Convection Marangoni driven

Figure 5.30 Schematic of Marangoni-driven convection inside a droplet due to temperature difference...

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... 

Figure 3.3. Various features of diffusion and convection associated with crystal growth in solution (a) in a beaker and (b) around a crystal. The crystal is denoted by the shaded area. Shown are the diffusion boundary layer (db) the bulk diffusion (D) the convection due to thermal or gravity difference (T) Marangoni convection (M) buoyancy-driven convection (B) laminar flow, turbulent flow (F) Berg effect (be) smooth interface (S) rough interface (R) growth unit (g). The attachment and detachment of the solute (solid line) and the solvent (open line) are illustrated in (b).

Single crystal silicon is one of the important fundamental materials for the modern photovoltaic industry. The Czochralski method of growing single crystal silicon is affected by the thermocapillary convection. Temperature and concentration gradients at the free surface of the melt give rise to surface tension-driven Marangoni flow, which can lead to crystal defects, if it is sufficiently large. 

Reichenbach, J. and Linde, H., Linear perturbation analysis of surface-tension-driven convection at a plane interface (Marangoni instability), J. Colloid Interface Sci., 84, 433 143, 1981. Nepomnyashchy, A.A., Velarde, M.G., and Colinet, R, Interfacial Phenomena and Convection, CRC Press/Chapman Hall, London, 2002. 

In the opposite case, when the surfactant is soluble in the continuous phase, the Marangoni effect becomes operative and the rate of film thinning becomes dependent on the surface (Gibbs) elasticity (see Equation 5.282). Moreover, the convection-driven local depletion of the surfactant monolayers in the central area of the film surfaces gives rise to fluxes of bulk and surface diffusion of surfactant molecules. The exact solution of the gives the following... 

Finally, we consider the problem of Marangoni instability, namely convection in a thin-fluid layer driven by gradients of interfacial tension at the upper free surface. This is another problem that was discussed qualitatively in Chap. 2, and is a good example of a flow driven by Marangoni stresses. 

Three examples of shallow-cavity flows that we consider in this section are sketched in Fig. 6-7. At the top is the case in which all four boundaries are solid walls, the fluid is assumed to be isothermal, and the motion is driven by tangential motion of the lower horizontal boundary. In the middle, a generalization of this problem is sketched in which the fluid is still assumed to be isothermal and driven by motion of the lower horizontal boundary, but the upper boundary is an interface with air that can deform in response to the flow within the cavity. Finally, the lower sketch shows the case in which fluid in the shallow cavity is assumed to have an imposed horizontal temperature gradient, produced by holding the end walls at different, constant temperatures, and the motion is then driven by Marangoni stresses on the upper interface. In the latter case, there will also be density gradients that can produce motion that is due to natural convection, but this contribution is neglected here (however, see Problem 6-13.)... 

C. V. Stemling and L. E. Scriven, Interfacial turbulence hydrodynamic instability and the Marangoni effect, AIChE J. 5, 514 (1959) L. E. Scriven and C. V Stemling, On cellular convection driven by surface tension gradients effects of mean surface tension and surface viscosity, J. Fluid Mech. 19, 321 (1964). 

Problem 12-15. Stability of a Fluid Layer in the Presence of Both Marangoni and Buoyancy Effects. A fluid layer is heated from below. It has a rigid, isothermal boundary at the bottom, but its upper surface is a nondeforming fluid interface. There are now two potential mechanisms for instability when the fluid is heated from below buoyancy-driven and surface-tension-gradient-driven convection. Determine the eigenvalue problem (i.e., the ODE or equations and boundary conditions) that you would need to solve to predict the linear instability conditions. Is the principle of exchange of stabilities valid Discuss how you would approach the solution of this eigenvalue problem. 

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. 

Recently, Sen and Davis (20) studied capillary flow in bounded cavities in which d/ is small, end effects are present, and the flow is very slow and the cavity is heated from the side. Cowley and Davis (21) studied the high Marangoni number Thermocapillary analogue of a buoyancy driven convection problem solved by Roberts (22). Later, we shall make some comparisons between our results for the deflection of the surface and those of Sen and Davis (20). 

Lewis Pratt, in 1953, were the first to report that the observed Marangoni convection in their experimental ternary systems was beneficial to hquid-hquid extraction processes because it increased mass transfer rates. The effect of density gradients on interfacial convection was studied by several researchers including Berg Morig (1969), who investigated the interaction between buoyancy and interfacial tension driven effects in ternary systems. The combined interfacial convection was also seen to be beneficial to mass transfer processes. 

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,... 

The presence of surfactants, which adsorb at the liquid-vapor interface and reduce the surface tension, can also have a large impact on the evaporation-driven pattern formation. Inhomogeneities in the surfactant distribution create a surface tension gradient and a corresponding (additional) Marangoni flow. With respect to the Rayleigh-Benard convection, it has been shown that the surfactant-driven flow can favor the formation of convection cells and considerably alter the deposition patterns [7]. 

Surface tension is affected both by chemical concentration and by temperature. Figure 9.10 shows how a hot spot can cause convection by locally lowering the surface tension. The cooler fluid has a higher surface tension and draws the warm fluid towards itself. If the temperature gradient is perpendicular to the interface, both buoyancy-driven convection and Marangoni (surface-tension-driven) convection are possible (Antar and Nuotio-Antar, 1993). 

Fundamental studies on the kinetics of crystal nudeation and growth in glassy melts under the influence of surface tension-driven (Marangoni) convection that, under Ig conditions, would be superseded by gravity-driven (B nard or Rayleigh) convection. 

If there is a free interface between fluids, gradients in concentration and/or temperature parallel to the interface cause gradients in the surface (interfacial) tension, which cause convection [85]. This convection, also known as Marangoni convection, is especially noticeable in thin layers (or weightlessness) in which buoyancy-driven convection is greatly reduced. 

Now, it is important to note that in the case of the Benard-Marangoni convection in a liquid layer with a deformable interface, as was previously shown by Takashima (1981a) through a linear stability analysis (see the 7), there exist two monotonous modes of surface tension driven instability. 

If a system lacks an interface between different fluids, such a monomer/air interface, then only buoyancy-driven convection will occur. If a free interface exists, then we will see that gradients in the interfacial tension can cause fluid motion — a process called Surface-Tension Induced Convection or Marangoni convection. This will be especially important in "microgravity". (How the condition of apparently zero gravity is achieved is discussed in chapter 2.)... 

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