Practical Asymptotics

387 Pages·2001·13.602 MB·English

by  H. K. Kuiken (auth.), H. K. Kuiken (eds.)

Practical Asymptotics
Practical Asymptotics
Edited by
H.K. KUlKEN
Eindhoven University of Technology,
Eindhoven, The Netherlands
Reprinted from Journal of Engineering Mathematics, Volume 39, Nas. 1-4 (March 2001)
SPRINGER-SCIENCE+ BUSINESS MEDIA, B.V.
ISBN 978-94-010-3827-0 ISBN 978-94-010-0698-9 (eBook)
DOI 10.1007/978-94-010-0698-9
TABLE OF CONTENTS
Practical asymptotics
byH. K. Kuiken 1-2
Shearflow over aparticulate or fibrousplate
by C. Pozrikidis 3-24
Current-voltage characteristics from an asymptotic analysis of the MOSFETequa-
tions
by E. Cumberbatch, H.Abebe and H. Morris 25-46
Separating shearflowpast asurface-mounted blunt obstacle
by S. Bhattacharyya, S.C.R. Dennis and F.T. Smith 47-62
Microwavejoining oftwo long hollow tubes: an asymptotic theory and numerical
simulations
by G.A. Kriegsmann and J.Luke 63-78
Fastcomputation of limit cycles in an industrial application
byS.Gueron and N.Liron 79-86
Asymptotic analysis of the steady-state and time-dependent Bermanproblem
by1.R. King and S.M.Cox 87- 130
Generation ofwater wavesand boresby impulsive bottom flux
byM. Landrini and P.A. Tyvand 131-170
On the asymptotic analysis of surface-stress-driven thin-layer flow
byL. W.Schwartz 171-188
Matched asymptotic expansions and the numerical treatment of viscous-inviscid
interaction
by A. E.P.Veldman 189-206
Stokes flow around an asymmetric channel divider; a computational approach us-
ing matlab
by J.D.Fehribach and A.M.J. Davis 207-220
The frozen-field approximation and the Ginzburg-Landau equations ofsupercon-
ductivity
by H.G. Kaper and H. Nordborg 221-240
Analytical approximations tothe viscous glass-flow problem in the mould-plunger
pressing process, including an investigaton ofboundary conditions
by S.W. Rienstra and T.D.Chandra 241-259
Asymptotic adaptive methods for multi-scale problems in fluid mechanics
by R. Klein, N. Botta, T. Schneider, C. D. Munz, S. Roller, A. Meister,
L. Hoffmann and T. Sonar 261-343
Asymptotic analysis of the flow of shear-thinning foodstuffs in annular scraped
heat exchangers
by A.D.Fitt and C. P.Please 345-366
The evolution oftravelling waves from chemical-clock reactions
by S.J. Preece,J.Billingham and A. C. King 367-385
Journal ofEngineering Mathematics 39: 1-2,200I.
©2001 KluwerAcademicPublishers.
Practical asymptotics
H. K. KUIKEN
DepartmentofMathematics andComputing Science,Eindhoven UniversityofTechnology,P.O.Box513,
5600MB Eindhoven,TheNetherlands
Received25July 1998,accepted inrevisedform 26September 2000
Abstract. The termpractical asymptotics isexplained.Itisargued that many practical problems are amenable
toand will benefit from the asymptotic approach. Fifteen papers by authors who are active in a wide range of
disciplines demonstratethis.Itisarguedthattheteachingofasymptoticmethods shouldremainanintegralpartof
anysophisticatednumericscurriculum, raising itfarabovethelevelofmerenumbercrunching.
Keywords: asymptotics, asymptotics teaching,non-dimensionalisation,numerics,real-world problems.
A few years ago this journal published a special issue entitled Large-Scale Numerical Mod
elling ofProblemsInvolving theNavier-Stokes Equations (32(1997) 101-280).That appeared
to be a very timely publication. Indeed, these days applied mathematics seems, at first sight
anyway, to become more and more dominated by direct numerical simulation. Admittedly,
this leads to new insights which, itwould seem, could not havebeen attained by other means.
The purpose of that issue was tohighlight that particular aspect.
The interest of large sections ofthe community of applied mathematicians being focussed
intheabove direction nowadays, theolder andbroad subject ofasymptotics seems tobelosing
the popularity it once enjoyed and the impact it had. This is partly because many believe
that asymptotics deals with exceptional cases which are usually outside the practical domain.
