Marangoni number critical

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. 

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. 

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. 

Y. Fujita and Q. Bai, Critical Heat Flux of Binary Mixtures in Pool Boiling and its Correlation in Terms of Marangoni Number, Proc. EUROTHERM Seminar No. 48 Pool Boiling 2, D. Gorenflo, D. B. R. Kenning, and C. Marvillet eds., Edizioni ETS, Pisa, Italy, pp. 319-326,1996. 

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. 

The neutral stability curves, when Bi = 0 and Bo = 0.1, are shown in above Fig. 7.1 as a sample case. The values of Cr are given in the figure. Since the region below each curve represents stable state, the lowest point of each curve gives the critical Marangoni number Mac and the corresponding critical wave number kc. When Cr 0 (i.e. in the case of a non-deformable free surface, and in a such case Fr2—> 0 also but Bo 0), the result of Pearson s (1958) ... 

It follows from this latter fact that the highest point of each curve gives the critical Marangoni number Mac, for the onset of overstability and the corresponding critical wave number kc. When, according to (2.23), Ma = y d P/p (Vq) < 0, and upside of the layer is a free surface (Bo > 0), then it is necessary that (see (1.3) ) ... 

In the particular case of a zero G (zero Bond, Bo, number), the critical Marangoni number is zero and relation (7.17) yields ... 

The critical Marangoni number is calculated as a function of the Rayleigh, Prandtl, and Biot numbers together with the aspect ratio (i.e., the ratio of the container radius to the mean depth of a liquid. 

The stability criteria for multiple nonlinear steady states near the critical Marangoni number are determined. 

The transitions between two steady convective states of equal probability corresponding to the binary double proper values for the critical Marangoni number are predicted and analyzed. 

By setting different Sc and Bi, the relationship between Ma and kx can be obtained from/(Ma, Bi, Sc, kx) = 0 as shown by the curves in Figs. 8.11 and 8.12. In these figures, any points above the curve are unstable and induce Marangoni convection, while any points below are stable without Marangoni convection. The minimum point of the curve represents the critical Marangoni number Maa- It is also shown in Figs. 8.13 and 8.14 that Ma is affected by both Sc and Bi of the process. 

The intensity of Marangoni convection and Rayleigh convection can be represented by the Marangoni number Ma and Rayleigh number Ra. The onset of convection and orderly interfacial structure only when Ma and Ra reach its critical value Macr and Roa. When Ma and Ra number further increase to a certain extent, the system turns to stable at fully turbulence or chaos state. 

Critical Marangoni number increases with increasing values of Biot number. For a fixed temperature boundary condition at the wall surface and for Bi = 0, critical Marangoni number 80. For fixed heat flux condition and Bi = 0, critical Marangoni number 40. 

They have defined the Marangoni number by Eq. (4). Mic is the critical Marangoni number which denotes the upper limit of mass tr nisfer enhancement, i.e. at Ma < Ma, 1 =1 is valid. The critical Marangoni numbra-... 

To determine the mass transfer enhancement factor R it is advisable to define the Marangoni number by equation (4) using Eq. (7). The critical Marangoni number has to be considercxl as a fonctiem only of the gas - liquid phase resistance ratio B. 

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