Gradients of surface tension

Equations (5-30) and (5-31) are plotted in Fig. 5.29. In agreement with numerical predictions (BIO, HI, H6), no boundary layer separation is predicted when there are no gradients of surface tension at the surface (H8). 

The theory of effective viscosity has been developed by Betherton [172], Hirasaki and Lawson [173], Falls et al. [171] and Kovscek and Radke [153]. It was shown that the effective viscosity is a sum of three terms the first accounts for the contribution of the slugs of liquid between bubbles, the second is the resistance against surface deformation in the advancement of bubbles through the capillaries (pores) and the third is the gradient of surface tension caused by the withdrawal of the surfactant (from the bubble front to the bubble back). The experimental data of Falls et al. [171], Hirasaki and Lawson [173], Ettinger and Radke [166] for bead packs and Berea core agree with the calculations from Hirasaki and Lawson s models [173],... 

Because T(1> T] there is a gradient of surface tension that drives motion in the interface in a direction that tends to maintain the flow perturbation... 

The subject of the present section is the capillary motion in a vertical narrow tube [2]. The moistening flow and the flow due to a surface gradient of surface tension will be considered, respectively, in Sections 17.3 and 17.5. 

One can adduce many examples of flows induced by the gradient of surface tension. It is sufficient to recall a phenomenon well known from the ancient times - the damping of ocean waves that takes place when oil is poured into the ocean, spreading with a thin layer over the water surface. Another familiar phenomenon is the dancing of a ball of solid camphor on the water surface. All these phenomena are caused by a change of S. 

The heating of an emulsion produces temperature gradients, which in their own turn cause thermocapillary migration of the droplets driven by thermally excited gradients of surface tension (170—172) ... 

Equation 47, with the modifications introduced by Eq. 48, physically represents a rather simphfied situation. In reality, a number of other complicating factors may be present, resulting in further modifications in the equation of capillary advancement. For example, the capillary walls can adsorb surfactants from the bulk solution, leading to a continuous variation in the surface tension coefficient. The adsorption of the surfactant, riv, can lead to an axial gradient of surface tension, following the Gibbs equation ... 

The first and second terms of this equation represent the effects of gradients of surface tension and smface viscosity, respectively. Using the velocity profile across the thickness of the film found in deriving the result of (a), show that the last term in the equation represents the shear stress atz = h. 

J F Lancaster The stability of meridional liquid flow induced by a gradient of surface tension or by electromagnetic forces Int. Inst. Welding, Doc 212-682-87 (1987). 

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