However,this isamisconception! Tomake this clear, we offerthe following argument:
Having non-dimensionalised your problem, consideratypical dimensionless group
ND = (XIk,(Xkz2(Xk33•••,
where the k, are real-valued exponents that can beboth positive and negative. Weknow that,
within agiven setofunits, eachofthephysical parameters can assume valuesthat mayvary
(Xi
widely depending upon the problem in hand, i.e.(Xi '" lO'",where the integers n, can range
from large negative to large positive values.Thus,
ND '" lOP,
and we can ask ourselves: what is the probability that p '" O? Ofcourse, p is most likely to
beless than -lor larger than 1,which means that the problem can be simplified through the
application of asymptotic techniques, without seriously affecting the practical usefulness of
the reduced model.The technique, orrather the craft, by which the reduction isaccomplished
iscalled PracticalAsymptotics. It is a highly intuitive process and is based, to a large extent,
onphysical reasoning.
Papers which demonstrate the power of practical asymptotics should involve both large
scale computations on the full model and computations based on the asymptotically reduced
2 H. K. Kuiken
model. Acomparison between thetwoshould then illustrate aconsiderable reduction incom
puting effort in terms of cpu time.Furtheremphasis should be put on the explicit representa
tionoftrends,physical relationships, rules, laws, etc.,asexpressed by mathematical formulae
resulting from the asymptotics with numerically generated coefficients. This, then, should
contrast with the practice of large-scale numerics which usually results in the presentation of
numbers, (colour) plates orcontour plots relating to specific examples.
In the present issue fifteen papers have been collected, each seeking to demonstrate the
usefulness and practical nature of the asymptotic approach for the discipline from which it
arose. These papers address subjects as varied as: the production of glass bottles, semicon
ductors, surface-tension-driven flows, microwave-assisted joining of tubes, viscous-inviscid
interaction, industrial limit cycles, heat exchangers used in the production of foodstuffs, wa
ter waves generated by bottom topography, channel flows, chemical-clock reactions, Stokes
flows, separated flows,superconductivity, flowsoverfibrous plates, multiscale problems aris
ing inlow-Mach-number flows;awide range of subjects indeed!
We hope that this issue will demonstrate the continuing usefulness and validity of the
asymptotic approach in reducing the complexity of mathematical models to apractical mini
mum, i.e., without unduly sacrificing their accuracy, thus achieving large reductions in com
puting effort and increased physical understanding. Webelieve thattheteaching ofasymptotic
methods should remain an integral part of the academic curriculum as an indispensable tool
for the numericist wishing to tackle problems ofever-increasing complexity.
JournalofEngineeringMathematics 39: 3-24,200I.
©2001KluwerAcademicPublishers.
Shear flow over a particulate or fibrous plate
C.POZRIKIDIS
DepartmentofMechanicalandAerospaceEngineering UniversityofCalifornia,SailDiego,LaJolla,
California92093-0411, U.S.A.(e-mail:[email protected], InternetURL:http://stokes.ucsd.edu/c..,pozrikidis)
Received: 15September 1999,acceptedinrevisedform 10April2000
Abstract. Simple shearflowoveraporousplateconsistingofaplanararrayofparticlesisstudiedasamodelof
flowovera membrane.The mainobjective istocompute theslipvelocity defined with reference tothe velocity
profilefarabove the plate,andthedrift velocity induced by theshear flowunderneath the plate.Thedifference
betweenthesetwovelocitiesisshowntobeproportional tothethicknessoftheplate.Whenthegeometryofthe
particlearrayisanisotropic,thedirectionsoftheslipanddrift velocityaregenerally different fromthedirection
ofthe overpassing shear flow. An integral formulation is developed todescribe flowover a plate consisting of
a periodic latticeofparticles witharbitrary shape,and integralrepresentations for the velocity and pressure are
developedintermsofthedoubly-periodicGreen'sfunctionofthree-dimensionalStokesflow.Basedontheintegral
representation,asymptotic expressions for the slipand drift velocityarederived todescribe the limitwhere the
particlesize issmallcomparedtothe inter-particleseparation, andnumerical resultsare presented for spherical
andspheroidalparticlesofarbitrarysize.Theasymptoticresultsarefoundtobeaccurateoveranextendedrangeof
particlesizes.Tostudythelimitofsmallplateporosity,theavailablesolutionforshearflowoveraplanewallwith
acircularorificeisusedtodescribeflowoveraplatewithahomogeneousdistributionofcircularperforations,and
expressionsfortheslipanddriftvelocityarederived.Correspondingresultsarepresentedforaxialandtransverse
shearnowoveraperiodicarrayofcylindersarrangeddistributedinaplane.Streamlinepatternillustrationsconfirm
thatanegativedriftvelocityisduetotheonsetofeddiesbetweenclosely-spacedparticles.
