Modelling and Perturbation Methodssjoerdr/papers/aom.pdf · •Constructing models Other possible...

Modelling and Perturbation Methods

S.W. Rienstra

Eindhoven University of Technology

21 Mar 2002

T

L

10:56 21 Mar 2002 2 version: 21-03-2002

Chapter I

The art of modelling

The mathematical solution of a “real world” problem starts with the modelling phase, where theproblem is described in a mathematical representation of its primitive elements and their relations. Asthe solution is not served by unnecessary complexity, we are interested in an adequate mathematicaldescription with the lowest number of essentially independent parameters and variables.

A very important aspect in the modelling is therefore the identification of a hierarchy of importance,to distinguish between the important, the less important, and the unimportant effects. From on thishierarchy we may decide which aspects can be included and which can be neglected in the model.

1 Introduction

Modelling in the context of applied mathematics is something like

Describing a real-world problem in a mathematical way by what is called a MODEL, suchthat it becomes possible to deploy mathematical tools for its solution. The accuracy ofthe description should be limited, in order to make the model not unnecessary complex.The model should be based on first principles and elementary relations, such that it hasreasonable claims to predict both quantitative and qualitative aspects of the solution.

This is evidently a very loose definition. Apart from the question what is meant with: a problem beingdescribed in a mathematical way, there is the confusing paradox that we only know the precision ofour model, if we can compare it with a better model, but this better model is exactly what we try toavoid as it is usually unnecessarily complex! In general we don’t know the problem and quality of itsaccompanying model well enough to be absolutely sure that the sought description is both consistent,complete and sufficiently accurate for the purpose, ànd not too formidable for any treatment. A modelis, therefore, to a certain extent a vague concept.

On the other hand, however, modelling plays a key rôle in applied mathematics, since mathematicscannot be applied to any real world problem without the intermediate steps of modelling. So modelling isarguably the main or nearly the mainraison d’être of applied mathematics. Therefore, a more structuredapproach is necessary, which is the aim of the present chapter.

Some people definemodelling as the process of translating a real-world problem into mathematical terms.We will not do so, as this definition is too wide to include the subtle aspects of “limited precision”.

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2. MODELS

A model may be too crude, but also it may be too refined. It is too crude if it just doesn’t describethe problem considered, or if the numbers it produces are not accurate enough to be acceptable. It istoo refined if it includes irrelevant effects that make the problem untreatable, or make the model socomplicated that important relations or trends remain hidden.

Therefore we will introduce the word “mathematizing”, defined as the process of translating a real-worldproblem into mathematical terms. It is a translation in the sense that we translate from the inaccurate,verbose “everyday” language to the language of mathematics. For example, the geometrical presence ofobjects in space may be described parametrically in a suitable coordinate system. In a similar way, anyproperties that are known and expected to play a rôle may be formulated, in some suitable accuracy, bya field or function in time and/or space, explicitly or implicitly, for example as a differential equation.

A very important point to note is the fact that such a formulation isalways at some level simplified. Theearth can be a point or a sphere in astronomical applications, or an infinite half-space or just absent inproblems of human scale. Based on the level of simplification, or, put in another way, of sophisticationor accuracy, we can associate an inherent hierarchy to the set of possible descriptions.

Mathematizing is an elementary but not trivial step. In fact, it forms the single most important step in theprogress of science. It requires the distinction, naming, and exact specification of the essential relevantelementary objects and their interrelations, where mathematics acts as a language in which the problem isdescribed. If theory is available for the mathematical problem obtained this way, the problem consideredmay be subjected to the strict logic of mathematics, and reasoning in this language will transcend overthe limited and inaccurate ordinary language.

Mathematizing is therefore not only important scientifically in general, it is also important from a morerestricted mathematical point of view, because it provides the link between the mathematical world andthe real world. This is important for mathematics in general because the most important, most chal-lenging, and therefore probably most interesting mathematical problems are those originating from areal-world problem.

The ultimate goal for mathematizing a problem is a deeper understanding and a more profound analysisand solution of the problem. Now a refined translation is usually more accurate but also more complicatedand more difficult – if not impossible! – to analyse and solve than a simpler one. Therefore, not everymathematical translation is a good one.

We will call a good mathematical translation amodel or mathematical model if it is lean in the sense, thatit describes our problem quantitatively or qualitatively in a suitable or required accuracy with aminimalnumber of essentially different parameters and variables. (We say “essentially different”, in view ofa reduction that is always possible by writing the problem in dimensionless form. See Buckingham’stheorem I.7 below.)

Again, this definition is rather subjective, as it greatly depends on the context of the problem consideredand our knowledge and resources. So there will rarely be one “best” model. At the same time, it showsthat modelling, even if relying significantly on intuition, is part of the mathematical analysis.

2 Models

We will distinguish the following three classes of models.

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CHAPTER I. THE ART OF MODELLING

• Systematic models.Other possible names arereducing models or asymptotic models. Here one starts to use availablecomplete models, which are adequate, but so over-complete that effects are included which areirrelevant, uninteresting, or negligibly small, and thereby making the mathematical problemunnecessarily complex. By using available additional information (order of magnitude of theparameters) assumptions can be made which minimize in a systematic way theover-completemodel into agood model: small parameters are taken zero, large parameters become infinite, analmost symmetry becomes a full symmetry.

Example I.1 Examples of asymptotic models are numerous. Any ordinary flow is (usually)described by a model which is reduced from the full (i.e. compressible, viscous) Navier-Stokes equations. For the sake of argument we take a simple convection-diffusion problem.For a temperature fieldT and given velocity fieldv in spacex and timet and diffusioncoefficientα we have the assumed “over-complete” model

∂T

∂ t+ v ·∇T = α∇2T, (2.1)

which is, however, difficult to solve. If we have reasons to assume that the diffusion termmay be ignored, we obtain the reduced problem

∂T

∂ t+ v ·∇T = 0, (2.2)

which is far more attractive than the full problem, as it may be solved exactly along thestreamlines as

d

dtT (ξ(t), t) = 0 → T (ξ (t), t) = constant, where

dξ

dt= v. (2.3)

�

• Constructing modelsOther possible names arebuilding block models or lumped parameter models. Here we buildour problem description step by step from low to high, from simple to more complex, by addingeffects and elements like building blocks, until the required accuracy or adequacy is obtained.

Example I.2 The following problem of air release by a simple air pump may be an exampleof a building block model. Consider a pump of cross sectionS and lengtha(t), whichdepends on the piston position. Under pressure, the enclosed volume of airSa(t) leavesthe pump through a small hole, forming a jet of cross sectionSj and (mean) velocityv j .From timet = 0, a spring pushes against the piston with a forceF = λa(t). Assuming anyinertia effects of the piston to be much smaller than the inertia of the flow, the piston forceis balanced by a pressure increase fromp∞ to p0 inside the pump. SoF = S(p0 − p∞).Assuming incompressible air of densityρ0, conservation of mass tells us that {the variationinside} =−{what goes out}. In formula this is

d

dt

(ρ0Sa(t)

)= −ρ0v j S j −→ v j = − S

Sj

da

dt(2.4)

From Bernoulli’s law, relating pressurep and velocityv by p+ 12ρ0v

2 = constant, and notingthat the pressure inside the jet is equal to the atmospheric pressurep∞, we can deduce that

p0 = p j + 12ρ0v

2j = p∞ + 1

2ρ0v2j −→ 1

2ρ0v2j =

λa

S(2.5)

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2. MODELS

Together we have the model

da

dt= −K

√a, with K = Sj

S

√2λ

ρ0S(2.6)

which is easily solved by

a(t) = L(1− 1

2

K t√L

)2along 0≤ t ≤ 2

√L

K. (2.7)

�

• Canonical models.Other possible names arecharacteristic models or essential models. Here an existing model isfurther reduced to describe only the essence of a certain aspect of the problem considered. Thesemodels are particularly important if the mathematical analysis of a model from one of the othercategories is lacking available theory. The development of such theory is usually hindered bytoo much irrelevant details. These models are useful for the understanding, but usually far awayfrom the original full problem setting and therefore not suitable for direct industrial application.

Example I.3 The Navier Stokes equation describing the behaviour of incompressible viscousflow in terms of a velocity and pressure field is in general very complex.

∂

∂ tv + v·∇v = − 1

ρ∇ p + ν∇2v (2.8)

Especially the coupling between the non-linear and viscous terms yielding instabilities andturbulence is complicated and difficult to analyse. Therefore, Burgers proposed to considerthe following very simplified version of it, where the pressure gradient has been neglected,and only behaviour in one dimension is taken into account. This equation

∂

∂ tu + u

∂

∂xu = ν

∂2

∂x2u (2.9)

is called Burgers’ equation [36]. A great deal of insight is obtained by the remarkable trans-formation

u = −2νφ−1 ∂

∂xφ (2.10)

found independently by Cole (1951) and Hopf (1950), by which the nonlinear equation isreduced to a linear equation, related to the heat equation

∂

∂ tφ − ν

∂2

∂x2φ = C(t)φ (2.11)

whereC(t) is an arbitrary function oft . This equation is well understood and allows manyexact solutions. �

Note that an asymptotic model may start as a building-block model, which is only found at a later stageto be too comprehensive. Similarly, a canonical model may reduce from an asymptotic model if the latterappears to contain a particular, not yet understood effect, which should be investigated in isolation beforeany progress with the original model can be made.

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3 Non-dimensionalization and scaling

Modelling implies to decide which effects are relevant and should be included, and which are irrelevantand can be ignored. More in general, we may expect a hierarchy in relevance, from most dominant, vialess relevant and locally irrelevant to absolutely unimportant effects or contributions.

Relevant and irrelevant are rather vague qualifications. To make this operational we will relate them tosmall and large terms in our mathematical description (equations, etc.).

Small and large have no absolute meaning, as long as we have not defined our “measuring stick”. Toillustrate this we may imagine the following science fiction scenery. Suppose we are lost in outer space,with all planets, stars, and galaxies so far away that they are only seen as sizeless spots on our retina.Then, a rock drifts slowly into our field of vision. As long as we are not close enough for a stereoscopicview with both our eyes, we are not able to compare its size or distance with anything we know. Thereis no way to estimate if it is big and far away, or small and nearby. Only the rock itself is our scale ofreference.

A similar experience is found when we look into a microscope of unknown amplification. An object,visible but not recognizable, may be as big as an amoeba, or as small as a virus or a molecule. Re-interpreting the famous saying of Protagoras,Man is the measure of all things, nothing we observe issmall or large, fast or slow, in any absolute sense. It is only by comparison that these qualifications havea meaning.

The next question is: what do we use for comparing. We can use an absolute or universal measuringstick, like a meter or a kilogram, if we want to archive the observations in order to be able to reproducethem exactly again. However, we use a natural scale, like typical sizes in the problem itself, if we wantto classify the type of phenomena.

Inherent scaling

When we model, we need to understand the problem in advance to a certain degree, such that we areable to formulate the relevant physical laws and relations. Therefore, in modelling the natural scaling isthe appropriate one to use. We introduce for all our dependent and independent variables typical values,taken from the problem in question. For example, a lengthL for the independent spatial variablex , anda velocityV for the dependent variablev, and thus an intrinsic timeL/V for time coordinatet .

Dimensionless numbers

When more than one problem parameters in the same units is available, for example a lengthL and widthD, or a timeL/V and a frequencyω, it is inevitable that if one is selected for the scaling, the combinationwith the other gives us a new parameter, likeD/L or ωL/V , which is now independent of the units(meters, seconds) and is therefore called adimensionless parameter. Incidentally, this meaning of theword “dimension” has nothing to do with the mathematical meaning of the number of independent basisvectors in a vector space. Dimensionless parameters are very important for a systematic classification oftypes of problems. They measure the relative importance of certain effects in an absolute way.

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3. NON-DIMENSIONALIZATION AND SCALING

Principle of Dimensional Homogeneity

If the model is a proper one, reflecting the intrinsic relations between the variables, it should not dependon the arbitrary use of meters or inches, etc. Terms which are added or subtracted should have the sameunits. The equations should be dimensionally homogeneous.

