Expansion in the Inner Region

Let us suppose, based upon the estimates of (9-28), that the dimensionless form (9-24) of the thermal energy equation is valid within the region 1 r ()(Pe ). In other words, we suppose that within this so-called inner region, the sphere radius is an appropriate characteristic length scale as we have assumed in the nondimensionalization leading up to (9 24). Hence, within this region, the dimensionless temperature field can be represented in the form (9 25) with ( o = 1 /r, that is, 

If fi (Pe) = Pe, as assumed in the regular perturbation expansion (9 25), then, of course, the governing equation for 0 is still (9 26), and the general solution for 0 is still (9 27). However, we do not know a priori what formal and the othcr/ (/L) in (9 32) should take this must be determined as part of the solution of the problem. 

The fact that the first term in (9 32) is still 1 /r is a consequence of the fact that ( o must satisfy Laplace s equation, the boundary condition 0O = 1 at r = 1, and also be a decreasing function of r in order that the total heat flux is conserved for increasing values of r. The only solution of V20o = 0 with the latter two properties is ( o = 1 /r. 

It should be noted, however, that the inner region does not extend to r — 00. Hence boundary conditions cannot be imposed on any of the terms of (9 32) in the limit r — 00, and it is more or less accidental that the first term of (9 32) is consistent with the original boundary condition (9 8b) for r 00. The most that we can require of the approximate 

We shall return shortly to attempt to obtain a second approximation for 9, corresponding to the 9 term in (9 32). Before doing this, however, it is necessary to consider a first approximation to 6 for the outer region. 

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