-active solute transfer occurs across the interface. In particular, they predicted that oscillatory instability demands suitable conditions cmcially dependent on the ratio of viscous and other (heat or mass) transport coefficients at adjacent phases. This was the starting point of numerous theoretical and experimental studies on interfacial hydrodynamics (see Reference 4, and references therein). Instability of the interfacial motion is decided by the value of the Marangoni number, Ma, defined as the ratio of the interfacial convective mass flux and the total mass flux from the bulk phases evaluated at the interface. When diffusion is the limiting step to the solute interfacial transfer, it is given by... 

Whatever the boundary conditions, the aspect and the regnlarity of the pattern depend on the shape, the size of the container, and the local value of the Marangoni number. 

The dimensionless parameter that appears in this boundary condition is known as the Marangoni number and is defined as... 

Figure 12-8. Neutral stability curves from Eq. (12-300) for three different values of Bi. The critical Marangoni number for each Bi is the minimum value over the range of possible values for a.

Now, for each value of Bi, we can plot the neutral stability curve, as shown in Fig. 12-8 for Bi = 0, 2, and 4. The critical Marangoni numbers for these three cases are approximately 80, 160, and 220. As noted earlier, the system is stabilized by increase of Bi because this leads toward an isothermal interface, and thus cuts the available Marangoni stress to drive convection. The critical wave numbers for these three cases are, respectively, 2.0, 2.3, and 2.5. 

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. 

Bratukhin, Yu. K., Flow of an inhomogeneously heated fluid past a gaseous bubble at small Marangoni numbers, J. Eng. Phys. Thermophys., Vol. 32, No. 2, 1977. 

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