Example I.4 Consider the following model of a quantityx satisfying the equation

ax2+ bx + c = 0. (3.1)

Assume thatx denotes a length, with units in meters, denoted by[x] = m, andc is a velocity withunits in meters per second, or[c] = m/s. If the equation is dimensionally homogeneous with[ax2] =[bx] = [c], the units of the other parametersa andb cannot be else but[a] = [c]/[x2] = 1/ms and[b] = [c]/[x] = 1/s. Therefore, we can scale time and length on several combinations as follows

x = c

bX, a = b2

cα −→ αX2 + X + 1= 0, (3.2a)

or (if ac > 0)

x =√( c

a

)X, b = √

(ac)β −→ X2 + βX + 1= 0, (3.2b)

or

x = b

aX, c = b2

aγ −→ X2 + X + γ = 0. (3.2c)

The constantsα, β, andγ are dimensionless constants, parametrizing the respective reduced problem.It should be noted that any of these scalings are equivalent (no information is lost), but they are notequally useful. The preferred reduction is the one in whichx is scaled on a value typically occurringin the situation considered, andX is henceforth of order unity. So a careful inspection of the range ofnumerical values ofx and the parametersa, b, c is essential. Only then the dimensionless parameter,finally left, can tell us more about the behaviour ofX . �

Example I.5 Consider an object of typical sizeL has initially a temperature distributionT (x,0) =T0(x). The temperatureT satisfies a heat diffusion equation with diffusion constantα. The edges ofthe object are kept at a constant temperatureTedge= 0. We scalex on L, the only length scale in theproblem. As the problem is linear, it is not really necessary to scaleT , but we could use the mean, ormaximum value ofT0. There is no explicit time scale in the problem, for example from an externalsource. The only parameter with the dimension of time isL2/α. If we assume a balance betweendecay and diffusion, the typical decay time (the half-life, say) is just of the order ofL2/α, which istherefore the natural time to scale on.

∂T

∂ t= α∇2T,

Tedge= 0, T (x,0) = T0(x)

t = t0τ

x = Lξ

}1

t0

∂T

∂τ= α

L2∇2ξ T

T

L

�

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Example I.6 A piece of metal of sizeL is heated by applying an electric field with potentialφ andtypical voltageV . This heat source is given|∇φ|2 (see for an extensive description example I.12).From a balance of the storage, dissipation and source terms, it follows that the generated heat isdissipated through the metal with a typical decay time ofO(ρcL2/k), and the final temperature ofthe stationary state is typicallyO(σV 2/k).

ρc∂T

∂ t= k∇2T + σ |∇φ|2, T (x, y,0) = 0

T = T0θ

t = t0τ

x = Lξ

φ = Vψ

ρcT0

t0

∂θ

∂τ= kT0

L2∇2

ξ θ +σV 2

L2|∇ξψ|2

L

�

3.1 Dimensional Analysis

In any description of reality, the variables & parameters have physical dimensions and are therefore di-mensionally related. For example, any relation or physical law is only universally valid if the expressionor expressions are dimensionally homogeneous, and independent of the physical dimensions used. Thisfact alone implies that the number of essentially independent groups of variables & parameters is lessthan the number of variables & parameters.

Buckingham’s �-theorem

Suppose we have a physical variableq0, described byn variables & parameters:q1, q2, . . . , qn, in rindependent physical dimensions (kg, m, sec, A, K,. . . ): d1, d2, . . . , dr . [q] denotes the dimension, orunit of scale, ofq. It is not even necessary that there is an explicit model, only some relation like

q0 = f (q1, q2, . . . , qn). (3.3)

In general, f is any combination of algebraical and transcendental expressions. Any transcendentalfunctions like sin and exp are dimensionless, so the argument should be dimensionless too. Hence, wecan assume thatf can be written as

q0 = qγ11 qγ2

2 . . . qγnn �(R1, . . . , Rm) (3.4)

where� depends onm dimensionless groups ofq1, . . . , qn of the formR j = qα11 qα2

2 . . . qαnn .

As the number (m) of essentially different dimensionless groups tells us something about the complexityof the model involved (and for a good model this shouldn’t be too high; see the above sections I.1,2), weare interested to determinem, i.e. to determine how many dimensionless groups are at most possible.

From thePrinciple of dimensional homogeneity each group has the same dimension, so

[R j ] = [qα11 qα2

2 . . . qαnn ] = · · · = 1 . (3.5)

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From the same Principle the dimension[q j ] of eachq j may be written as

[q j ] = dµ1 j

1 dµ2 j

2 . . . dµr jr =

r∏i=1

dµi j

i (3.6)

So we have n∏

j=1

qα j

j

= n∏

j=1

r∏i=1

dµi j α j

i =r∏

i=1

d∑n

j=1 µi j α j

i = 1 (3.7)

In other words, the products∏n

j=1 qα j

j are the possible dimensionless groups that describe our problem.The number of possible groups is the number of (non-trivial) solutions of

n∑j=1

µi jα j = 0 for i = 1,2, . . . , r and anyα j , (3.8)

or in matrix notationµ11 µ12 . . . µ1n

µ21...

. . .

µr1 µrn

α1

α2...

αn

=

00...

0

(3.9)

Hence, we have ann − r dimensional (non-trivial) solution space, as the number of homogeneous solu-tions is equal to the dimensionn of the solution space minus the rank of the matrixk.

In the same way we find forq0

[q0] = dµ101 dµ20

2 . . . dµr0r =

r∏i=1

dµi0i (3.10)

that the possibleγ1, . . . , γn satisfyµ11 µ12 . . . µ1n

µ21...

. . .

µr1 µrn

γ1

γ2...

γn

=

µ10

µ20...

µn0

(3.11)

and thus have in generaln − r + 1 solutions: one solution of the inhomogeneous problem andn − rsolutions of the homogeneous problem.

This yields Buckinghams’�-theorem (� stands for the products).

Theorem I.7 (Buckingham’s�-theorem). If a physical quantity is described byn variables & param-eters inr dimensions, the number of essentially different problem parameters is at mostn − r . Thenumber of possible dimensionless groups, including the primary quantity, is at mostn − r + 1. Di-mensionless groups consisting of a combination of only problem parameters are calleddimensionlessnumbers. Groups consisting of a combination of parameters and several variables are calledsimilarityvariables.

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Example I.8 Consider the dragD - the reaction force due to the surrounding flow - of a sphere ofradiusa moving with velocityV in a viscous fluid with viscosityη and densityρ. We assume, as ourmodel, that the dragD is only dependent ofρ, V , η anda. Of course, this is only true for relativelylow velocities, an infinite medium, a relatively large sphere, etc., but this is our basic assumption.

Now we verify the dimensions of the parameters

[D] = kg m/s2, [ρ] = kg/m3, [V ] = m/s, [η] = kg/m s, [a] = m (3.12)

and conclude that we can expressD in at most 4−3= 1 dimensionless constant. The common formis the Reynolds number:Re = ρV L/η. The drag may be scaled on the pressure difference betweenfront and back of the sphere12ρV 2a2, or on the viscous friction due to wall shear stressaVη. Thisleads to the following functional relations

D = 12ρV 2a2F(Re ) = aVη G(Re), whereRe = ρV L

η. (3.13)

The first one is the proper scaling for nearly inviscid flow (Re large), and the second one for veryviscous flow (Re small). �

Example I.9 A famous example, originally due to G.I. Taylor [2], is the analysis of the shock frontpropagation of a very intense (e.g. nuclear bomb) explosion. From physical considerations the radiusof the shock wave frontR depends, during the early stages of the explosion when the pressure insidethe shock wave is much higher than outside, only on the time intervalt since the explosion, the initialenergyE and the initial air densityρ0. Since[R] = m, [t] = s, E = kg m2/s2, andρ0 = kg/m3, wehave only 3− 3= 0 dimensionless groups. In other words, we can expressR as

R = constant( E

ρ0

)1/5t2/5. (3.14)

The full solution to the appropriate gas dynamical problem showed that the constant has a value closeto unity. �

Example I.10 In the convection problem

∂u

∂ t+ U0

∂u

∂x= 0, u(x,0) = F(x) (3.15)

there is the length scaleU0t and a length scale, sayL, hidden in the initial profileF(x), asx cannotoccur on its own. The dimensions ofu andF are the same, sayF0, and we writeF(x) = F0 f ( x

L ).We scalex = Lξ , t = L

U0τ , u = F0ν, andν(ξ,0) = f (ξ), to get

∂ν

∂ t+ ∂ν

∂ξ= 0, (3.16)

with solutionν(ξ, τ ) = f (ξ − τ ). �

Similarity solutions

If the problem contains no other length scale than the spatial variablex itself and no other time scalethan the time variablet itself, dimensionless groups can only occur by combinations ofx andt . Thus,dimensional analysis leads naturally to similarity solutions. This is easiest explained by an example.

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Example I.11 Consider the following semi-infinite one-dimensional heat conduction problem.

∂T

∂ t= α

∂2T

∂x2, with T (x,0) = 0,

∂

∂xT (0, t) = −Q0 (3.17)

xx = 0Scaling:

• As there is no length scale inx or t , the inherent length scale can only be√αt .

• The only temperature in the problem isQ0x or Q0√αt .

Therefore, we suppose:

T (x, t) = Q0xg(η), where the similarity variableη is given by η = x√4αt

. (3.18)

It follows that

12ηg′′ + (1+ η2)g′ = 0 (3.19)

with solution

T (x, t) = Q0x

[1

η√π

exp(−η2)− 1+ erf(η)

]

= Q0

[√4αt

πexp(−η2)− x(1− erf(η))

](3.20)

where erf(x) = 2√π

∫ x

0e−ξ2

dξ . Note that there exists no stationary solution! �

Example I.12 Consider the edge singularity of the (time-dependent) temperature field generated ina homogeneous and isotropic conductor by an electric field. The electric current densityJ and theelectric fieldE satisfy Ohm’s lawJ = σ E, whereσ is the electric conductivity, i.e. the inverse ofthe specific electric resistance. For an effectively stationary current flow the conservation of electriccharge leads to a vanishing divergence of the electric current density,∇· J = 0. The electric fieldE satisfies∇ × E = 0, and therefore has a potentialφ, with E = −∇φ, satisfying∇·(σ∇φ) = 0.The electric conductivityσ is a material parameter which is quite strongly dependent on temperature.Nevertheless, to make progress we will assume a constantσ , independent ofT . This, then, leads tothe Laplace equation forφ

∇2φ = 0. (3.21)

The heat dissipated as a result of the work done by the field per unit time and volume (Ohmic heating)is given by Joule’s lawJ ·E, and leads to the heat-source distribution

σ |∇φ|2. (3.22)

Since energy is conserved, the net rate of heat conduction and the rate of increase of internal energyare balanced by the heat source, which yields the equation for temperatureT

ρC∂T

∂ t= k∇2T + σ |∇φ|2. (3.23)

The thermal conductivityk, the densityρ and the specific heat of the materialC are mildly dependenton temperature, so we assumed these parameters constant.