Keywords: Stokesflow,integralequations,shear flow,porousplate,membrane.
1. Introduction
Animportant areaofresearch inbiomechanics concerns theeffect ofashear flowon theequi
librium shapes of, and mass transport through, membranes enclosing vesicles and biological
cells [1-3].The membranestypically consistoflipid bilayers,sometimes resting onanetwork
of proteins comprising the cytoskeleton, separating the vesicle or cytoplasmic fluid from the
ambient plasmaorbufferfluid.From thepointofviewofhydrodynamics,amembrane maybe
regarded asascreenorsieve,andthe problem offlowthrough orover itmay bestudied within
the more general context of flowover aporousor irregular plate separating two semi-infinite
regionsoccupied bythesameordifferent fluids.This moregeneral problem ispertinent tosev
eral other areas of biofluiddynamics and mainstream engineering fluid mechanics involving,
forexample,flowoverarrangements of cilia orbundled tubes.
Previous studies of the thermodynamics and hydrodynamics of polymeric and biological
membraneshavemodeled thenetwork ofthe fundamental molecularunitscomprising amem
brane as a planar lattice of particles over which the smaller molecules of the solute slide [4,
5].The theory seekstopredict thespecies diffusivityand toestablish the relationship between
the pressure drop and the geometrical properties of the membrane expressed in terms of the
hydrodynamic resistance for flow normal to the membrane. Ishii [6] studied flow normal to
4 C:. lJozrikitiis
a planar lattice ofspherical particles in the asymptotic limit where the particle size is small
compared tothe particle separation.Tothe author'sknowledge, shearflowover aporousplate
or particulate plane has not been addressed explicitly by previous authors, although three
related classes ofproblems have been discussed in detail.
The firstclassofproblemsinvolvesflowoveranirregularsurface with smallorlargeampli
tude protuberancespossiblyoffractal nature [7,8]. Issuesofinterestincludetheestablishment
ofthe physical origin oftheno-slip boundary condition,thecomputation ofthe slipvelocityas
afunction ofthe morphologyofthe surface irregularities,and the study oftransport properties
in terms ofeffective heat or masstransfer coefficients.The second classofproblems involves
shear flow over a semi-infinite particulate matrix modeling a porous medium. Reviews and
numerical simulationsfortwo- andthree-dimensional arrangements were presentedbyLarson
and Higdon [9,10] and Sangani and Behl [11]. Theirresults illustrated the dependence of the
slip velocity on the geometry ofthe porous medium microstructure, and helped to establish
a theoretical foundation for Brinkman's equation governing the structure ofthe flow near the
surface of a porous medium, also pointing out its limitations. The third class of problems
involves shear flow over a wall with a circular hole or side pore, possibly in the presence of
suction that drives the fluid through the hole or into the pore, with applications to particle
entrainment and particle screening [12-15].
Inthispaper,westudyshearflowoveraplanarlatticeofparticlesformingaporoussurface.
When the thicknessoftheplate iscomparableto,or larger than,thegapsbetween the particles,
regions ofrecirculating flow develop in the intervening spacing, and the flow decays rapidly
underneath the array to give a virtually quiescent lower fluid. When, however, the particle
size is smaller than the inter-particle separation,the shear flow penetrates the lower fluid,and
a uniform drift velocity is established underneath the array. If the particle shape or lattice
geometry is anisotropic, the direction of the drift velocity is generally different from that of
the overpassing shear flow.Far above the lattice, the flowreduces to simple shear flow with a
macroscopic slip velocity similarto that established inshear flowovera porous material. The
main goal of the present work is to illustrate the dependence ofthe slip and drift velocity on
the particle shape and lattice geometry.
The asymptotic and numerical studies are based on integral representations of periodic
Stokes flow using an appropriate Green's function. In implementing the numerical procedure
for solving an integral equation of the first kind for the traction over a particle surface, a
general method isdeveloped for removing theeigenfunction of the single-layerStokeshydro
dynamic potential. Tocomplement the results for three-dimensional flow over a planar array,
tworelatedconfigurationsarealsoconsidered.Thefirstconfigurationinvolves shearflowover
aflatplate of zero thickness containing a homogeneous distribution ofcircularperforations of
smallsize, which isrelevant to shear flowover aporous plate with a large solidarealfraction.
The second configuration involves longitudinal and transverse shear flow over an array of
cylinders. Comparisons of the results for three-and two-dimensional configurations illustrate
once again the fundamental differences in the nature ofthe corresponding flows.
