Preface to fourth Special Issue on Practical Asymptotics

Abstract

The aim of this Special Issue on Practical Asymptotics, the fourth such issue published in the Journal of Engineering Mathematics, is to collect a number of outstanding examples of studies in which formal asymptotics is used to provide invaluable information about mathematical models resulting from important practical problems. The following papers certainly meet that goal, and together illustrate how powerful these approaches can be, even today in the computer age.This issue has a strong fluid-mechanics flavour, with studies on thin-film flows arising in the process of blade coating [1], dewetting from a hydrophobic substrate [2], and also viscous films that form inside the rim of a horizontal rotating cylinder [3]. Other topics in the fluid-mechanics discipline included here are high-Reynoldsnumber branching and network flows [4], time-dependent flows through curved pipes [5], low-Reynolds-number flow between two parallel rotating disks [6] and past a rotating cylinder [7], the classical dam-breaking problem [8], and premixed-flame propagation [9]. A few illustrative applications of these works are to blood flow in human arteries, coating processes in industry, or viscous pumps operating on the microscale. In addition, there is a paper that falls within the discipline of solid mechanics, studying edge wrinkling of thin elastic plates [10]. Finally, the issue contains two papers of a theoretical nature, one considering uniformly valid polynomial solutions to boundary-value problems [11], and the other concerned with 'beyond-all-orders' approaches to studying the asymptotic behaviour of spectral coefficients [12].The analysis contained in this issue comprises a variety of perturbation methods, including repeated use of matched asymptotic expansions (with the usual suspects of inner and outer problems, corner regions, interior layers, etc.), boundary-layer theory, quite sophisticated multi-scale approaches, WKB analysis, exponential asymptotics, and more. In the usual manner, these techniques are made possible by careful consideration of the various mathematical models, and subsequent exploitation of small (or large) length-and/or time-scales, or small (or large) nondimensional parameters. Examples of such parameters in the present issue measure surface tension, rotation rates between plates, activation to thermal energies, fluid fluxes, weak and strong gravity effects, and aspect ratios or curvatures in problem geometry. Typically the asymptotic limits considered in these papers are singular in nature, with the most obvious examples arising when small parameters multiply higher-order derivatives in the governing equations, or when solutions themselves develop singularities in finite time. A browse through the following pages demonstrates how asymptotic methods link closely with topics such as linear stability theory, and often lead to,

Applied Mathematical Modelling of Continuous Casting Processes: A Review

“…There are several motivations for this hybrid asymptotic/numerical approach. The first is that it has been applied successfully in other areas of science and technology, as witnessed by the growth of an activity known as practical asymptotics [3][4][5][6][7][8]; note, however, that this is not identical to the model order-reduction approach [9], which is also prevalent in industrial mathematics. The second is the Moore's law effect in modelling [10]: namely, that, although computational power doubles every 18 months, it will still be many years before all the length scales in a casting process will be numerically resolved [11].…”

Preface to the sixth special issue on “Practical Asymptotics”

“…In the future, with further increases in computational power, these claims will undoubtedly become more frequent. It is the role of this series of Special Issues on Practical Asymptotics [1][2][3][4][5] to dispel this myth.Computational studies should be guided by a theoretical framework in order to gain more comprehensive insight. The most important contribution of asymptotics is to provide this theoretical framework for understanding, interpreting, developing and solving mathematical models.…”

“…In the future, with further increases in computational power, these claims will undoubtedly become more frequent. It is the role of this series of Special Issues on Practical Asymptotics [1][2][3][4][5] to dispel this myth.…”

Preface to the fifth special issue on practical asymptotics

“…However, this is a misconception!" The Guest-Editors of the following, second, third and fourth issues on practical asymptotics, argued that asymptotic solutions and asymptotic methods remain and will be valuable because they "provide important complements to numerical simulation" [2], "can offer ways to systematically study problems that may otherwise be inaccessible" [3] and "may provide quantitative and/or qualitative information that computer simulations can not" [4]. It is seen that the issues on practical asymptotics are a forum for discussion "Does the asymptotic analysis have a future in the era of supercomputers?".…”