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Since we are only interested in the role of the edge, the conductor is modelled, in cylindrical(r, θ)-coordinates, as an infinite wedge-shaped two-dimensional region (without any geometrical lengthscale) 0≤ θ ≤ ν with an electric field with potential

φ(x, y) = (ν/π)A rπ/ν cos(θπ/ν), (3.24)

while the temperature distributionT due to the heat generated by this source is then given by

ρC∂T

∂ t= k∇2T + σ A2r2π/ν−2 (3.25)

with boundary and initial conditions

∂T

∂θ= 0 at θ = 0, θ = ν, T (x, y, t) ≡ 0 at t = 0. (3.26)

Since there are no other (point) sources inr = 0, we have the additional condition of a finite field atthe origin: 0≤ T (0,0, t) < ∞. Boundary conditions and the symmetric source imply thatT is afunction oft andr only, so that equation (3.23) reduces to

ρC∂T

∂ t= k

(∂2T

∂r2+ 1

r

∂T

∂r

)+ σ A2r2π/ν−2. (3.27)

Owing to the homogeneous initial and boundary conditions, the infinite geometry, and the sourcebeing a monomial inr , homogeneous of the order 2π/ν − 2, there is no length scale in the problem

other than√( kt

ρC

), while the temperatureT can only scale onσ A2

k

( ktρC

)πν . This indicates that a

similarity solution is possible of the following form

T (r, t) = σ

4kA2(4kt

ρC

)π/ν

h(X), X = ρcr2

4kt, (3.28)

whereX is a similarity variable, reducing equation (3.27) to

Xh′′ + (1+ X)h′ − (π/ν)h = −Xπ/ν−1. (3.29)

This equation may be recognized as an inhomogeneous confluent hypergeometric equation in−X ,which has the regular solution

h(X) = constant×1F1(−π/ν; 1; −X)− (ν/π)2Xπ/ν (3.30)

where1F1(a; b; z) is the regular confluent hypergeometric function.

From the asymptotic expansion of1F1(−π/ν; 1; −X) and the initial-value condition, the integrationconstant is found to be(ν/π)�(π/ν). Putting everything together, we have the solution

T (x, y, t) = σν2A2

4π2k

[(4kt

ρC

)π/ν

�(1+ π

ν

)1F1

(−π

ν; 1; −ρCr2

4kt

)− r2π/ν

]. (3.31)

�

4 Scaling of the compressible Navier-Stokes equations

One of the most fruitful uses of scaling is the hierarchy it provides in comprehensive and rich models.In most applications, such over-complete models are not truly adapted to the problem, and therefore nota true, “lean” model as we discussed above. From a suitable, inherent scaling the order of magnitude

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4. SCALING OF THE COMPRESSIBLE NAVIER-STOKES EQUATIONS

of the various contributions or effects can be estimated, leading to a hierarchy on which we can base anasymptotic modelling.

We will give here a (maybethe) most important example, the scaling and reduction of the compressibleNavier-Stokes equations.

The original laws of mass, momentum and energy conservation, written in terms of pressurep, densityρ, velocity vectorv, viscous stress tensorτ , internal energye, and heat flux vectorq, are given by

mass: ∂∂t ρ +∇·(ρv) = 0 (4.1a)

momentum: ∂∂t (ρv) +∇·(ρvv) = −∇ p +∇·τ (4.1b)

energy: ∂∂t (ρe + 1

2ρv2)+∇·(ρev + 1

2ρv2v) = −∇·q −∇·(pv)+∇·(τv) (4.1c)

Depending of the application, it is often convenient to introduce enthalpyh = e+ p/ρ, or entropys andtemperatureT via the fundamental law of thermodynamics for a reversible process

T ds = de + pdρ−1 = dh − ρ−1dp. (4.2)

With ddt = ∂

∂t + v ·∇ for the convective derivative, the above conservation laws may be reduced to

mass: ddt ρ = −ρ∇·v (4.3a)

momentum: ρ ddt v = −∇ p +∇·τ (4.3b)

energy : ρ ddt e = −∇·q − p∇·v + τ :∇v (4.3c)

ρ ddt h = d

dt p −∇·q + τ :∇v (4.3d)

ρT ddt s = −∇·q + τ :∇v. (4.3e)

For an ideal gas we have de = CV dT and dh = CPdT such that

energy 1: ρCVddt T = −∇·q − p∇·v + τ :∇v (4.3f)

energy 2: ρCPddt T = d

dt p −∇·q + τ :∇v (4.3g)

For an incompressible fluid we haveT ds = CPdT with

energy 3: ρCPddt T = −∇·q + τ :∇v (4.3h)

CV is the heat capacity or specific heat at constant volume.CP is the heat capacity or specific heat atconstant pressure [18]. In general (for an ideal gas) they may depend on temperature. For a perfect gasthey are constants. For an incompressible fluid like a liquidCV is practically equal toCP .

Constitutive equations

These 5 equations are less than the number of unknowns. Therefore, the mathematical description is notcomplete without some “closure” relations, describing properties of the matter and certain instantaneous(both in time and space) interactions between material parameters. These relations are called the consti-tutive equations. They are not as basic as the conservation laws, and their form will depend on the modeladopted. In fact, this part of the model, the constitutive equations, may be called: the physical model.We will consider here an ideal, heat conducting and viscous gas.

Ideal gas relation: p = ρRT, (4.4)

Fourier’s heat flux model:q = −κ∇T, (4.5)

Newton’s viscous stress tensor:τ = η(∇v +∇v†)− 23η(∇·v)I . (4.6)

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Scaling, non-dimensionalisation:

Assume that we have the following typical scales for velocity, pressure, density and temperature.

v with v0,�v; p with p0,�p; ρ with ρ0,�ρ; T with �T ; x with L; t with f −1

where a subscript “0” refers to the primary variable, and� denotes a typical difference. The typical timeis written as the inverse of frequencyf . Note that sometimes we have more candidates for a scaling pa-rameter. For example, the typical frequency may be enforced by an external source or instability, or maybe inherited from the intrinsic hydrodynamic frequencyv0/L, or the inverse diffusion timeκ/ρ0CP L2.

The scaled equations are now, symbolically, with energy equation 2,

f �ρ ,v0�ρ

L= ρ0�v

L

fρ0�v ,ρ0v0�v

L= �p

L,η�v

L2

fρ0CP�T ,ρ0CPv0�T

L= f �p ,

v0�p

L,κ�T

L2,η�v2

L2

From gas laws we know that the typical sound speed squaredc20 scales on bothp0/ρ0 as�p/�ρ (not

exactly, only in order of magnitude). Furthermore, we will take here the common situation where�v =O(v0), andκ andη are constants. With a rescaling such that all coefficients become dimensionless, wehave

f L�p

v0ρ0c20

,�p

ρ0c20

= 1

f L

v0, 1 = �p

ρ0v20

,η

ρ0v0Lf L

v0, 1 = f L�p

ρ0CPv0�T,

�p

ρ0CP�T.

κ

ρ0CPv0L,

v0η

ρ0CP L�T

4.1 Some dimensionless groups with their common names:

REYNOLDS : Re =ρ0v0L

η, PRANDTL : Pr =

CPη

κ, MACH : M =

v0

c0

STROUHAL : Sr =f L

v0, FOURIER : Fo =

κ f −1

ρ0CP L2, EULER : Eu =

�p

ρ0v20

HELMHOLTZ : He =f L

c0, ECKERT : Ec =

v20

CP�T, PECLET : Pe =

ρ0CPv0L

κ

The Prandtl numberPr depends only of the material, and is for most gases and fluids of order 1. Manydimensionless number are related. For example,Pe = Pr Re , He = Sr M , andSr Fo Pr Re = 1.

A Fourier numberFo = 1 corresponds to a typical time, determined by the heat diffusion process. AStrouhal numberSr = 1 corresponds to a scaling frequencyf = v0/L, corresponding to hydrodynam-ical convection processes. An Euler numberEu = 1 corresponds to a pressure scaled on hydrodynamicpressure variations,ρ0v

20, while the combinationEu M2 = 1 corresponds to a pressure scaled on pressure

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variations due to compressible effects,ρ0c20. If the reference speed is taken equal to the sound speed, we

haveM = 1.

The scaled equations are now (written in dimensionless variables)

Eu M2(Sr ∂

∂t ρ + v ·∇ρ) = −ρ∇·v (4.7a)

Sr ρ ∂∂t v + ρv ·∇v = −Eu∇ p + 1

Re∇·τ (4.7b)

Sr ρ ∂∂t T + ρv ·∇T = Eu Ec

(Sr ∂

∂t p + v ·∇ p) + 1

Pe∇2T + Ec

Reτ :∇v (4.7c)

4.2 Asymptotic reductions of the Navier-Stokes equations

When M, Re , Sr , Ec , etc. become small or large, we may derive from equations (4.7) various reducedmodels by assuming the respective terms to vanish or dominate completely, for example:

incompressible : ∇·v = 0,

inviscid : ρ0(

∂∂t v + v ·∇v

)+∇ p = 0

highly viscous : ∇ p = η∇2v

sound waves :∂∂t ρ + ρ0∇·v = 0

ρ0∂∂t v +∇ p = 0

}−→ ∂2

∂t2 p − c20∇2 p = 0

convection-diffusion : ρCP

(∂∂t T + v ·∇T

) = κ∇2T

(4.8)

Potential flow

Another important reduction, not immediately obtainable from small parameter considerations, is thescalar velocity potential that may be introduced for irrotational flow:

if ∇×v = 0, there exists a velocity potentialφ with v = ∇φ.

In incompressible flow this potential is independent of pressure (except indirectly via boundary condi-tions) and satisfies Laplace’s equation∇2φ = 0.

Bernoulli’s law

In incompressible inviscid flow the momentum equation can be integrated along a streamline, leading toan equation, known as Bernoulli’s law, that describes conservation of mechanical energy density:

v ·∇v = 12∇|v|2+ (∇ × v)× v,

integrate along streamline, noting that[(∇ × v)× v]·ds = 0:

0=∫ [

12ρ0∇|v|2+ ρ0(∇ × v)× v +∇ p

]·ds = 12ρ0|v|2+ p − constant.

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Exercises

1. The dragD of a moving ship, due to viscous effects and wave generation, depends on its lengthL, velocity V , water viscosityη, gravity g and water densityρ. The dimensional units are[D] =kg m/s2, [L] = m, [V ] = m/s,[η] = kg/ms,[g]=m/s2, and[ρ]= kg/m3. By how many dimensionlessgroups is the problem completely described? Give an example of such a set of dimensionless groups(these are not uniquely defined).

2. Analyse example I.12 in terms of Buckingham’s theorem. Verify the dimensional groups, includingtheir number.

3. Determine the small or large dimensionless numbers corresponding to the above reduced models(4.8). Be aware of possible combined limits.

4. Reconsider example I.1. Make the problem complete by adding boundary and initial conditions.Make the problem dimensionless by scaling on the inherent time and length scales. Determine con-ditions, in terms of a dimensionless number, for which the diffusion term can be neglected. Note thatthe order of the differential equation is then reduced from 2 to 1. What are the consequences for theboundary and/or initial conditions?

5. Simplify, by suitable scaling of the independent variables, the equations of examples?? (i) and (iii)such that the coefficients become just equal to 1.

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Chapter II

Perturbation methods

Inherent in any modelling is the hierarchy in importance of the various effects that constitute themodel. Therefore, certain effects in any modelling will be small. Sometimes small but not smallenough to be ignored, and sometimes small but in a non-uniform way such that they are importantlocally.

For an efficient solution, and to obtain qualitative insight, it make sense to utilize this “smallness”.Methods that systematically exploit such inherent smallness are called “perturbation methods”.

Perturbation methods have a long history. Before the time of the numerical methods and the com-puter, perturbation methods were the only way to increase the applicability of available exact solu-tions to difficult, otherwise intractable problems. Nowadays, perturbation methods have their use asa natural step in the process of systematic modelling, the insight it provides in the nature of singu-larities occurring in the problem and typical parameter dependencies, and sometimes the speed ofpractical calculations.

1 Introduction

We have seen above that a real-world problem can be described by a hierarchy of models, such that ahigher level model is more comprehensive and more accurate than one from a lower level. Now supposethat we have a fairly good model, describing the dominating phenomena in good order of magnitude. Andsuppose that we are interested in improving on this model by adding some previously ignored aspects oreffects. In general, this implies a very abrupt change in our model. The equations are more complex andmore difficult to solve.

So it seems that with our model, necessarily also our solution method is abruptly more complex. Some-times, this is an argument to retain the simpler model. This is, however, only true if we are merelyinterested in exact or numerically “exact” solutions. We have to keep in mind that an exact solution of anapproximate model is not better than an approximate solution of an exact model. So there is absolutelyno reason to demand the solution to be more exact than the corresponding model.

