The Spherical-Cap Limit

As already stated, the limiting form of the governing mass transfer problem for this limit of insoluble surfactant is (7-270). Thus, in this case, we do not need to consider either the bulk transport or surfactant adsorption-desorption processes and the problem is greatly simplified. The governing equation (7-270) requires that either us or T be zero at every point on the drop interface. To verify this fact, we may note that the surfactant interface concentration is axisymmetric so that the solution of (7-270) reduces to the form 

Note that the polar angle is again defined as 0 = n at the front stagnation point and 0 = 0 at the back. We thus see that either the interface velocity is zero or that T = 0 and the tangential-stress must be continuous across the interface [see, e g., (7-264)]. This physical picture has been called the spherical-cap model. It is sketched in Fig. 7 20. The problem then is to determine the velocity fields in the two fluids and determine the critical angle 0C. 

Up to the point of applying the boundary conditions at the surface of the drop, this problem is actually identical to the problem of a rising drop under the action of buoyancy with a clean fluid interface. The solutions, satisfying axisymmetry, and the uniform flow at infinity were given previously as (7 213) and (7 215). Now, at the drop interface, the normal velocity vanishes  

Incorporating this condition, we can now write the general solutions in the form 

The additional boundary conditions are continuity of the tangential velocity, 

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