Singular Perturbation Theory: A Viscous Flow out of Göttingen

O’Malley, Robert E.

doi:10.1146/annurev.fluid.060909.133212

Cited by 14 publications

(8 citation statements)

References 50 publications

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“…In particular, the quest for a model that can describe separated boundary layers spanned three quarters of a century. Also, as shortly touched upon, it strongly stimulated the development of singular perturbation theory, e.g., [24,58,59].…”

Section: Epiloguementioning

confidence: 99%

Entrainment and boundary-layer separation: a modeling history

Veldman

2017

J Eng Math

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For many decades since Prandtl's presentation of the boundary-layer equations, all attempts to calculate separated boundary layers ended up in a break-down of the numerical algorithm. It was not until the late 1970s that the first successful calculations were reported. The paper describes the history and philosophy of the steps that led to understanding where the break-down originates and how to circumvent it. Especially, the role of Head's entrainment will be highlighted.

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Section: Epiloguementioning

confidence: 99%

Entrainment and boundary-layer separation: a modeling history

Veldman

2017

J Eng Math

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“…Upon substituting this decomposition for Ψ H into (2.19)-(2.22), we obtain that Ψ H (ρ, θ ; S) is C 1 as ρ → 0 and satisfies the nonlinear problem 4 is the best least-squares quartic polynomial to the numerical data for R(S). The three-term 5 is the best quintic polynomial to the numerical data for C 2 (S). The one-term approximation (2.27) is the dashed curve, which provides a somewhat poor approximation unless S is rather small where L 0s is the linearized Oseen operator.…”

Section: Lemmamentioning

confidence: 99%

“…The development and application of singular perturbation methods to various problems in fluid mechanics is discussed in the classic text of [4]. For a recent survey of the history of singular perturbations applied to boundary layer problems in fluid mechanics see [5]. A comprehensive recent survey on asymptotic and renormalization group methods applied to slow viscous flow problems is given in [6].…”

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confidence: 99%

A hybrid asymptotic-numerical method for calculating drag coefficients in 2-D low Reynolds number flows

Hormozi

Ward

2014

J Eng Math

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Steady-state incompressible low-Reynolds-number fluid flow past a cylindrical body in an unbounded two-dimensional domain is a singular perturbation problem involving an infinite logarithmic expansion in the Reynolds number ε as ε → 0. The central difficulty with applying a conventional matched asymptotic approach to this problem is that only the first few terms in the infinite logarithmic expansion of the drag coefficient and of the flow field can be calculated analytically. To overcome this difficulty, a hybrid asymptotic-numerical method that incorporates all logarithmic correction terms is implemented for three low-Reynolds-number flow problems. In particular, for a nanocylinder of circular cross section with surface roughness, modeled by a Navier boundary condition involving a sliplength parameter, a hybrid asymptotic-numerical method is formulated and implemented to determine an approximation to the drag coefficient that is accurate to all powers of −1/ log ε. A similar analysis is done to determine a corresponding approximation of the drag coefficient for a porous cylinder, where the flow inside the cylinder is modeled by the Brinkman equation. For both the nano-and porous-cylinder problems, the hybrid asymptotic-numerical method is extended to calculate the first transcendentally small correction term to the Stokes flow near a body. This term, which governs weak upstream/downstream asymmetry in the Stokes flow, is extrapolated to finite ε to predict the formation of any eddies near the body. Finally, the hybrid method is used to determine the drag coefficient, valid to within all logarithmic terms, for two identical cylinders of circular cross section in tandem alignment with the free stream. An extension of the theoretical framework to more general slow viscous flow problems is discussed.

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“…Within the 20th century, the use of small parameters and perturbation techniques for differential equations have permeated effectively all areas of science and engineering, while more recently also quantitative modelling in the social sciences tends to rely on differential equation modelling. For some pointers to the vast literature, we refer to the books [25,27,107,122,128,133,145,177,181,186,193,209,213,214], where classical cases of ordinary and partial differential equations (ODEs and PDEs) with one small parameter are considered from a number of different viewpoints.…”

Section: Introductionmentioning

confidence: 99%

A General View on Double Limits in Differential Equations

Kuehn,

Berglund,

Bick

et al. 2021

Preprint

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In this paper, we review several results from singularly perturbed differential equations with multiple small parameters. In addition, we develop a general conceptual framework to compare and contrast the different results by proposing a three-step process. First, one specifies the setting and restrictions of the differential equation problem to be studied and identifies the relevant small parameters. Second, one defines a notion of equivalence via a property/observable for partitioning the parameter space into suitable regions near the singular limit. Third, one studies the possible asymptotic singular limit problems as well as perturbation results to complete the diagrammatic subdivision process. We illustrate this approach for two simple problems from algebra and analysis. Then we proceed to the review of several modern double-limit problems including multiple time scales, stochastic dynamics, spatial patterns, and network coupling. For each example, we illustrate the previously mentioned three-step process and show that already double-limit parametric diagrams provide an excellent unifying theme. After this review, we compare and contrast the common features among the different examples. We conclude with a brief outlook, how our methodology can help to systematize the field better, and how it can be transferred to a wide variety of other classes of differential equations.

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Singular Perturbation Theory: A Viscous Flow out of Göttingen

Cited by 14 publications

References 50 publications

Entrainment and boundary-layer separation: a modeling history

Entrainment and boundary-layer separation: a modeling history

A hybrid asymptotic-numerical method for calculating drag coefficients in 2-D low Reynolds number flows

A General View on Double Limits in Differential Equations

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