2. Shearflowover a planarparticle lattice
Consider infinite simple shear flow over a two-dimensional lattice of identical rigid particles
positioned at the vertices of a regular lattice that is parallel to the xy plane, as shown in
Shearflowover aparticulate orfibrousplate 5
z
y
/
x
Figure I. Shearflowoveraplanarlatticeofparticles.Farbelowthelattice,thevelocitytendstotheuniformdrift
velocitywhosedirection and magnitude aredetermined bythe particle shape and size, the latticegeometry,and
thedirectionoftheshearflow.
Figure 1.Thelattice isdescribed by two base vectors 31and 32, so that the designated centers
oftwo particles labelled n and m are related by
(I)
where i1and ii are two integers.
Far above the lattice, as z tends to + 00, the flow reduces to simple shear flow with shear
rate y in the direction ofthe unit vector e that is parallel to the xy plane. Thus, the slope of
the velocity isrequired toexhibit the asymptotic behavior
. du
hm - -+ ye (2)
z--->+oodz
while the pressure tends to aconstant value denoted by p"?". Farbelow the lattice, asztends
to - 00, the shear stress isrequired to vanish,
du
lim - -+ 0 (3)
z--->-oodz
and the pressure tends to a constant value denoted by p Integration ofthe preceding two
- 00.
equations with respect to z produces the asymptotic forms
lim u -+ yze +U+oo +edt (4)
z--->+oo
and
lim u -+ U-oo +edt, (5)
z~-oo
whereU+oo and U-oo aretwoconstant velocities parallel to the xy plane,bothtobecomputed
aspart of the solution, and 'edt' stands for exponentially decaying terms. The magnitude and
direction ofthe slip velocity U+oo depend on the definition ofthe origin ofthe z axis; in the
present case, the origin coincides with the designated particle centers [9, 16]. The magnitude
and directionofthe uniformdriftvelocity induced bythe shearflow underthe lattice,U-oo,on
6 C.Pozrikidis
the otherhand, isindependent ofthe origin ofthe zaxis. Neither the slip velocity nor the drift
velocity is necessarily oriented in the direction ofthe shear flow. In practice, the shear flow
may be generated by the translation ofa flat plate located at z = zp with velocity Upparallel
+
to the particle lattice. Using the asymptotic form (4), we find Up = Yzpe U"?", which
provides us with an expression for evaluating ye and U+oo, when another linear relationship
between them has been established.
Returning to the problem ofinfinite shear flow, we perform a force balance overacontrol
volume that is confined between (a) four planes that are perpendicular to the x)' plane and
enclose one lattice cell and thus one particle, (b) two planes that are parallel to the .ry plane
located far above or below the particle lattice, and (c) the surface ofthe enclosed particle, to
obtain
F == [ f(x) dS(x) = flYAe+A(p-OO - p+OO)ez, (6)
}Particle
where F is the force exerted on one particle, f = a .n is the hydrodynamic traction, a is the
stress tensor, n is the unit vector normal to the particle pointing into the fluid, /l- is the fluid
viscosity, A is the area of one lattice cell, and e, is the unit vector pointing along the z axis.
The motion ofthe fluid isgoverned by the equations ofStokesflow [17]
'\7·u=o (7)
which are to be solved subject to (a) the far-field conditions expressed by Equations (4) and
(5), (b) the no-slip and no-penetration condition u = 0 on the particle surface, and (c) the
periodicity condition
(8)
where iI and i: are two integers.Ourmain objective is to compute the slip and drift velocities
asfunctions ofthe particle shape and size and ofthe lattice geometry, in the limit ofvanishing
Reynolds number.
2.1. INTEGRAL FORMULATION
To prepare the ground for the integral formulation, we introduce the Green's function ofthe
equations ofStokes flow describing the doubly-periodic flow due toa two-dimensional lattice
of point forces that is identical to the particle lattice shown in Figure I. The velocity and
pressure field induced by the point forces at the point x are expressed by
I
III(x) = --Gij(x,xo) bj, (9)
8Jr/l-
where G is the velocity Green's function tensor, P is the pressure Green's function vector,
and b is the strength of a point force; one arbitrarily selected point force is located at XQ.
The Green's functions for the velocity and pressure satisfy the periodicity condition shown in
Equation (8).Moreover, as z - Zotends to +00, we require the asymptotic behavior
8Jr 8Jr
lim G(x,xo) -+ - - (z - zo)J +edt, lim P(x, xo) -+ -e +edt, (10)
z->+oo A z->+oo A z
where J is the identity matrix but with the third diagonal component corresponding to the z
axis set equal to zero. As z - Zo tends to - 00, all components ofthe Green's functions are