Stability critical Marangoni number

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

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. 

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) ... 

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

The interactions between an oil phase and foam lamellae are extremely complex. Foam destabilization in the presence of oil may not be a simple matter of oil droplets spreading upon foam film surfaces but may often involve the migration of emulsified oil droplets from the foam film lamellae into the Plateau borders where critical factors, such as the magnitude of the Marangoni effect in the pseudoemulsion film, the pseudoemulsion film tension, the droplet size and number of droplets may all contribute to destabilizing or stabilizing the three phase foam structure. 

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