Marangoni stress

A solvent dissolution, a vapor adsorption, any kind of surface-active substance exchange between the surface and the adjacent subphase, or heating makes the surface tension locally vary, thus generating Marangoni stresses and convection. Then, gravitocapillary waves (wavelength X and amplitude q) excited and sustained by the Marangoni effect in the shallow water waves approximation can be described by the equation ... 

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. 

Figure 6-7. Three configurations for the shallow-cavity problem (a) Four isothermal solid walls with motion driven by tangential motion of the lower wall (b) the same problem as (a) except, in this case, the upper surface is an interface that may deform because of the flow (c) the configuration is the same as (b), except, in this case, the lower wall is stationary and the motion in the cavity is assumed to be driven by Marangoni stresses caused by nonuniform interface temperature that is due to the fact that the end walls are at different temperatures.

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.)... 

The fluid dynamical boundary conditions are similar to those applied in the previous problem, with two notable exceptions. First, u = 0 at z = 0 (i.e., the lower boundary is stationary). Second, the tangential-stress condition is modified to account for the presence of Marangoni stresses that are due to gradients of the interfacial tension at the fluid interface,... 

The characteristic velocity is determined by the ratio of the characteristic tangential (Marangoni) stress, 0(PAT/L), which drives this motion to the viscous forces ()(p,uc/d) that derive from this motion. The definition (6 212) also allows us to return to the condition for neglect of buoyancy forces compared with Marangoni forces as a potential source of fluid motion in the thin cavity. To do this, we introduce the thermal expansion coefficient, which we denote as a, so that the characteristic density difference Ap = O(paAT). Then the condition (Apge2t2/puc) 1 can be expressed in the form... 

If we think about the physics of establishing the immobile cap at the rear of the bubble, it is necessary that the local Marangoni stress balance the net shear stress on the interface. In effect, this means that the condition (7-258) must be satisfied at all points on the surface where the cap exists,... 

This means that there are no surfactant concentration gradients at this level of approximation, and the Marangoni stress condition, (7 299), is reduced to continuity of the viscous shear stress from the two fluids ... 

As expected, we see that this interface contribution produces a gradient from a minimum concentration at the front of the drop (where i] = 1) to a maximum concentration at the back of the drop (where / = - ). Now, the Marangoni stress term on the right-hand side of the shear-stress condition, (7 299), at 0(1i2) is nonzero ... 

If we combine (7-323) with the zero-order solution, (7-305), we see, as expected, that the presence of surfactant on the interface retards the flow. This is consistent with our qualitative expectations based on the fact that the surfactant concentration increases as we move from the front to the back of the drop. However, one surprising feature of the solution (7-323) is that there is no dependence on the viscosity ratio. This flow is established as a consequence of the shear-stress balance, (7-320). Clearly, the shear-stress difference [the left-hand side of (7-320)] does depend on the viscosity ratio however we see from (3-322) that the Marangoni stress that drives the flow also depends on viscosity ratio in precisely the same form. 

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. 

Surfactants are either present as impurities that are difficult to remove from the system or are added deliberately to the bulk fluid to manipulate the interfacial flows [24]. Surfactants may also be created at the interface as a result of chemical reaction between the drop fluid and solutes in the bulk fluid [25, 26]. Surfactants usually reduce the surface tension by creating a buffer layer between the bulk fluid and droplet at the interface. Non-uniform distribution of surfactant concentration creates Marangoni stress at the interface and thus can critically alter the interfacial flows. Surfactants are widely used in numerous important scientific and engineering applications. In particular, surfactants can be used to manipulate drops and bubbles in microchannels [2, 25], and to synthesize micron or submicron size monodispersed drops and bubbles for microfluidic applications [27]. 

From the Frumkin equation (9.61) and the Marangoni stress balance, a differential equation for the surfactant distribution may be obtained... 

The Marangoni stress balance, given by Eq.(8.38), can be used as the basis for this estimation. Eq. (8.38) derived for small Re, must be generalised taking into account that the vorticity is proportional to (Clint et al., 1978),... 

As a consequence, it is necessary to introduce in Equation 20.5 a tangential stress V y along the tangent to the interface at every point (Marangoni stress). 

Hu, H., Larson, R.G. Analysis of the effects of Marangoni stresses on the microfiow in an evaporating sessile droplet Langmuir 21, 3972-3980 (2005)... 

However, drop formation is not the only situation during which a new surface is created. Indeed, while the volume of fluid inside a drop must be ccmserved, its shape may vary and with it the surface area of its interface. For instance, a spherical drop (of minimal surface area) may be deformed by the external flow to form an oval or other shape. This creation of a new surface is coupled with the presenc e of Marangoni stresses as we shall see below, the surfac e variaticMi can create Marangoni stresses, but it can also be caused by uneven distribution of surfactant. 

Further, from the considerations of global mass balance, one may note that the rate of volume of fluid pumped by the Marangoni stress, Qi, must be the same as the rate of volume displaced by the moving bubble, Q2. Since b a, one can neglect 0 b a) terms and approximate Qx as... 

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