Solution by domain perturbation for

This brings the inertial and convective terms, 0(eRe) and O(ePe) into the problem at0(8) Thus, the governing equations at 0(8) are (6-216). In the preceding subsection, we obtained a solution for 5 = 0(1), but then evaluated the resulting function h(x) in the limit 5 C 1. In effect, this means that we assume that Eqs. (6-215) are the governing equations toat least the 0(S2) term that we actually calculated. Thisistruixprovided 0(82) 0(sRe, ePe). Here, we suppose that 0(8) = 0(eRe, ePe) for Re, Pe = 0(1). 

The domain perturbation anaylsis largely follows that from the preceding problem. We assume that 

Although we will ultimately consider the special (distinguished) limit where 5 = e, we initially retain 8 as a small parameter that is independent of e. 

The interface boundary conditions at z = h are first converted to asymptotically equivalent boundary conditions (for 8 ) applied at the unperturbed surface at z = 1. These conditions are similar to those derived in the preceding section. The kinematic and normal-stress conditions are, in fact, unchanged, and are thus given (in the present notation) as 

We begin with the 0(1) problem by integrating (6-215d), and applying the insulating boundary condition (6 217d) This gives 

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