Expansion in the Outer Region

The most effective approach, as we have already seen in previous problems, is rescaling. In this case, we rescale the original nondimensionahzed radial variable r in the very general form 

For any nonzero m, this newly rescaled radial variable p is just the original dimensional radial variable, r nondimensionahzed with respect to the new length scale 

All that remains to completely specify the form of the governing equation in this outer region is to determine the parameter m and thus specify the characteristic length scale i for this region. To do this, we recall that conduction and convection terms are expected to remain in balance in this outer portion of the domain as Pe - 0. We see from (9-35) that this implies that 

Thus the appropriate characteristic length scale t in this outer portion of the domain is just kI Uoo. Although this choice was recognized immediately in this problem as an obvious possibility (and thus could have been guessed a priori), we shall see eventually that the rescaling of variables is not usually so obvious. 

With m = 1, the thermal energy equation, (9-35), can now be rewritten in the dimensionless form appropriate to the outer region of the domain  

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In Appendix 5 we show (see Equation A5.9) that Equation 3.180 has the proper asymptotic behavior at z -°o. In this sense, the problem under consideration is completely mathematically correct it is possible to match asymptotic expansions in the outer region (meniscus) and inner region in a vicinity of the moving apparent three-phase contact line. In the following text we undertake the matching procedure numerically. 

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