Let’s go back to our “fairly good”, improved model. The effects we added are relatively small; otherwise,the previous lower level model was not fairly good as we assumed, but just completely wrong. Usually,this smallness is quantified by small dimensionless parameters occurring in the equations and (or) bound-ary conditions. This is a generic situation. The transition between a higher-level and a lower-level theory

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1. INTRODUCTION

is characterized by the presence of one or more inherent modelling parameters, which are large or small,and yield in the limit a simpler description.

Examples are: simplified geometries reducing the spatial dimension, small amplitudes allowing lin-earization, low velocities and long time scales in flow problems allowing incompressible description,small relative viscosity allowing inviscid models, zero or infinite lengths rather than finite lengths, etc.

If we accept approximate solutions, where the approximation is based on the inherent small or largemodelling parameters, we do have the possibilities to gradually increase the complexity of a model, andstudy small but significant effects in the most efficient way.

The methods utilizing systematically this approach are called “perturbations methods”. The approxima-tion constructed is almost always an asymptotic approximation, i.e. where the error reduces with thesmall or large parameter.

Usually, a distinction is made between regular and singular perturbations. A (loose definition of a)regular perturbation problem is where the approximate problem is everywhere close to the unperturbedproblem. This, however, depends of course on the domain of interest and, as we will see, on the choiceof coordinates.

If a problem is regular without any need for other than trivial reformulations, the construction of anasymptotic solution is straightforward. In fact, it forms the usual strategy in modelling when terms arelinearised or effects are neglected. The more interesting perturbation problems are those where thisstraightforward approach fails.

We will consider here four methods relevant in the presented modelling problems. The first two areexamples of regular perturbation methods, but only after a suitable coordinate transformation. They arecalled the method of slow variation, where the typical axial length scale is much greater than the trans-verse length scale. The next one is the Lindstedt-Poincaré method or the method of strained coordinates,for periodic processes. Here, the intrinsic timescale (∼ the period of the solution) is unknown and hasto be found.

The other methods are of singular perturbation type, because there is no coordinate transformation pos-sible that renders the problem into one of regular type. The first one is the method of multiple scalesand may be considered as a combination of both methods above, as now several (long, short, shorter)time scales act together. This cannot be repaired by a single coordinate transformation. Therefore, theproblem is temporarily reformulated into a higher dimensional problem by taking the various lengthscales apart. Then the problem is regular again, and can be solved. A refinement of this method is theWKB method, where the coordinate transformation of the fast variable becomes itself slowly varying.The last singular perturbation method considered here is the method of matched asymptotic expansions.Also here different scalings are necessary, but they don’t exist together, but in spatially distinct regions(boundary layers).

Before we can introduce the methods, we have to define our terminology of asymptotic approximationsand asymptotic expansions.

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CHAPTER II. PERTURBATION METHODS

2 Asymptotic approximations and expansions

2.1 Asymptotic approximations

In order to describe qualitatively the way a functionf of parameterε behaves near a point of interest,sayε = 0 (equivalent to any other value by a simple translation), we have the so-called order symbolsO, o, andOs . Oftenε = 0 is the lower limit of a parameter range, and we have the tacit assumption thatε ↓ 0.

Definition II.1.

1. f (ε) = O(φ(ε)) as ε → 0 if there are constantsK andε1 (independent ofε) such that

| f (ε)| ≤ K |φ(ε)| for 0 < ε < ε1.

2. f (ε) = o(φ(ε)) as ε → 0 if for every δ > 0, there is anε1 (independent ofε) such that

| f (ε)| ≤ δ|φ(ε)| for 0 < ε < ε1.

3. f (ε) = Os(φ(ε)) as ε → 0 if f (ε) = O(φ(ε)) and f (ε) �= o(φ(ε)).

Theorem II.2.

1. If f = o(φ) asε → 0, then also f = O(φ).

2. If the limit limε→0

f (ε)

φ(ε)exists as a finite number, thenf = O(φ) as ε → 0.

3. If the limit limε→0

f (ε)

φ(ε)exists as a finite number�= 0, then f = Os(φ) as ε → 0.

4. If limε→0

f (ε)

φ(ε)= 0, then f = o(φ) as ε → 0.

Example II.3

ε sin(ε) = Os(ε2), ε → 0,

ε cos(ε) = O(1), ε → 0,

εn = o(1), ε → 0, for any positiven,

e−1/ε = o(εn), ε → 0, for any positiven.

e−1/ε is called a transcendentally (TST) or exponentially small term (EST) and can be ignored asymp-totically against any power ofε.

Definition II.4. φ(ε) is an asymptotic approximation tof (ε) asε → 0 if

f (ε) = φ(ε)+ o(φ) as ε → 0,

sometimes more compactly denoted byf ∼ φ.

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2. ASYMPTOTIC APPROXIMATIONS AND EXPANSIONS

If f andφ depend onx, this definition remains valid pointwise, i.e. forx fixed. It is, however, useful toextend the definition to uniformly valid approximations.

Definition II.5. If f (x; ε) andφ(x; ε) are continuous functions forx ∈ D and 0< ε < a, we callφ(x; ε) a uniformly valid asymptotic approximation tof (x; ε) for x ∈ D asε → 0 if for any positivenumberδ there is anε1 (independent ofx andε) such that

| f (x; ε)− φ(x; ε)| ≤ δ|φ(x; ε)| for x ∈ D and 0< ε < ε1.

We write: f (x; ε) = φ(x; ε)+ o(φ) uniformly in x ∈ D as ε → 0. Note thatD may depend onε.

Example II.6 Consider D = [0,1], 0 < ε < 1. Then we have cos(εx) = 1+ o(1) as ε →0 uniformly inD, since for any givenδ we can chooseε1 =

√δ, such that| cos(εx)−1| ≤ ε2x2 ≤

ε21 = δ.

Example II.7 Although cos(x/ε) = O(1) uniformly in x ∈ [0,1] for ε → 0, there is no constantKsuch that cos(x/ε) = K + o(1).

Example II.8 x + sin(εx)+ e−x/ε = x + εx + O(ε3) asε → 0 for all x �= 0, but not uniformly inx ∈ [0,1]. If x = 0, the otherwise exponentially small term is not small anymore and equal to unity.

0

0.2

0.4

0.6

0.8

1

y

0.2 0.4 0.6 0.8 1x

In the figure we may compare the originalx+sin(εx)+e−x/ε and its non-uniform asymptotic approx-imationx + εx for ε = 0.01. Note that difference between the original function and its non-uniformasymptotic approximation is typically large in a neighbourhood ofx = 0, while the size of thisneighbourhood isx = O(ε). This neighbourhood is an example of a boundary layer. The occurrenceand behaviour of boundary layers will be discussed in more detail in the next section.

2.2 Asymptotic expansions

Definition II.9. The sequence{µn(ε)}∞n=0 is called an asymptotic sequence, ifµn+1(ε) = o(µn(ε)), asε → 0, for eachn = 0,1,2, · · · .

Example II.10 Examples of asymptotic sequences (asε → 0) are

µn(ε) = εn, µn(ε) = ε12n

, µn(ε) = tann(ε), µn(ε) = ln(ε)−n,

µn(ε) = ε p ln(ε)q where p = 0,1,2..., q = 0...p and n = 12 p(p + 3)− q. (2.1)

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Definition II.11. If {µn(ε)}∞n=0 is an asymptotic sequence, thenf (ε) has an asymptotic expansion ofNterms with respect to this sequence, given by

f (ε) ∼N−1∑n=0

anµn(ε)

wherean are independent ofε, if

f (ε)−M∑

n=0

anµn(ε) = o(µM ) as ε → 0

for eachM = 0, . . . , N − 1. µn(ε) is called a gauge-function. Ifµn(ε) = εn, we call the expansion anasymptotic power series.

Definition II.12. Two functions f and g are asymptotically equal up ton terms, with respect to theasymptotic sequence{µn}, if f − g = o(µn). If the remaining error is clear from the context, this issometimes denoted asf ∼ g.

Asymptotic expansions based on the same gauge functions may be added. They may be multiplied if theproducts of the gauge functions can be asymptotically ordered.

In contrast to ordinary series expansions, defined for an infinite number of terms, we consider in asymp-totic expansions only a finite (N ) number of terms. ForN → ∞ the series may either converge ordiverge, but this is irrelevant for the asymptotic behaviour. In addition it may be worthwhile to note thatit is not necessary for a convergent asymptotic expansion to converge to the expanded function.

For given{µn(ε)}∞n=0, an can be determined uniquely by the following recursive procedure (providedµn

are nonzero forε near 0 and each of the limits below exist)

a0 = limε→0

f (ε)

µ0(ε),

a1 = limε→0

f (ε)− a0µ0(ε)

µ1(ε),

...

aN−1 = limε→0

f (ε)−∑N−2n=0 anµn(ε)

µN−1(ε).

Example II.13 Given different asymptotic sequences, a function may have different asymptoticexpansions.

tan(ε) = ε + 13ε

3+ 215ε

5+ O(ε7)

= sin(ε)+ 12 sin3(ε)+ 3

8 sin5(ε)+O(sinh7(ε))

= ε cos(ε)+ 56(ε cos(ε))3+ 161

120(ε cos(ε))5+O((ε cos(ε))7).

Example II.14 The asymptotic expansionN∑

n=1

n! εn diverges asN →∞ if ε �= 0..

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Example II.15 Different functions may have the same asymptotic expansion.

cos(ε) = 1− 12ε

2+ 124ε

4+O(ε6).

cos(ε)+ e−1/ε = 1− 12ε

2+ 124ε

4+O(ε6).

Note that both asymptotic expansions, considered as regular power series inε, converge to cos(ε)rather than cos(ε)+ e−1/ε.

Theorem II.16. An asymptotic expansion vanishes only if the coefficients vanish, i.e.

{µ0(ε)a0+ µ1(ε)a1+ µ2(ε)a2+ . . . = 0 (ε → 0)

} ⇔ {a0 = a1 = a2 = . . . = 0

}.

Proof: {µn} are asymptotically ordered, so bothµ0a0 = −µ1a1 − . . . = O(µ1) andµ1 = o(µ0). Sothere is a positive constantK such that for any positiveδ there is anε-interval where|µ0a0| < δK |µ0|,which is only possible ifa0 = 0. This may now be repeated fora1, etc.

2.3 Perturbation problems

The assumed existence of an asymptotic expansions yields a class of methods to solve otherwise in-tractable problems depending on a typically small parameter. Such methods are called perturbationmethods.

If a(ε) is implicitly given as the solution of an algebraic equation

F (a; ε) = 0 (2.2)

and botha(ε) andF (a; ε) have an asymptotic series expansion with the same gauge functions,a(ε)may be determined asymptotically by the following perturbation method. We expanda, substitute thisexpansion inF , and expandF to obtain

a(ε) = µ0(ε)a0+ µ1(ε)a1+ . . . , (2.3a)

F (a; ε) = µ0(ε)F0(a0)+ µ1(ε)F1(a1, a0)+ µ2(ε)F2(a2, a1, a0)+ . . . = 0. (2.3b)

From theorem II.16 it follows that that each termFn vanishes, and the sequence of coefficients(an) canbe determined by induction:

F0(a0) = 0, F1(a1, a0) = 0, F2(a2, a1, a0) = 0, etc. (2.4)

It should be noted that finding the sequence of gauge functions(µn) is of particular importance. Thisdone iteratively. First the order of magnitude ofa should be determined by seeking the asymptotic scalinga(ε) = γ (ε)A(ε) which yields in the limitε → 0 a meaningfulA = Os(1). This is called a distinguishedlimit, while the reduced equation forA(0) is called a significant degeneration. (There may be more thanone.) The first term of the gauge functions that occurs is nowµ0(ε) = γ (ε), while a0 = A(0). Nowthe procedure may be repeated for the new unknowna(ε)− µ0(ε)a0, and so on. Usually, the rest of thesequence(µn) can be guessed from the structure of the defining equationF = 0.

We illustrate this procedure by the following example.

