Effect of buoyancy forces on a boundary layer near a free boundary

https://link.springer.com/content/pdf/10.1007/BF00910372.pdf

This chapter presents an introduction to asymptotic analysis. Early in the 19th century, Abel, Cauchy, and others undertook the task of placing mathematical analysis on firmer foundations. One result was the introduction of a complete ban on the use of divergent series, although it appears that this step was taken somewhat reluctantly. No way of rehabilitating the use of divergent series was forthcoming during the next half century. Two requirements for a satisfactory general theory were the following: (1) that it apply to most of the known series; and (2) that it permit elementary operations, including addition, multiplication, division, substitution, integration, differentiation, and reversion. Neither requirement would be met if, for example, one confines to series expansions whose remainder terms are bounded in magnitude by the first neglected term. Both requirements were satisfied eventually by Poincare by defining what he called asymptotic expansions.

Effect of buoyancy forces on a boundary layer near a free boundary

References

Introduction to asymptotic analysis

Asymptotics of the flows of a liquid with a free boundary with vanishing viscosity

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