Surface Tension Variations

The work currently being conducted by Satyanarayan, Kumar, and Kuloor (S3) indicates that the effect of surface tension is more involved than hitherto appreciated. Some of their data are presented in Figs. 6. and 7. They find that at very small orifice diameters or at very large flow rates, the surface tension variation has negligible influence on the bubble volume. For higher orifice diameters, the influence is more pronounced at small flow rates, as is evident from Fig. 7. 

Princen [64] discovered that the yield stress, xQ, was strongly dependent on <(>, increasing sharply with increasing phase volume. x was also found to depend linearly on the surface tension. Variation with the mean droplet radius, however, did not match theoretical predictions this was reportedly due to the presence of a finite film thickness between adjacent droplets. 

The amount of pigment utilized depends on the color and the hiding power required of the coating. The flow additive often is introduced to relieve surface tension variations between the coating and substrate, to eliminate pinholes or crater formation. Solvents are added as necessary to achieve flow under application conditions. 

Surface tension variations of surface tension from one liquid to another. 

An estimate of the total desorption flow from the surface of a strongly retarded region in the neighbourhood of the rear pole of the bubble is derived as follows. When electrostatic retardation of adsorption-desorption kinetics does not exists, the results of Chapter 8 [Eq. (8.145)] can be applied. For ionic surfactant, the equation for surface tension variation somewhat differs from that for non-ionic surfactant. With regard to these differences, the following estimate of desorption flow results. 

Clearly, then, we must know the concentration, temperature, and charge distributions at the interface in order to define the surface tension variation required to solve the hydrodynamic problem. However, these distributions are themselves coupled to the equations of conservation of mass, energy, and charge through the appropriate interfacial boundary conditions. The boundary conditions are obtained from the requirement that the forces at the interface must balance. This implies that the tangential shear stress must be continuous across the interface, and the net normal force component must balance the interfacial pressure difference due to surface tension. 

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. 

YIH, C-S. 1968. Fluid motion induced by surface tension variation. Phys. Fluids 11, 477-480. 

During the 1870 s, Carlo Marangoni, who was apparently aware of Carra-dori s work but not of Thompson s, formulated a rather complete theory of surface tension driven flow (M2, M3). He noted that flow could result from surface tension variations as they are caused by differences in temperature and superficial concentration, and that, conversely, variations in temperature and concentration could be induced by an imposed surface flow. Marangoni ascribed several new rheological properties to the surface (notably surface viscosity, surface elasticity, and even surface plasticity), while remarking that perhaps some of these properties could be associated only with surface contamination. Most present-day authors ascribe the first explanation of surface tension driven flow to Marangoni, and term such flow a Maragoni effect. ... 

For an air/liquid system a measure of the surface-tension variation resulting from the imposed periodic area variation in the Langmuir trough is performed. If both dilational viscous = (f) and dilational elastic j = e (f) data are needed, and if a Langmuir-type trough is used, then one barrier can be oscillated and another barrier can be used to adjust the extent of the interfacial area. The calculation of the complex modulus, , requires complete scans at different frequencies. 

Faidley, R. W. Panton, R. L. Measurement of liquid jet instability induced by surface tension variations. Exp. Therm. Fluid Sci. 3, 383-387 (1990). 

At short times (f 0), the second term may be neglected and F(f) increases simply by diffusion from the bulk. The surface tension variation at those times can be written as... 

Surface tension variations can also be produced by adding surfactants on the interface. These surface active materials (e.g., soap) typically consist of a hydrophilic head group and a hydrophobic tail. Therefore, the presence of surfactants in solution is energetically unfavorable and one gains in free energy if the molecules align along the free surface, which is the equilibrium situation. The creation of a layer of surfactant molecules on the interface then lowers the surface tension of the system. 

Surface tension variations affect the mobility of the fluid-fluid interface and cause Marangoni flow instabilities. Surfactant-laden flows exhibit surface tension variations at the gas-liquid or liquid-liquid contact line due to surfactant accumulation close to stagnation points [2,53]. For gas-liquid systems, these Marangoni effects can often be accounted for by assuming hardening of the gas bubble, i.e. by replacing the no-shear boundary condition that is normally associated with a gas-liquid (free) boundary with a no-slip boundary condition. It should be noted that such effects can drastically alter pressure drop in microfluidic networks and theoretical predictions based on no-shear at free interfaces must be used with care in practical applications [54]. 

The hydrodynamics of the experimental system can be described theoretically. Such approach is very important for correct interpretation of the experimental results, and for their extrapolation for the conditions not attainable in the existing experimental system. With the mathematical model the parametric study of the system is also possible, what can reveal the most important factors responsible for the occurrence of the specific transport phenomena. The model was presented in details elsewhere [2]. It was based on the equations of the momentum and mass transfer in the simplified two-dimensional geometry of the air-water-surfactant system. Those basic equations were supplemented with the equation of state for the phopsholipid monolayer. The resultant set of equations with the appropriate initial and boundary conditions was solved numerically and led to temporal profiles of the surface density of the surfactant, T [mol m ], surface tension, a [N m ], and velocity of the interface. Vs [m s ]. The surface tension variation and velocity field obtained from the computations can be compared with the results of experiments conducted with the LFB. 

Accordingly, a hypothesis was formulated, which seeks the theoretical rationale of appearance of the specific flow structures inside the liquid layer. It is known, that under certain circumstances, the surface tension variations may lead to the flow instability and to the induction of convection cells [e.g., 5, 6]. Our preliminary theoretical analysis of the hydrodynamic stability of the system [7] indicated that it is possible, that for the Reynolds numbers exceeding the critical value, convection cells inside the hypophase can be formed. This should lead to the significant increase of the mass transfer rate. 

Dilational elasticity gives the surface tension variation of a liquid surface with respect to the unit fraction area change, and it is a measure of the ability of the surface to adjust its surface tension to an instantaneous stress. As the dilational elasticity measures the ability of the surface to develop surface tension gradient, this property characterizes the film and foam stability. The elasticity of the film is proportional to the dilational modulus from which the film was formed. Films from solutions with higher elasticity have higher... 

The surface tension variation in Eq. (12) can be expressed with the help of the complex dilatational modulus e(icb) [9-11]... 

In the preceding section, we have examined thermocapillary and diffiisocapillary flows. Not all such flows are stable, and in fact, surface tension variations at the interface can be sufficient to cause an instability. Rayleigh number is mostly used to describe the instability of buoyancy-driven flow defined as... 

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