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Example II.17 Consider the zeros forε → 0 of the polynomial

x3− εx2+ 2ε3x + 2ε6 = 0.

From the structure of the problem it seems reasonable to assume that the solutionsx (1), x (2), x (3)

have an asymptotic expansion in powers ofε. However, not clear is the order of magnitude of theleading order term:

x(ε) = εn(X0+ εX1 + ε2X2 +O(ε3))

Therefore, we have to determinen first. This is done by balancing the terms. We scale

x = εn X (ε), X = O(1)

and we seek thosen that produce under the limitε → 0 a non-trivial limit. We compare asymptoti-cally the coefficients in the equation that remain after scaling:

ε3n X3− ε1+2n X2 + 2ε3+n X + 2ε6 = 0.

In order to have a meaningful (or “significant”) degenerate solutionX (0) = Os(1) at least two termsof the equation should be asymptotically equivalent, and at the same time of leading order whenε → 0. So this leaves us with the task to compare as a function ofn the powers 3n,1+ 2n,3+ n,6.Consider the figure II.1 The drawn lines denote the logarithm of the powers ofε, that occur in

1 2 3

1

2

3

4

5

6

n

3n

1+ 2n

3+ n

6

Figure II.1: Analysis of distinguished limits

the coefficients of the equation considered. At the crossings of these lines, denoted by the openand closed circles, we find the candidates of distinguished limits, i.e. the points where at least twocoefficients are asymptotically equivalent. Finally, only the closed circles are the distinguished limits,because these are located along the lower envelope (thick broken line) and therefore correspond toleading order terms whenε → 0.

We have now three cases.

n = 1.

ε3X3 − ε3X2+ 2ε4X + 2ε6 = 0, or X3 − X2 + 2εX + 2ε3 = 0.

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If we assume the expansionX = X0+ εX1+ . . ., we have finally

X30 − X2

0 = 0, 3X20X1− 2X0X1+ 2X0 = 0, etc.

and soX0 = 1, andX1 = −2, etc.. Note: notX0 = 0 because that would change the orderof the scaling!

n = 2.

ε6X3 − ε5X2+ 2ε5X + 2ε6 = 0, or εX3 − X2 + 2X + 2ε = 0.


−X20 + 2X0 = 0, etc.

and soX0 = 2, etc..n = 3.

ε9X3 − ε7X2+ 2ε6X + 2ε6 = 0, or ε3X3 − εX2 + 2X + 2= 0.


2X0+ 2= 0, etc.

and soX0 = −1, etc..

Not always is it so easy to guess the general form of the gauge functions. Then all terms have to beestimated iteratively, by a similar process of balancing as for the leading order term. See exercise 2.

An asymptotic expansion of a function of variablex and small parameterε of the following form

f (x; ε) =N−1∑n=0

an(x; ε)µn(ε)+O(µN ), an = O(1),

appears to be too general to be of practical use. The restriction thatan depend ofx only appears to befruitfull. This is called

2.4 Asymptotic expansions of Poincaré type

Definition II.18. If {µn(ε)}∞n=0 is an asymptotic sequence, andf (x; ε) has an asymptotic expansion ofN terms with respect to this sequence, given by

f (x; ε) ∼N−1∑n=0

an(x)µn(ε)

where shape functionsan(x) are independent ofε, this expansion is called a Poincaré expansion.

Definition II.19. If a Poincaré expansion is uniform inx on a given domainD (the ε-intervals onlydepend on theδ’s, not onx) this expansion is called a regular expansion. If not, the expansion is calleda singular expansion1.

1There is no uniformity in the literature on the definition of regular and singular expansions.

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Regular expansions may be differentiated to the independent variablex.

Example II.20 The following asymptotic power series expansion withD = R and the gauge func-tions given byεn

cos(x + ε) = cos(x)− ε sin(x)− 12ε

2 cos(x)+ 16ε

3 sin(x)+O(ε4).

is uniform since cos(x) and sin(x) are bounded for allx ∈ R. It follows that it is a regular expansion.

Example II.21 The following expansion

cos(x + ε) e−x/ε + sin(x + ε) = sinx + ε cosx +O(ε2)

is a uniform, and therefore regular expansion on any interval[A,∞), whereA > 0. However, it isa non-uniform, and therefore singular expansion on[0,∞). In fact, on any interval[Aεα,∞) it isregular ifα < 1, and singular ifα ≥ 1.

The role of the independent coordinate

It is absolutely vital for the appreciation of the perturbations methods to be considered below, to note therole of the choice of the independent variablex in a Poincaré expansion. By suitable linear coordinatetransformations of the typex = λ(ε) + δ(ε)ξ we can change and optimize the domain of uniformity,and filter out asymptotically specific behaviour that belongs to one length scale. This filter property isespecially useful when asymptotically analysing models.

Example II.22

• sin(x+εx+ε2x) = sin(x)+εx cos(x)+ε2(x cosx− 12x2 sinx)+O(ε3), which is only uniform

on an interval[0, A], but if we introduceξ = (1+ε)x , we have sin(ξ +ε2x) = sin(ξ)+O(ε2)

uniform in x ∈ [0, Aε−1] for any positive constantA.

• sin(εx + ε) = εx + ε +O(ε3), which is only uniform on a finite interval and, moreover, doesnot show any of the inherent periodicity. If we introduceX = εx , we get the much bettersin(X + ε) = sin(X)+O(ε) which is even uniform inR.

• e−x/ε = 0+ o(εn) is a singular expansion onx > 0, but if we introduceξ = x/ε it becomesthe regular expansione−ξ = O(1) on ξ > 0.

• On x > 0 we have2π

arctan( xε)+ sin(εx)/(1+ x2) = 1+ ε

(x/(1+ x2)− 2/πx

)+ O(ε2) inx , = 1+ ε2

(sin(X)/X2 − 2/π X

) + O(ε3) in X = εx , and= 2π

arctan(x)+ ε2x + O(ε3) inξ = x/ε.

• e−εx sin(x√

1+ ε ) = sinx+εx(12 cosx−sinx)+O(ε2), which is only uniform on an interval

[0, A]. This cannot be improved by a single other choice of independent variable. However, ifwe introducetwo variables,x1 = εx andx2 = (1+ 1

2ε)x , we get the much bettere−x1 sinx2−18ε

2x2 e−x1 cosx2+ O(ε4).

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3. REGULAR PERTURBATION PROBLEMS

3 Regular perturbation problems

If a function�(x; ε) is implicitly given by an equation (usually a differential equation with boundaryconditions), say

L(�, x; ε) = 0 on a domainD (3.1)

and both� andL(�, x; ε) have a regular asymptotic expansion onD with the same gauge functions,(3.1) is called a regular perturbation problem [15]. The shape functions�n are determined as follows.We expandL

L(�, x; ε) = µ0(ε)L0(�0, x)+µ1(ε)L1(�1,�0, x)+µ2(ε)L2(�2,�1,�0, x)+ . . . = 0. (3.2)

According to theorem II.16 each term vanishes, and the sequence(�n) can be determined by induction:

L0(�0, x) = 0, L1(�1,�0, x) = 0, L2(�2,�1,�0, x) = 0, . . . (3.3)

It should be noted that in many interesting cases the problem is only regular after a suitable coordinatetransformation. The major task when solving the problem is then to find this scaled or shifted coordinate.Practically important solution methods of this type are the method of slow variation, for geometricallystretched or slowly varying configurations, and the Lindstedt-Poincaré method, for solutions which areperiodic in time with an unknown,ε-dependent period.

If (3.1) is not a regular perturbation problem, we call it a singular perturbation problem. Practically im-portant solution methods for singular perturbation problems are the method of matched asymptotic ex-pansions, where regular expansions exist locally but not in the whole region considered, and the methodof multiple scales, where 2 or more distinct long and short length scales occur intertwined.

3.1 Method of slow variation

Suppose we have a functionφ(x; ε) of spatial co-ordinatesx and a small parameterε, such that thetypical variation in one direction, sayx , is of the order of length scaleε−1. We can express this behaviourmost conveniently by writingφ(x, y, z; ε) = �(εx, y, z; ε). Now if we were to expand� for smallε,we might, for example, get something like

�(εx, y, z; ε) = �(0, y, z;0)+ ε(x�x (0, y, z;0)+�ε(0, y, z;0))+ . . .

which is only uniform inx on an interval[0, L] if L = o(ε), and the inherent slow variation on the longerscale ofx = O(ε−1) would be masked. It is clearly much better to introduce the scaled variableX = εx ,and a (assumed) regular expansion of�(X, y, z; ε)

�(X, y, z; ε) = µ0(ε)φ0(X, y, z)+ . . . (3.4)

now retains the slow variation inX in the shape functions of the expansion.

This situation frequently happens when the geometry involved is slender [34]. The theory of one dimen-sional gas dynamics, lubrication flow, or sound propagation in horns (Webster’s equation) are importantexamples, although they are usually derived not systematically, without explicit reference to the slendergeometry. We will illustrate the method by an example of heat flow in a varying bar.

10:56 21 Mar 2002 28 version: 21-03-2002


Example II.23 Consider the stationary problem of the temperature distributionT in a long heat-conducting bar with surface normalnS and slowly cross sectionA. The bar is kept at a temperaturedifference such that a given heat flux is maintained, but is otherwise isolated. As there is no leakageof heat, the flux is constant. With spatial coordinates made dimensionless on a typical bar radius, wehave the following equations and boundary conditions

∇2T = 0, ∇T ·nS = 0,∫∫

A

∂T

∂xdσ = Q. (3.5)

Integrate∇2T over a slicex1 ≤ x ≤ x2, and apply Gauss’ theorem, to find that the axial fluxQ isindeed independent ofx . The typical length scale of diameter variation is assumed to be much largerthan a diameter. We introduce the ratio between a typical diameter and this length scale as the smallparameterε, and write for the bar surface

S = r − R(X, θ) = 0, X = εx . (3.6)

By writing R as a continuous function of slow variableX , rather thanx , we have made our formalassumption of slow variation explicit in a convenient and simple way, since∂

∂x R = εRX = O(ε).

The crucial step will now be the assumption that the temperature isonly affected by the geometricvariation induced byR. Any initial or entrance effects are ignored or have disappeared. As a resultthe temperature fieldT is a function ofX , rather thanx , and its axial gradient scales onε, as ∂

∂x T =O(ε).

Introduce the gradient∇S and the transverse gradient∇⊥S

∇S = −εRX ex + er − r−1Rθ eθ , ∇⊥S = er − r−1Rθ eθ . (3.7)

At the bar surfaceS = 0 the gradient∇S is a vector normal to the surface, while the transversegradient∇⊥S, directed in the plane of a cross sectionX = const., is normal to the circumferenceS(X = c, r, θ) = 0.

