The present study examines the flow past slender bodies possessing finite centre-line curvature in a viscous, incompressible fluid without any appreciable inertia effects. We consider slender bodies having arbitrary centre-line configurations, circular transverse cross-sections, and longitudinal cross-sections which are approximately elliptic close to the body ends (i.e. prolate-spheroidal body ends). The no-slip boundary condition on the body surface is satisfied, using a convenient stepwise procedure, to higher orders in the slenderness parameter (e) than has previously been possible. In fact, the boundary condition is satisfied up to an error term of O(e2) by distributing appropriate stokeslets, potential doublets, rotlets, sources, stresslets and quadrupoles on the body centre-line. The methods used here produce an integral equation valid along the entire body length, including the ends, whose solution determines the stokeslet strength or equivalently the force per unit length up to a term of O(e2). The O(e2) correction to the stokeslet strength is also found. The theory is used to examine the motion of a partial torus and a helix of finite length. For helical bodies comparisons are made between the present theory and the resistive-force theory using the force coefficients of Gray & Hancock and Lighthill. For the motion considered the Gray & Hancock force coefficients generally underestimate the force per unit length, whereas Lighthill's coefficients provide good agreement except in the vicinity of the body ends.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112080000687

An improved slender-body theory for Stokes flow

Singular Perturbations and Time Scales in Guidance and Control of Aerospace Systems: A Survey

The existence theory for steady vortex rings of small cross-section is used to derive asymptotic formulae that describe the shape and overall properties of such rings. A certain two-parameter family of rings is studied in detail, to a first approximation; for members of this family, the ratio ω/r (of vorticity to cylindrical radius) falls from a positive maximum at a central point of the core cross-section to a value at the core boundary that can be substantially smaller or even negative. The case of uniform ω/r is considered to a higher order of approximation, and the formulae given for this case appear to be useful for quite substantial cross-sections.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112072001107

Examples of steady vortex rings of small cross-section in an ideal fluid

In many singular perturbation problems multiple scales are used. For instance, one may use both the coordinate x and the coordinate $x^ * = \varepsilon ^{ - 1} x$. In a secular-type problem x and $x^ * $ are used simultaneously. This paper discusses layer-type problems in which $x^ * $ is used in a thin layer and x outside this layer. Assume one seeks approximations to a function $f(x,\varepsilon )$, uniformly valid to some order in $\varepsilon$ for x in a closed interval D. In layer-type problems one uses (at least) two expansions (called inner and outer) neither of which is uniformly valid but whose domains of validity together cover the interval D. To define “domain of validity” one needs to consider intervals whose endpoints depend on $\varepsilon $. In the construction of the inner and outer expansions, constants and functions of e occur which are determined by comparison of the two expansions “matching.” The comparison is possible only in the domain of overlap of their regions of validity. Once overlap is established, matching is easily carried out. Heuristic ideas for determining domains of validity of approximations by a study of the corresponding equations are illustrated with the aid of model equations. It is shown that formally small terms in an equation may have large integrated effects. The study of this is of central importance for understanding layer-type problems. It is emphasized that considering the expansions as the result of applying limit processes can lead to serious errors and, in any case, hides the nature of the expansions.

/pdf/basic-concepts-underlying-singular-perturbation-techniques-1zgjqyti15.pdf

Basic Concepts Underlying Singular Perturbation Techniques

The Korteweg–de Vries equation is shown to govern formation of solitary and cnoidal waves in rotating fluids confined in tubes. It is proved that the method must fail when the tube wall is moved to infinity, and the failure is corrected by singular perturbation procedures. The Korteweg–de Vries equation must then give way to an integro-differential equation. Also, critical stationary flows in tubes are considered with regard to Benjamin's vortex breakdown theories.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112070001611

Weakly non-linear waves in rotating fluids

The method of matched (or of ‘inner and outer’) asymptotic expansions is reviewed, with particular reference to two general techniques which have been proposed for ‘matching’; that is, for establishing a relationship between the inner and outer expansions, to finite numbers of terms, of an unknown function. It is shown that the first technique, which uses the idea of overlapping of the two expansions, can be difficult and laborious in some applications; while the second, which is the ‘asymptotic matching principle’ in the form stated by Van Dyke(13) can be incorrect. Two different sets of conditions sufficient for the validity of the asymptotic matching principle are then established, on the basis of assumptions about the structure of expressions which approximate to the desired function f(x,∈) for all relevant values of x. Finally, it is noted that in four classes of singular-perturbation problems for which complete and rigorous asymptotic theories exist, uniform approximations to the solutions have a structure which is a particular form of the general one assumed in this paper.

L. E. Fraenkel

Papers

On the method of matched asymptotic expansions