A Second Approximation in the Inner Region

The first approximation to the temperature field in the outer region, Eq. (9-51), has been constructed in such a way that it asymptotically matches the first approximation in the inner region, 6 = 1/r, in the region of overlap. However, we have already noted that the match between these two leading-order approximations is not perfect. Rather, there is an (HIP) mismatch, which can be made arbitrarily small in the limit Pe - 0 but is nonetheless nonzero. To determine the precise form of this mismatch, we investigate the approximate form of (9-51) lor p 1. In particular, if we expand the exponential and retain the first two terms, we find that 

When written in terms of the inner variable r = p /Pe, this becomes 

The first term matches identically with the pure conduction, leading-order approximation in the inner region, 1 /r, but the second term clearly represents an ()(Pe) mismatch between the first approximations in the two regions. The third term from the power-series approximation of the exponential is O(Pep), but we see from (9 52) that this leads to a mismatch that is 0(Pe)2 when expressed in terms of inner variables. A term of 0(Pe2) cannot appear until at least the third approximation in the inner region. 

to determine the correct form for 6 in the inner region, we first apply the boundary condition 

Using (9 55) for large r and (9 52), this matching condition reduces to 

---