Inside the bar we have the rescaled heat equation

ε2TX X + ∇2⊥T = 0. (3.8)

At the wall the boundary condition of vanishing heat flux is

∇T ·∇S = ε2TX SX +∇⊥T ·∇⊥S = 0 at S = 0. (3.9)

The flux condition, for lucidity rewritten withQ = εq, is given by∫∫A

∂T

∂Xdσ = q. (3.10)

This problem is too difficult in general, so we try to utilize in a systematic manner the small parameterε. Since the perturbation terms areO(ε2), we assume the asymptotic expansion

T (X, r, θ; ε) = T0(X, r, θ)+ ε2T1(X, r, θ)+O(ε4). (3.11)

After substitution in equation (3.8) and boundary condition (3.9), further expansion in powers ofε2

and equating like powers ofε, we obtain to leading order a Laplace equation in(r, θ)

∇2⊥T0 = 0 with ∇⊥T0·∇⊥S = 0 at S = 0. (3.12)

An obvious solution isT0 ≡ 0. Since solutions of Laplace’s equation with vanishing normal deriva-tives at the boundary are unique up to a constant (here: a function ofX), we have

T0 = T0(X). (3.13)

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3. REGULAR PERTURBATION PROBLEMS

We could substitute this directly in the flux condition, to find thatAT0X = q. We do, however, notneed this global conservation law, and it is interesting to see this result emerging from the equations.To obtain an equation forT0 in X we continue with theO(ε2)-equation and corresponding boundarycondition

∇2⊥T1+ T0X X = 0, ∇⊥T1·∇⊥S = − T0X SX . (3.14)

The boundary condition can be rewritten as

∇⊥T1·n⊥ = T0X RX

|∇⊥S| =T0X RRX√

R2 + R2θ

(3.15)

wheren⊥ = ∇⊥S/|∇⊥S| is the transverse unit normal vector. By integrating equation (3.14) over across sectionA of areaA(X), using Gauss’ theorem, and noting thatA = ∫ 2π

012 R2 dθ , and that a

circumferential line element is given by d! = (R2 + R2θ )

1/2dθ , we obtain

∫∫A∇2⊥T1+ T0X X dσ =

∫∂A∇⊥T1·n⊥ d!+ A T0X X

= T0X

∫ 2π

0RRX dθ + A T0X X = AX T0X + A T0X X = d

dX

(A

d

dXT0

)= 0. (3.16)

The finally obtained equation can be solved easily. Note that we recovered the conservation law ofheat fluxAT0X = q. Finally we have

T0(X) =∫ X q

A(z)dz + Tref (3.17)

It should be noted we did not include in our analysis any boundary conditions at the ends of the bar.It is true that here the present method fails. The found solution is uniformly valid onR (sinceR(X)

is assumed continuous and independent ofε), but only as long as we stay away from any ends. Nearthe ends the boundary conditions induce transverse gradients ofO(1) which makes the prevailinglength scale againx , rather thanX . This region is asymptotically of boundary layer type, and shouldbe treated differently (see below). �

Example II.24 Quasi 1-D gas dynamics. Consider a slowly varying duct with irrotational inviscidisentropic flow, described (in dimensionless form) by the velocity potentialφ and densityρ satisfyingthe equation for mass conservation and the compressible form of Bernoulli’s equation

∇·(ρ∇φ) = 0, 1

2

∣∣∇φ∣∣2+ ργ−1

γ − 1= E . (3.18)

The parameterγ is a gas constant (1.4 for air) andE is a given problem parameter. Using thesame notation as in the previous example, the duct wall is given byS(εx, r, θ) = 0, while at theimpermeable wall∇φ ·∇S = 0. The mass flux, the same at any cross sectionA, is given by∫∫

Aρφx dσ = F. (3.19)

Introduce the slow variableX = εx , and assumeφ andρ to depend essentially onX , rather thanx .Due to the scaling, the dimensionless axial flow velocityφx , the densityρ, the cross sectional areaA, the fluxF and the thermodynamical constantE areO(1). So we have to rescaleφ and write

φ(x, y, z; ε) = ε−1�(X, y, z; ε). (3.20)

10:56 21 Mar 2002 30 version: 21-03-2002


The equations for� andρ become

ε2 ∂∂X

(ρ ∂

∂X �)+∇⊥·(ρ∇⊥�) = 0, 1

2�2X + 1

2ε−2∣∣∇⊥�∣∣2+ ργ−1

γ − 1= E, (3.21a)

with boundary condition

∇� ·∇S = ε2�X SX +∇⊥� ·∇⊥S = 0 at S = 0. (3.21b)

We assume the expansion

�(X, y, z; ε) = �0(X, y, z)+O(ε2), ρ(X, y, z; ε) = ρ0(X, y, z)+O(ε2). (3.22)

From Bernoulli’s equation it follows that|∇⊥�0|2 = 0, so�0 = �0(X), and henceρ0 = ρ0(X).From the mass flux equation we get�0X = F/ρ0 A, while finally ρ0 is to be determined (in generalnumerically) from the algebraic equation

F2

2ρ20 A2

+ ργ−10

γ − 1= E . (3.23)

3.2 Lindstedt-Poincaré method

When we have a functionf , depending onε, and periodic int with fundamental frequencyω(ε), we canwrite f like a Fourier series

f (t; ε) =∞∑

n=−∞An(ε)einω(ε)t (3.24)

If amplitudes and frequency have an asymptotic expansion, say

An(ε) = An,0 + εAn,1 + . . . , ω(ε) = ω0+ εω1+ . . . , (3.25)

we have a natural asymptotic series expansion forf of the form

f (t; ε) =∞∑

n=−∞An,0 einω0t + ε

∞∑n=−∞

(An,1 + inω1t An,0

)einω0t + . . . (3.26)

This expansion, however, is only uniform int on an interval[0, T ], whereT = o(ε−1). On a largerinterval, for example[0, ε−1], the asymptotic hierarchy in the expansion becomes invalid, becauseεt =O(1).

Remark.The corresponding terms∼ t sint, t cost are called “secular terms” (secular = occurringonce in a century,saeculum = generation, referring to their astronomical origin).

It is therefore far better to apply first a coordinate transformationτ = ω(ε)t , introduceF(τ ; ε) =f (t; ε), and expandF , rather thanf , asymptotically.

In practical situations, the frequencyω is of course unknown, and to be found. When constructing thesolution we have to allow for an unknown coordinate transformation

τ = (ω0+ εω1+ ε2ω2+ . . .)t (3.27)

(where we may takeω0 = 1) and determine the coefficients from the additional condition that theasymptotic hierarchy should be respected as long as possible. In other words, secular terms should notoccur.

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4. SINGULAR PERTURBATION PROBLEMS

Example II.25 Consider the pendulum

θ + K 2 sin(θ) = 0, with θ(0) = ε, θ ′(0) = 0. (3.28)

After the transformationτ = ωt and noting thatθ = O(ε), we have

ω2θ ′′ + K 2(θ − 16θ

3+ . . .) = 0, (3.29)

We expand

ω = 1+ ε2ω1 + . . . , θ = εθ0+ ε3θ1+ . . . (3.30)

and find, after substitution, the equations for the first 2 orders

θ ′′0 + K 2θ0 = 0, θ0(0) = 1, θ ′0(0) = 0 (3.31a)

θ ′′1 + K 2θ1 = −2ω1θ′′0 + 1

6 K θ30 , θ1(0) = 0, θ ′1(0) = 0 (3.31b)

The solution forθ0 is obviously

θ0 = cos(K τ ), (3.32)

leading to the following equation forθ1

θ ′′1 + K 2θ1 = 2K(ω1K + 1

16

)cos(K τ )+ 1

24K cos(3K τ ). (3.33)

Now it is essential to observe that the right-hand-side consists of two forcing terms: one with fre-quency 3K and one withK , the resonance frequency of the left-hand-side. This resonance wouldlead to secular terms, as the solutions will behave likeτ sin(K τ ) andτ cos(K τ ). Therefore, in orderto suppress the occurrence of secular terms, the amplitude of the resonant forcing term should vanish,which yields the unknownω1, and subsequentlyθ1

ω1 = − 1

16K→ θ1 = 1

48K

(cos(K τ )− cos(K τ )3) (3.34)

�

4 Singular perturbation problems

4.1 Matched asymptotic expansions

Very often it happens that a simplifying limit applied to a more comprehensive model gives a correctapproximation for the main part of the problem, but not everywhere: the limit isnon-uniform. This non-uniformity may be in space, in time, or in any other variable. For the moment we think of non-uniformityin space, let’s say a small region nearx = 0.

If this region of non-uniformity is crucial for the problem, for example because it contains a boundarycondition, or a source, we are able to utilize the pursued limit and have to deal with the full problem (atleast locally). This, however, does not mean that no use can be made of the inherent small parameter.The local nature of the non-uniformity itself gives often the possibility of another reduction. In such acase we call this a couple of limiting forms, “inner and outer problems”, and are evidence of the fact thatwe have apparently physically two connected but different problems as far as the dominating mechanismis concerned. Depending on the problem, we now have two simpler problems, serving as boundaryconditions to each other via continuity ormatching conditions.

10:56 21 Mar 2002 32 version: 21-03-2002


Non-uniform asymptotic approximations

Suppose that a given sufficiently smooth function�(x; ε), with 0≤ x ≤ 1, 0< ε ≤ ε1, does not have auniform limit ε → 0, x → 0. Typically, such a function will depend, apart fromx , on combinations likex/δ(ε), whereδ = o(1).

Assume that this function does not have a regular asymptotic expansion on the whole interval[0,1] butonly on partial intervalsx ∈ [η(ε),1], whereη = o(1) andδ = o(η). We call this expansion the outerexpansion, principally valid in the “x = O(1)”-outer region [6, 17], but extendible to[O(η),1].

�(x; ε) =n∑

k=0

µk(ε)ϕk(x)+ o(µn) ε → 0, x = O(1). (4.1)

Consider now the transformation to the stretched coordinate

ξ = x

δ(ε). (4.2)

Assume that the function� = #(ξ ; ε) has a non-trivial regular asymptotic expansion on partial intervalsξ ∈ [0, ζ(ε)/δ(ε)], whereη(ε) < ζ(ε). We call this expansion the inner expansion, principally valid inthe “ξ = O(1)”-inner region, but extendible to[0,O(ζ )].

#(ξ ; ε) =m∑

k=0

λk(ε)ψk(ξ)+ o(λm) ε → 0, ξ = O(1). (4.3)

Example II.26

�(x; ε) = arctan( xε)+ sin(x + ε) = π

2 + sinx + ε cosx − εx +O(ε3/2) on 1

2ε1/2 ≤ x ≤ 1

#(ξ; ε) = arctan(ξ)+ sin(εξ + ε) = arctan(ξ)+ ε(ξ + 1)+ O(ε3/2) on 0≤ ξ ≤ 2ε−1/2

The adjective non-trivial is essential: the expansion must be “significant”, i.e. different from the outer-expansion inϕn rewritten in the inner variableξ . This determines the choice of the inner variableξ =x/δ(ε). The scalingδ(ε) is the asymptoticallylargest gauge function with this property. We call theexpansion for# the inner expansion or boundary layer expansion, the regionξ = O(1) or x = O(δ)

being the boundary layer with thicknessδ, andξ the boundary layer variable. Boundary layers may benested, and may occur at internal points of the domain of�. Then they are called “internal layers”. Theassumptionη < ζ , i.e. that inner and outer expansion may be extended to regions that overlap, is calledthe “overlap hypothesis”.

Suppose,�(x; ε) has an outer-expansion

�(x; ε) =n∑

k=0

µk(ε)ϕk(x)+ o(µn) (4.4)

and a boundary layerx = O(δ) with inner-expansion

#(ξ ; ε) =m∑

k=0

λk(ε)ψk(ξ)+ o(λm) (4.5)

10:56 21 Mar 2002 33 version: 21-03-2002


and suppose that both expansions are complementary, i.e. there is no other boundary layer in betweenx = O(1) and x = O(δ), then the overlap-hypothesis says that both expansions represent the samefunction in an intermediate region of overlap. This overlap region may be described by a stretchedvariablex = η(ε)σ , asymptotically in betweenO(1) andO(δ), so: δ � η � 1. In the overlap regionboth expansions match, which means that asymptotically both expansions are equivalent and reduceto the same expressions. A widely used and relatively simple procedure is Van Dyke’s matchings rule[33, 6]: the outer-expansion, rewritten in the inner-variable, has a regular series expansion, which isequalto the regular asymptotic expansion of the inner-expansion, rewritten in the outer-variable. Suppose that

n∑k=0

µk(ε)ϕk(δξ) =m∑

k=0

λk(ε)ηk(ξ)+ o(λm) (4.6a)

n∑k=0

λk(ε)ψk(x/δ) =n∑

k=0

µk(ε)θk(x)+ o(µn) (4.6b)

then the expansion ofηk back toxn∑

k=0

λk(ε)ηk(x/δ) =n∑

k=0

µk(ε)ζk(x)+ o(µn) (4.7)

is such thatζk = θk for k = 0, · · · , n.

The idea of matching is very important because it allows one to move smoothly from one regime intothe other. The method of constructing local, but matching, expansions is therefore called “MatchedAsymptotic Expansions” (MAE) [17].

