Asymptotic approximation regular terms

Examining these solutions, we see that the temperature becomes nonuniform and the velocity profile nonlinear. These results are to be expected from a qualitative point of view. In a sense, the most important conclusion is that the regular asymptotic expansion in terms of the small parameter Br provides a method to obtain an approximate solution of the highly nonlinear boundary-value problem to evaluate the influence of weak dissipation, which can clearly be applied to other problems. 

We saw in the previous example lor R, <approximate solution, (4-4), which we obtained by taking the limit R, —> 0 in the exact equation, (4-1), was just the first term in an asymptotic solution for Rr approximate equation and solution (4-17) that we obtain by taking the limit Rm -> oo in (4-16) will also be the first term in a formal asymptotic expansion of H for Rm y>> 1. Assuming this expansion is regular, it will take the form... 

Hence we seek an asymptotic solution for the leading-order terms in the thin-film approximation, i.e., u<(>>p<(l>w<(>>, as a regular perturbation in 8, i.e.,... 

The first approximation of the deviation from sphericity is contained in the function f>(, t), and we have anticipated, from the preceding arguments, that the magnitude of term is proportional to Ca. The presumption that the expansion for /(x) proceeds in whose powers of Ca is based on the fact that domain perturbation techniques almost always lead to a regular (rather than singular) asymptotic structure. 

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... 

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