Interface Marangoni number

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 ... 

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. 

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... 

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. 

Nakache E., and Raharimalala S. (1988). Interfacial Convection Driven by Surfactant Compounds at Liquid Interfaces Characterisation by a Solutal Marangoni Number. In Velarde M G, editor. Physicochemical Hydrodynamics Interfacial Phenomena. Plenum Press, New York and London,... 

Fig. 3. Influence of the electric field at the interface of two unmixing electrolyte solutions upon the instabilization of phase boundary. The curves 1, 2, 3 correspond to three fixed values of the Marangoni number (Ma=5 20 50) for...

The hydrodynamic instability leading to convective flow in the biochemical systems is driven by unbalanced forces (surface tension) at the liquid/gas interface, mainly caused by temperature gradients due to evaporative cooling [1,4]. Our experiments show that the chemical composition of the solution has to be accounted for as well. The significant parameters for the onset of pattern formation are the thermal and the solutal Marangoni numbers. Both are also important for spatial patterning in biochemically reactive liquid layers. 

E, Nakache, S. Raharimalala, Interfacial convection driven by surfactant compounds at liquid interfaces. Characterization by a solutal Marangoni number , in Proc. on Physicochemical Hydrodynamics Interfacial, Spain, July 1986. M.G. Verlade and B. Nichols, eds., Plenum Press, New York, 1987... 

With an open interface several other parameters enter the problem. Surface tension tractions must be considered if there is variation of surface tension with either temperature or solute (an impurity). This is accounted with the inclusion of the thermal and solutal Marangoni numbers (the latter is usually called the Elasticity),... 

Thus, to a first-order approximation with positive Marangoni numbers the deformation of the interface reduces the region of finite amplitude instability. 

Figure 2(b).Zero-gravity Benard-Marangoni convection. First-order corrections in the thermal Marangoni numbers produced by the deformation of the interface.To compute the actual Marangoni number one uses Equation (28). 

In this case, the Ac in Marangoni number, which represents the intensity of Marangoni convection, can be expressed by the interfacial solute concentration difference per unit length of interface as follows ... 

Similar to the temperature-driven Marangoni convection, concentration-driven Marangoni convection is also present in microfluidics. When surfactants are added to the fluid, they migrate to the interface, and a gradient of interface concentration may occur, leading to a convective motion at the interface. In the case of concentration-driven convection, the nondi-mensional Marangoni number is defined as... 

A more common source of Marangoni effects in systems of interest to chemical engineers is surfactants, as discussed in Chap. 2. This is particularly pertinent to the motion of gas bubbles (or drops) in water, or in any liquid that has a large surface tension (the surface tension of a pure air-water interface is approximately 70 dyn/cm). Experiments on the motion of gas bubbles in water at low Reynolds numbers show the perplexing result illustrated in Fig. 7-18. For bubbles larger than about 1 mm millimeter in diameter, the translation velocity is approximately equal to the predicted value for a spherical bubble with zero shear stress at the interface, that is,... 

In this section, we consider the classic problem of a fluid layer of depth d, with an upper surface that is an interface with air that is maintained at an ambient temperature 7o. The fluid layer is heated from below, and we shall assume that the lower fluid boundary is isothermal with temperature T (> To). This problem sounds exactly like the Rayleigh-Benard problem with a free upper surface. However, we consider the fluid layer to be very thin (i.e., d small) so that the Rayleigh number, which depends on d3, is less than the critical value for this configuration. Nevertheless, as previously suggested, the fluid layer may still undergo a convective motion that is due to Marangoni instability. 

---