Constructing asymptotic solutions

The most important application of this concept of inner- and outer-expansions is that approximate solu-tions of certain differential equations can be constructed for which the limit under a small parameter isapparently non-uniform.

The main lines of argument for constructing a MAE solution to a differential equation+ boundaryconditions are as follows. Suppose� is given by the equation

D(�′,�, x; ε) = 0 + boundary conditions, (4.8)

where�′ = d�/dx . Then we try to construct an outer solution by looking for “non-trivial degenerations”of D underε → 0, that is, findµ0(ε) andν0(ε) such that

limε→0

ν−10 (ε)D(µ0ϕ

′0, µ0ϕ0, x; ε) = D0(ϕ

′0, ϕ0, x) = 0 (4.9)

has a non-trivial solutionϕ0. A seriesϕ = µ0ϕ0 + µ1ϕ1 + · · · is constructed by repeating the processfor D − ν0D0, etc.

Suppose, the approximation is non-uniform. For example, not all boundary conditions can be satisfied.Then we start looking for an inner-expansion if we have reasons to believe that the non-uniformity is ofboundary-layer type. Presence, location and size of the boundary layer(s) are now found by the “corre-spondence principle”, that is the (heuristic) idea that if� behaves somehow differently in the boundarylayer, the defining equation must also be essentially different. Therefore, we search for “significant de-generations” or “distinguished limits” ofD. These are degenerations ofD underε → 0, with scaledxand�, that contain the most information, and without being contained in other, richer, degenerations.

10:56 21 Mar 2002 34 version: 21-03-2002


Example II.27 Under the limitε → 0, the equationεy ′ + y = sinx, y(0) = 1 reduces toy = sinxwith y(0) �= 1. After the scalingx = εξ , the equation reduces to the essentially differentyξ + y = 0.

The next step is then to select from these distinguished limits the one(s) allowing a solution that matcheswith the outer solution and satisfies any applicable boundary conditions.

Symbolically:

find x0, δ(ε), λ(ε), κ(ε)with x = x0+ δξ , �(x; ε) = λ(ε)#(ξ ; ε)such that B0(ψ

′0, ψ0, ξ ) = lim

ε→0κ−1D(δ−1λ# ′, λ#, x0 + δξ ; ε) has the “richest” structure,

and there exists a solution of

B0(ψ′0, ψ0, ξ ) = 0 (4.10)

satisfying boundary and matching conditions. Again, an asymptotic expansion may be constructed in-ductively, by repeating the argument. It is of practical importance to note that the order estimateλ of �in the boundary layer is often determined a posteriori by boundary or matching conditions.

Example II.28 A simple example to illustrate some of the main arguments is

D(ϕ′, ϕ, x; ε) = εd2ϕ

dx2 +dϕ

dx− 2x = 0, ϕ(0) = ϕ(1) = 2. (4.11)

The leading order outer-equation is evidently (withµ0 = ν0 = 1)

D0 = dϕ0

dx− 2x = 0 (4.12)

with solution

ϕ0 = x2+ A. (4.13)

The integration constantA can be determined by the boundary conditionϕ0(0) = 2 at x = 0 orϕ0(1) = 2 at x = 1, but not both, so we expect a boundary layer at either end. By trial and errorwe find that no solution can be constructed if we assume a boundary layer atx = 1, so, inferring aboundary layer atx = 0, we have to use the boundary condition atx = 1 and find

ϕ0 = x2+ 1. (4.14)

The structure of the equation suggests a correction ofO(ε), so we try the expansion

ϕ = ϕ0+ εϕ1+ ε2ϕ2+ · · · . (4.15)

This results forϕ1 into the equation

dϕ1

dx+ d2ϕ0

dx2= 0, with ϕ1(1) = 0 (theO(ε)-term of the boundary condition), (4.16)

which has the solution

ϕ1 = 2− 2x . (4.17)

Higher orders are straightforward:

dϕn

dx= 0, with ϕn(1) = 0 (4.18)

10:56 21 Mar 2002 35 version: 21-03-2002


leading to solutionsϕn ≡ 0, and we find for the outer expansion

ϕ = x2+ 1+ 2ε(1− x)+O(εN ).i (4.19)

We continue with the inner expansion, and find withx0 = 0,ϕ = λψ, x = δξ

ελ

δ2

d2ψ

dξ2 +λ

δ

dψ

dξ− 2δξ = 0. (4.20)

Both from the matching (ϕouter→ 1 for x ↓ 0) and from the boundary condition (ϕ(0) = 2) we haveto conclude thatϕinner= O(1) and soλ = 1. Furthermore, the boundary layer has only a reason forexistence if it comprises new effects, not described by the outer solution. From the correspondenceprinciple we expect that new effects are only included if(d2ψ/dξ2) is included. Soεδ−2 must be atleast as large asδ−1, the largest ofδ−1 andδ. From the principle that we look for the equation withthe richest structure, it must be exactly as large, implying a boundary layer thicknessδ = ε. Thus wehaveκ = ε−1, and the inner equation

d2ψ

dξ2 +dψ

dξ− 2ε2ξ = 0. (4.21)

From this equation it wouldseem that we have a series expansion without theO(ε)-term, since theequation for this order would be the same as for the leading order. However, from matching with theouter solution:

ϕouter→ 1+ 2ε + ε2(ξ2 − 2ξ)+ · · · (x = εξ, ξ = O(1)) (4.22)

we see that an additionalO(ε)-term is to be included. So we substitute the series expansion:

ϕ = ψ0 + εψ1 + ε2ψ2+ · · · . (4.23)

It is a simple matter to find

d2ψ0

dξ2 + dψ0

dξ= 0, ψ0(0) = 2 → ψ0 = 2+ A0(e−ξ −1) (4.24a)

d2ψ1

dξ2+ dψ1

dξ= 0, ψ1(0) = 0 → ψ1 = A1(e−ξ −1) (4.24b)

d2ψ2

dξ2 + dψ2

dξ= 2ξ, ψ2(0) = 0 → ψ2 = ξ2 − 2ξ + A2(e

−ξ −1) (4.24c)

where constantsA0, A1, A2, · · · are to be determined from the matching condition that outer expan-sion (4.19) forx → 0:

1+ x2+ 2ε − 2εx + · · · (4.25)

must be functionally equal to inner expansion (4.23) forξ →∞:

2− A0− εA1+ x2− 2εx − ε2A2+ · · · . (4.26)

A full matching is obtained if we choose:A0 = 1, A1 = −2, A2 = 0.

Logarithmic switchback

It is not always evident from just the structure of the equation what the necessary expansion will looklike. Sometimes it is well concealed, and we are only made aware of an invalid initial choice by thewarning of matching failure. In fact, it is also the matching process itself that reveils us the requiredsequence of scaling functions. An example of such a back reaction is known aslogarithmic switch back.

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Example II.29 Consider the following problem fory = y(x; ε) on the unit interval.

εy ′′ + x(y ′ − y) = 0, y(0; ε) = 0, y(1; ε) = e. (4.27)

The outer solution appears to have the expansion

y(x; ε) = y0(x)+ εy1(x)+O(ε2). (4.28)

By trial and error, the boundary layer appears to be located nearx = 0, so the governing equationsand boundary conditions are then

y ′0− y0 = 0, y0(1) = e, (4.29a)

y ′n − yn = −x−1y ′′n−1, yn(1) = 0 (4.29b)

with general solution

yn(x) = An ex +∫ 1

xz−1 ex−z y ′′n−1(z) dz (4.30a)

such that

y0(x) = ex (4.30b)

y1(x) = − ex ln(x) (4.30c)

etc.

The boundary layer thickness is found from the scalingx = εmt (note thaty = O(1) because of the

matching with the outer solution) leading to the significant degeration ofm = 12, or x = ε

12 t . The

boundary layer equation fory(x; ε) = Y (t; ε) is thus

Y ′′ + tY ′ − ε12 tY = 0, Y (0; ε) = 0. (4.31)

The obvious choice of expansion ofY in powers ofε12 is not correct, as the found solution does not

match with the outer solution. Therefore, we consider the outer solution in more detail for smallx .Whenx = ε

12 t , we have for the outer solution

y(εt; ε) = 1+ ε12 t − 1

2ε ln(ε)+ ε(12t2 − ln(t))

− 12ε

32 ln(ε) − ε

32 t ln(t) − 1

4ε2 ln(ε)t2 + O(ε2) (4.32)

So apparently we need at least

Y (t; ε) = Y0(t)+ ε12 Y1(t)+ ε ln(ε)Y2(t)+ εY3(t)+ o(ε) (4.33)

with equations and boundary conditions

Y ′′0 + tY ′0 = 0, Y0(0) = 0, (4.34a)

Y ′′1 + tY ′1 = tY0, Y1(0) = 0, (4.34b)

Y ′′2 + tY ′2 = 0, Y2(0) = 0, (4.34c)

Y ′′3 + tY ′3 = tY1, Y3(0) = 0, (4.34d)

with solution

Y0(t) = A0 erf( t√

2

)(4.35a)

Y1(t) = A1 erf( t√

2

)+ Y ′0(t)+∫ t

0Y0(z) dz = (A1+ A0t) erf

( t√2

)+ 2( 2π

) 12 A0(e−

12 t2−1)

(4.35b)

Y2(t) = A2 erf( t√2) (4.35c)

10:56 21 Mar 2002 37 version: 21-03-2002


Unfortunately,Y3 cannot be expressed in closed form. However, for the demonstration it is sufficientto derive the behaviour ofY3 for larget . As erf(z)→ 1 exponentially fast forz →∞, we obtain

tY1(t) = A0t2+ (A1− 2( 2π

) 12 A0)t + exponentially small terms (4.35d)

and thus after some algebra

Y3(t) = 12 A0t2+ (A1− 2

( 2π

) 12 A0)t − A0 ln(t)+ . . . (4.35e)

(The constant term cannot be determined because this depends on the value att = 0.) For matchingof the inner solution, we have to compare with expression (4.32), and have

A0+ ε12(

A1+ A0t − 2( 2π

) 12 A0

)+ ε ln(ε)A2+ ε(1

2 A0t2 + (A1− 2( 2π

) 12 A0)t − A0 ln(t)

)≡ 1+ ε

12 t − 1

2ε ln(ε) + ε(12t2 − ln(t)) (4.36)

and indeed, a full matching is achieved with

A0 = 1, A1 = 2( 2π

) 12 , A2 = −1

2. (4.37)

The role of matching

It is important to note that a matching is possible at all! Only a part of the terms can be matched byselection of the undetermined constants. Other terms are already equal, without free constants, and thereis no way to repair a possibly incomplete matching here. This is an important consistency check on thefound solution, at least as long as no real proof is available. If no matching appears to be possible, almostcertainly one of the assumptions made with the construction of the solution have to be reconsidered.Particularly notorious are logarithmic singularities of the outer solution, as we saw above.

Summarizing: matching of inner- and outer expansion plays an important rôle in the following ways:

i) it provides information about the sequence of order (gauge) functions{µk} and{λk} of the expan-sions;

ii) it allows us to determine unknown constants of integration;

iii) it provides a check on the consistency of the solution, giving us confidence in the correctness.

4.2 Multiple scales

Introduction

Suppose a functionφ(x; ε) depends on more than one length scale acting together, for examplex , εx , andε2x . Then the function does not have a regular expansion on the full domain of interest,x ≤ O(ε−2) say.It is not possible to bring these different length scales together by a simple coordinate transformation, likein the method of slow variation or the Lindsted-Poincaré method, or to split up our domain in subdomainslike in the method of matched asymptotic expansions.

Therefore we have to find another way to construct asymptotic expansions, valid in the full doamin ofinterest. The approach that is followed in the method of multiple scales ([22, 4]) is at first sight ratherradical: the various length scales are temporarily considered as independent variables:x1 = x, x2 =εx, x3 = ε2x , and the original functionφ is identified with a more general functionψ(x1, x2, x3; ε)depending on a higher dimensional independent variable.

