By Prof. Jean Cousteix, Prof. Jacques Mauss (auth.)

This publication provides a brand new approach to asymptotic research of boundary-layer difficulties, the Successive Complementary growth technique (SCEM). the 1st half is dedicated to a common entire presentation of the instruments of asymptotic research. It provides the keys to appreciate a boundary-layer challenge and explains the the way to build an approximation. the second one half is dedicated to SCEM and its functions in fluid mechanics, together with exterior and inner flows. some great benefits of SCEM are mentioned compared to the traditional approach to Matched Asymptotic Expansions. specifically, for the 1st time, the idea of Interactive Boundary Layer is absolutely justified.

With its bankruptcy summaries, specific derivations of effects, mentioned examples and completely labored out difficulties and ideas, the publication is self-contained. it really is written on a mathematical point available to graduate and post-graduate scholars of engineering and physics with an excellent wisdom in fluid mechanics. Researchers and practitioners will esteem it as a priceless monograph of their box of work.

*Asymptotic research and Boundary Layers* is a longer English variation of *Analyse asymptotique et couche limite* released within the Springer sequence Mathématiques et Applications.

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**Sample text**

This procedure is suggested by the desire to take into account the exponential term. A better approximation is expected because the variable X covers a domain closer to the origin than the variable x. Indeed, the condition Y0 |x=0 = 0 is satisﬁed. However, the condition at x = 1 is no longer satisﬁed since Y0 |x=1 = (1 − a) 1 − e−1/ε . This result can be surprising but it must be noted that X belongs to a very wide domain 1 0≤X≤ , ε and terms which are neglected when X is bounded can be non negliglible in the whole domain.

This domain is called the outer region. The solution is x (3) y0 (x) = β exp − 1 b (ξ) dξ a (ξ) . Region 2: x ∈ Dε . The solution can have very fast variations in this domain which is very small when ε is small. A boundary layer forms. To study the boundary layer, the ﬁrst step consists of deﬁning the variable adapted to the study of the domain Dε . The adapted variable is called the inner variable. 5) X= δ (ε) where δ(ε) is a strictly positive function, yet undetermined, which tends towards 0 as ε → 0.

For example, δ1 = ε and δ2 = ε 1 + sin2 (1/ε) are such that δ1 δ2 and δ2 /δ1 has no limit as ε → 0; it is concluded that the two functions δ1 and δ2 cannot be compared with relation R. Sometimes, on a more general set of order functions that do not satisfy the internal law, the elements of the subset E are called gauge functions. This point of view is not adopted here. It will be seen that the notion of “gauge function” is used here according to another meaning (Subsect. 3). 1 Order Functions. 4 Order of a Function Let ϕ (x, ε) be a real function of real variables x = (x1 , x2 , .