Introduction - Whiteheads Paradox

A natural question is to what extent the overall rate of heat transfer is modified by convection for small, but nonzero, values of the Peclet number. To obtain a more accurate estimate of Nu for this case, it would appear, from what has been stated thus far, that we must calculate added terms in the regular asymptotic expansion (9-15) for 0. To attempt this, we must substitute (9-15) into (9-7) and (9-8) to obtain governing equations and boundary conditions for the subsequent terms 6 . In this section, we consider only the second approximation, 9. The governing equation and boundary conditions derived from (9-7), (9-8), and (9-15) are 

To complete the specification of the problem for 9, we must specify a particular velocity field u. In the case of Re C 1, we can use the creeping-flow solutions of Chaps. 7 and 8, and it is again convenient to focus our attention on the case of a sphere in a uniform streaming flow, in which a first approximation to the velocity field is given by the Stokes solution, Eq. (7-158), from which we can calculate the velocity components by means of (7-102). 

In this case, the full thermal energy equation, (9-7), takes the form 

Hence the expansion (9-15) apparently can be written in the slightly simplified form [see Eqs. (9-14)] 

The homogeneous solution of this equation is just (9-18). A particular solution, corresponding to each of the terms on the right-hand side of (9-26) can be obtained in the form rsP (rj). Combining the homogeneous and particular solutions, we can express the general solution of (9-26) as 

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