Marangoni Effects on the Motion of Bubbles and Drops

In the remainder of this section an approximate analysis is provided of the thermocapillary problem of motion of a bubble in a gravitational field.28 Thus we suppose that we have 

The problem, then, is to solve the creeping-flow equations in the fluid, subject to the condition 

As usual, i] = cos 0. To evaluate By/Dr) in (7 240), we use the temperature distribution (7 237), which is now assumed to hold throughout the fluid, and the relationship (7-236) between the surface tension and temperature. The temperature distribution can be expressed in terms of spherical coordinates as 

Because the problem is axisymmetric about the z axis, we can solve the creeping-flow equation in terms of the streamfunction Hence, after applying the asymptotic condition 

The coefficients C and D must be determined by application of the boundary conditions (7-239) and (7-243) at the bubble surface r = 1. The first of these requires that f = 0 at r = 1. Hence, 

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