Thermocapillary stress

Statement of the problem. Let us consider a similar problem in which the upper boundary of the channel is free and the surface tension depends linearly on temperature. The balance of tangential stresses on the free surface will then involve the thermocapillary stresses. The corresponding boundary condition has the form... 

Figure 5.5. Drop motion due to temperature gradient. Thin arrows show the direction of thermocapillary stresses on the drop surface and of the flow induced by these stresses thick arrow is the direction of the drop motion (it is assumed that surface tension is a decreasing function of temperature)...

Radiation-induced thermocapillary motion of a drop. The temperature gradient is the simplest but not the unique method for bringing about the thermocapillary drift of a drop. If the drop is opaque and the fluid is transparent, one can move the drop by a light beam in a uniformly heated fluid. The radiation absorbed by the drop will heat it nonuniformly, thus producing thermocapillary stresses. For dcr/dT < 0, the drop will drift towards the warmer part, that is, towards the beam. 

Further, following [419], we consider in detail another mechanism of the surface tension variability produced in the process of motion. We consider a drop moving at a constant velocity, on whose surface an exothermic or endothermal reaction is involved. It is assumed that a surfactants takes part in the reaction being dissolved in the surrounding liquid. Let the liquid temperature and the concentration of surfactants be constant remote from the drop, while on the interface, the concentration of the surfactants (the reactant) is zero (the diffusion regime of reaction). In this symmetric situation, the temperature variability, and, consequently, the thermocapillary stresses can be produced only in the process of motion of liquids. 

This modified Marangoni number, m, is of order unity and corresponds to the most general case as viscous and thermocapillary stresses are of the same order in (185). This also means that Ma is of order of Ga. 

In the case of an open substrate, meaning that only one wall is in contact with the fluid, the initial observation is that a semi-infinite film of hquid subject to a temperature gradient will spread towards the cooler side of the substrate. This can be understood by referring back to Fig. 1 and noting that fluid is pulled towards the cool region, since the surface tension is higher there. A lubrication analysis yields a speed that is proportional to the thermocapillary stress a) and to the layer height h), and inversely proportional to the fluid viscosity ( ). 

Here, we consider only the simpler situation in which the surfactant is assumed to be relatively dilute so that it is mobile on the interface and contributes a change only in the interfacial tension, without any more complex dynamical or rheological effects. In this case, the boundary conditions derived for a fluid interface still apply. Specifically, the dynamic and kinematic boundary conditions, in the form (2 122) and (2-129), respectively, and the stress balance, in the form (2 134), can still be used. However, the interfacial tension, which appears in the stress balance, now depends on the local concentration of surfactant. We shall discuss how this concentration is defined shortly. First, however, we note that flows involving an interface with surfactant are qualitatively similar to thermocapillary flows. The primary difference is that the concentration distribution of surfactant on the interface is almost always dominated by convection and diffusion within the interface, whereas the... 

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. 

By (5.9.20), the no-slip and no-flow conditions hold on the hard surface, and a linear temperature distribution is maintained. Condition (5.9.21) says that the no-flow condition on the free surface and the condition of zero heat flux through the free surface must hold, and the balance of tangential thermocapillary and viscous stresses must be provided. Taking into account the quadratic dependence (5.9.19) of the surface tension on temperature, we rewrite the right-hand side of the last condition in (5.9.21) using the relation... 

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 Microreactor, Fig. 2 Droplet microreactor based on thermocapillary actuation, (a) Without a temperature gradient, the droplet is at equilibrium and does not move, (b) If heater 1 is on, the induced temperature gradient propels the droplet owing to the difference in surface stress between the sides of the droplet, (c) If the droplet moves out of the temperature field of heater 1, heater 2 is activated to propel the droplet further... 

Thermocapillary pumping (TCP) is a pumping concept based on thermocapillary forces, where a liquid droplet moves through a microchannel or on a planar surface. A temperature gradient leads to a difference of the surface stresses across a liquid droplet and propels it into a cooler region. 

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