The Method of Matched Asymptotic Expansions and Its Generalizations

Milton Van Dyke’s Perturbation Methods in Fluid Mechanics [490] was effectively both the earliest and the most influential book specifically about applied singular perturbations. (Some credit might be given earlier fluid dynamics textbooks, e.g., Hayes and Probstein [199]). Van Dyke extensively surveyed the large extant aeronautical and fluid dynamical literature, forcefully advocating and clarifying the so-called method of matched asymptotic (or inner and outer) expansions . Although Van Dyke acknowledged that Prandtl’s boundary layer theory was the prototype singular perturbation problem, he introduced the subject by describing incompressible fluid flow past a thin airfoil. The book’s highlight message, sometimes called Van Dyke’s magic rule , states: