Thermocapillary force

Thus the hydrodynamic force is larger than it would be in the absence of the thermocapillary (Marangoni) contribution to the shear stress at the bubble surface. As a consequence, the bubble moves slower. Indeed, at steady-state, the bubble velocity can be calculated from the overall force balance ... 

Statement of the problem. Let us consider the motion of a viscous fluid in an infinite layer of constant thickness 2h. The force of gravity is directed normally to the layer. The lower plane is a hard surface on which a constant temperature gradient is maintained. The nonuniformity of the temperature field results in two effects that can bring about the motion of the fluid, namely, the thermogravitational effect related to the heat expansion of the fluid and the appearance of Archimedes forces, and the thermocapillary effect (if the second surface is free) produced by tangential stresses on the interface due to the temperature dependence of the surface tension coefficient. 

These results show that thermocapillary forces generate a complicated circulation liquid motion in the layer, and, the flow changes its direction at the depth equal to 1/3 of the layer depth. Just as one can expect, the flow is symmetric with respect to the plane X = 0 with temperature To the fluid flows out from the near-bottom layer along this plane. 

Let us estimate the thermocapillary force applied to a drop and the velocity of the drop thermocapillary drift in the absence of gravitation. We assume the ambient fluid to be infinite and the nonuniform temperature field remote from the drop to be linear in X ... 

The first term Fy in (5.10.8) is just Hadamard-Rybczynski s result (2.2.15) for the drag of a drop in a translational flow. The second term Ft is the thermocapillary force acting on the drop in the external temperature gradient due to the Marangoni effect. 

We can find the drop velocity under a thermocapillary force in the absence of gravitation by setting the force F in (5.10.8) equal to zero. As a result, we obtain... 

The results (5.10.8) for the thermocapillary force Ft and (5.10.9) for the thermocapillary drift velocity, which were obtained under the assumption of a constant temperature gradient remote from the drop, prove to hold also for a varying gradient. These expressions can be rewritten in a vector form as follows [468] ... 

The corresponding problem was considered in [322, 389], The radiation in [389] was assumed to have the form of a plane-parallel beam being absorbed on the drop surface as on a black body, but freely passing through the exterior fluid. The temperature remote from the drop is assumed to be constant. For the thermocapillary force and for the velocity of thermocapillary drift of the drop in the absence of gravitation, the following expressions were obtained (J is the radiation flux power) ... 

In the limit case 0 -4 oo (high viscosity of the drop substance), the thermocapillary effect does not influence the motion, B -> the flow around the drop will be the same as for a hard sphere, and (5.11.3) implies the Stokes law (2.2.5). For m = 0 (no heat production or independence of the surface tension on temperature), the thermocapillary effect is absent, and (5.11.3) yields a usual drag force for a drop in the translational flow (2.2.15). 

In the preceding section, we have examined a variety of steady thermocapillary and diffusocapillary flows. Not all such flows are stable and in fact surface tension variations at an interface can be sufficient to cause an instability. We consider here the cellular patterns that arise with liquid layers where one boundary is a free surface along which there is a variation in surface tension. It is well known that an unstable buoyancy driven cellular convective motion can result when a density gradient is parallel to but opposite in direction to a body force, such as gravity. An example of this type of instability was discussed in Section 5.5 in connection with density gradient centrifugation. 

When the relative motion of the drop is driven by a body force or by thermocapillary migration (rather than by selfdiffusion), Eq. (101) is no longer valid. Instead, in Eq. (98) one has formally to substitute the following expression for... 

Thermocapillary droplet control is based on the principle that surface tension is a function of temperature. By using an array of subsurface heaters or other means of ctMitroUing the temperature distribution around the periphery of a droplet, an imbalance of forces can be imposed on the droplet resulting in motion toward the cold regions [8]. Temperature increases of less than approximately 6 K were used by Darhuber et al. to move droplets of several different liquids. Their fabrication of metal electrodes involved photo-hthographic patterning and metal evaporation followed by coating with silicon dioxide. 

There are two basic platforms for droplet microreactors the planar platform and the in-channel continuous platform. In a planar platform, the droplet can move freely on a planar surface, while the motion of the droplet in an in-channel continuous platform is restricted by microchannels. The actuation of droplets in a planar platform is based on nonmechanical concepts such as electrowettmg, thermocapillary forces, and magnetic forces. Most in-channel... 

Thermocapillary forces can be used to manipulate a droplet microreactor in the same way as electrowetting does [2]. However, the elevated temperature required may cause evaporation and, in the worst case, boiling of the droplet. ThermocapiUaiy stresses caused by spatial variations of the surface tension at a gas/liquid interface can induce spontaneous flow of a liquid film to a cooler positirai. The surface tension at the... 

Droplet reactors are basic components of digital microfluidics. There are still a number of opportunities in droplet reactor research. The future directions can be categorized into fundamentals and applications. Fundamental research could result in other platforms for droplet reactors. While electrowetting has been widely reported in the past, microfluidic platforms based on thermocapillary and other forces are still underrepresented. More research on the systematic design of droplet-based reactors is needed to secure industrial adaptation and commercial... 

Driven by the pressure gradient and the thermocapillary force, fluid 2 together with the droplet flows out of the domain via either the upper or the lower branch. It is possible that the droplet splits at the T-junction. The two outlets are maintained at the same pressure with a fuUy developed temperature profile. 

The Marangoni number represents the ratio of the thermocapillary force to the viscous force. The driving temperature difference AT is created by the applied heat flux q. For the purpose of nondimensionalization, the applied heat flux q is assumed to increase the fluid 2 temperature flowing out at the upper outlet by AT. With this. At can be expressed in terms of q as... 

Flow Bifurcation in MicroChannel, Fig. 6 Thermocapillary forces acting on a droplet at a T-junction... 

The mechanism in generating the circulations is similar to the case of Ma = 0. However, since now the interfacial forces (the resultant of capillary and thermocapillary forces) at the pointed tails of the detached daughter droplets are no longer of the same magnitude, the strength of the induced circulations is different. Although... 

To obtain a clearer picture of the switching mechanism, a more detailed flow field is required. Figure 9 shows the evolution of the droplet from t = 1.4063 to f = 1.7813 with all the velocity vectors plotted. For the present case, the secondary flow induced by the thermocapillary forces is much stronger. The induced secondary flow of a circulatory nature (as in the case of Ma = 40)... 

Equation 27 can be used with Eq. 24 to express Co(U ) as a sole function of b/a in the asymptotic matching condition. With Cq U ) already being obtained numerically, the ratio of bja can therefore be obtained from Eq. 24, for a given value of Ay = Ay/y = y f aly (which is the ratio of the thermocapillary force to the mean surface tension force). The numerical results can be fitted with the following power law expression [2] ... 

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