10:56 21 Mar 2002 38 version: 21-03-2002


Example II.30

φ(x; ε) = A(ε) e−εx cos(x + θ(ε)) becomesψ(x1, x2; ε) = A(ε) e−x2 cos(x1+ θ(ε)).

Since this identification is not unique, we may add constraints such that this auxiliary functionψ doeshave a Poincaré expansion on the full domain of interest. After having constructed this expansion, it maybe associated to the original function along the linex1 = x, x2 = εx, x3 = ε2x .

The typical problems that are treated by the method of multiple scales are slowly varying waves, affectedby small effects during a long time. Intuitively, it is clear that over a short distance (a few wave lengths)the wave only sees a constant conditions, and will propagate approximately as in the constant case, butover larger distances it will somehow have to change its shape in accordance with its new environment.

The technique, utilizing this difference between small scale and large scale behaviour is the method ofmultiple scales. As with most approximation methods, this method has grown out of practice, and workswell for certain types of problems. Typically, the multiple scale method is applicable to problems withon the one hand a certain global quantity (energy, power) which is conserved or almost conserved andcontrols the amplitude, and on the other hand two rapidly interacting quantities (kinetic and potentialenergy) controlling the phase.

An illustrative example

We will illustrate the method by considering a damped harmonic oscillator

d2y

dt2+ 2ε

dy

dt+ y = 0 (t ≥ 0), y(0) = 0,

dy(0)

dt= 1 (4.38)

with 0 < ε � 1. The exact solution is readily found to be

y(t) = e−εt sin(√

1− ε2 t)/√

1− ε2 (4.39)

A naive approximation for smallε and fixedt would give

y(t) = sint − εt sint +O(ε2) (4.40)

which appears to be not a good approximation for larget for the following reasons:

1) if t = O(ε−1) the second term is of equal importance as the first term and nothing is left over of theslow exponential decay;

2) if t = O(ε−2) the phase has an error ofO(1) giving an approximation of which even the sign may bein error.

In the following we shall demonstrate that this type of error occurs also if we construct a straightforwardapproximate solution directly from equation (4.38). However, knowing the character of the error, wemay then try to avoid them. Suppose we can expand

y(t; ε) = y0(t)+ εy1(t)+ ε2y2(t)+ · · · . (4.41)

Substitute in (4.38) and collect equal powers ofε:

O(ε0) : d2y0

dt2+ y0 = 0 with y0(0) = 0,

dy0(0)

dt= 1,

O(ε1) : d2y1

dt2+ y1 = −2

dy0

dtwith y1(0) = 0,

dy1(0)

dt= 0,

10:56 21 Mar 2002 39 version: 21-03-2002


then

y0(t) = sint, y1(t) = −t sint, etc.

Indeed, the straightforward, Poincaré type, expansion (4.41) that is generated breaks down for larget ,whenεt ≥ O(1). As is seen from the structure of the equations foryn, the quantityyn is excited (by the“source”-terms−2dyn−1/dt) in its eigenfrequency, resulting in resonance. The algebraically growingterms of the typetn sint andtn cost that are generated are called in this context:secular2 terms.

Apart from being of limited validity, the expansion reveals nothing of the real structure of the solution:a slowly decaying amplitude and a frequency slightly different from 1. For certain classes of problems itis therefore advantageous to incorporate this structure explicitly in the approximation.

Introduce the slow time scale

T = εt (4.42)

and identify the solutiony with a suitably chosen other functionY that depends on both variablest andT :

y(t; ε) = Y (t, T ; ε). (4.43)

The underlying idea is the following. There are, of course, infinitely many functionsY (t, T ; ε) thatare equal toy(t, ε) along the lineT = εt in (t, T )-space. So we have now some freedom to prescribeadditional conditions. With the unwelcome appearance of secular terms in mind it is natural to think ofconditions, chosen such that no secular terms occur when we construct an approximation.

Since the time derivatives ofy turn into partial derivatives ofY

dy

dt= ∂Y

∂t+ ε

∂Y

∂T, (4.44)

equation (4.38) becomes forY

∂2Y

∂t2+ Y + 2ε

(∂Y

∂t+ ∂2Y

∂t∂T

)+ ε2

(∂2Y

∂T 2+ 2

∂Y

∂T

)= 0. (4.45)

Assume the expansion

Y (t, T ; ε) = Y0(t, T )+ εY1(t, T )+ ε2Y2(t, T )+ · · · (4.46)

and substitute this into equation (4.45) to obtain to leading orders

∂2Y0

∂t2+ Y0 = 0,

∂2Y1

∂t2+ Y1 = −2

∂Y0

∂t− 2

∂2Y0

∂t∂T,

2From astronomical applications where these terms occurred for the first time in this type of perturbation series: secular=occurring once in a century; saeculum= generation.

10:56 21 Mar 2002 40 version: 21-03-2002


with initial conditions

Y0(0,0) = 0,∂

∂tY0(0,0) = 1,

Y1(0,0) = 0,∂

∂tY1(0,0) = − ∂

∂TY0(0,0).

The solution forY0 is easily found to be

Y0(t, T ) = A0(T ) sint with A0(0) = 1, (4.47)

which gives a right-hand side for theY1-equation of

−2(

A0 + ∂ A0

∂T

)cost.

No secular terms occur (no resonance betweenY1 andY0) if this term vanishes:

A0 + ∂ A0

∂T= 0 −→ A0 = e−T . (4.48)

Note (this is typical), that we determinedY0 only on the level ofY1, but without having to solveY1 itself.

The present approach is by and large the multiple scale technique in its simplest form. Variations onthis theme are sometimes necessary. For example, we have not completely got rid of secular terms.On a longer time scale (t = O(ε−2)) we have inY2 again resonance because of the “source”:e−T sint ,yielding termsO(ε2t). We see that a second time scaleT2 = ε2t is necessary. From the exact solution wemay infer that if the solution is strictly periodic on the fast time scale, a fast time of strained coordinatetype may be worthwhile:τ = (1+ ε2ω1+ ε4ω4+ . . .)t .

Slowly varying fast time scale

The method fails when the slow variation is due to external effects, like a slowly varying problem pa-rameter.

Example II.31 Consider the problem

x + κ(εt)2x = 0, x(0; ε) = 1, x(0; ε) = 0. (4.49)

whereκ = O(1). It seems plausible to assume 2 time scales: a fast oneO(κ−1) = O(1) and a slowoneO(ε−1). So we introduce next tot the slow scaleT = εt , and rewritex(t; ε) = X (t, T ; ε). WeexpandX = X0 + εX1 + . . ., and obtainX0 = A0(T ) cos(κ(T )t − θ0(T )). Suppressing secularterms in the equation forX1 requiresA′0 = κ ′t − θ ′0 = 0, which is impossible.

In this case the fast variable is to be strained locally by a suitable strain function, as follows

τ =∫ t

ω(εt ′; ε)dt ′ = 1

ε

∫ T

ω(z; ε)dz, where T = εt, (4.50)

while for x(t; ε) = X (τ, T ; ε) we have

x = ωXτ + εXT and x = ω2Xττ + εω′Xτ + 2εωXτT + ε2XT T (4.51)

10:56 21 Mar 2002 41 version: 21-03-2002


Example II.32 Reconsider the problem II.31. After expandingX = X0 + εX1 + . . . andω =ω0 + εω1 + . . . we obtain

ω20X0ττ + κ2X0 = 0 (4.52a)

ω20X1ττ + κ2X1 = −2ω0ω1X0ττ − ω′0X0τ − 2ω0X0τT (4.52b)

The leading order solution isX0 = A0(T ) cos(λ(T )τ − θ0(T )), whereλ = κ/ω0. The right-handside of (4.52b) is then

2ω0A0λ2(ω1+ λ′τ − θ ′0) cos(λτ − θ0)+ (A0λ)

−1(ω0A20λ

2)′ sin(λτ − θ0).

Suppresion of secular terms requiresλ′ = 0. Without loss of generality we can takeλ = 1, orω0 = κ . Then we needω1 = θ ′0, which just yields thatλτ − θ0 = τ − θ0 = ε−1

∫ Tω0(z) dz. In other

words, we can takeω1 = 0 andθ0 = a constant. Finally we haveω0A20λ

2 = κ A20 = a constant.

For linear wave-type problems we may anticipate the structure of the solution and assume the WKBhypothesis (see [4, 15])

y(t; ε) = A(T ; ε)ei ε−1∫ T

0 ω(τ ;ε) dτ . (4.53)

The method is again illustrated by the example of the damped oscillator (4.38). After substitution andsuppressing the exponential factor, we get

(1− ω2)A + iε(2ω

∂ A

∂T+ ∂ω

∂TA + 2ωA

)+ ε2

(∂2A

∂T 2+ 2

∂ A

∂T

)= 0.

Note that the secular terms are now not explicitly suppressed. The necessary additional condition is herethat the solution of the present typeexists (assumption 4.53), and that each higher order correction is nomore secular than its predecessor. The solution is expanded as

A(T ; ε) = A0(T )+ εA1(T )+ ε2A2(T )+ · · ·ω(T ; ε) = ω0(T )+ ε2ω2(T )+ · · · . (4.54)

Note thatω1 may be set to zero since the factor exp(i∫ T

0 ω1(τ )dτ) may be incorporated inA. Substituteand collect equal powers ofε:

O(ε0) : (1− ω20)A0 = 0 → ω0 = 1,

O(ε1) : ∂ A0

∂T+ A0 = 0 → A0 = e−T ,

O(ε2) : 2i(∂ A1

∂T+ A1

)= (1+ 2ω2)e−T → ω2 = −1

2, A1 = 0.

The solution that emerges is indeed consistent with the exact solution.

Example II.33 The air-damped resonator. In dimensionless form this is given by

d2y

dt2+ ε

dy

dt

∣∣∣∣dy

dt

∣∣∣∣+ y = 0, with y(0) = 1,dy(0)

dt= 0. (4.55)

10:56 21 Mar 2002 42 version: 21-03-2002


By comparing the accelerationy ′′ with the dampingεy ′|y ′| it may be inferred that on a timescaleεtthe influence of the damping isO(1). So we conjecture a slow timescaleεt , and split up the timedependence in two by introducing the slow timescaleT and the dependent variableY

T = εt, y(t; ε) = Y (t, T ; ε), dy

dt= ∂Y

∂ t+ ε

∂Y

∂T,

and obtain for equation (4.55)

∂2Y

∂ t2 + Y + ε

(2∂2Y

∂ tT+ ∂Y

∂ t

∣∣∣∣∂Y

∂ t

∣∣∣∣)+O(ε2) = 0 (4.56)

Y (0,0; ε) = 0,( ∂

∂ t+ ε

∂

∂T

)Y (0,0; ε) = 0.

The error ofO(ε2) results from the approximation∂∂t Y + ε ∂

∂T Y ∂∂t Y , and is of course only valid

outside a small neighbourhood of the points where∂∂t Y = 0. We expand

Y (t, T ; ε) = Y0(t, T )+ εY1(t, T )+ O(ε2)

and find for the leading order

∂2Y0

∂ t2+ Y0 = 0, with Y0(0,0) = 1,

∂

∂ tY0(0,0) = 0 (4.57)

with solution

Y0 = A0(T ) cos(t −%0(T )), where A0(0) = 1, %0(0) = 0.

For the first order we have the equation

∂2Y1

∂ t2+ Y1 = −2

∂2Y0

∂ tT− ∂Y0

∂ t

∣∣∣∣∂Y0

∂ t

∣∣∣∣= 2

dA0

dTsin(t −%0)− A0

d%0

dTcos(t −%0)

+ A20 sin(t −%0)| sin(t −%0)|

(4.58)

with corresponding initial conditions. The secular terms are suppressed if the first harmonics of theright-hand side cancel. For this we use the Fourier series expansion

sint | sint| = − 8

π

∞∑n=0

sin(2n + 1)t

(2n − 1)(2n + 1)(2n + 3)(4.59)

and we obtain the equations

2dA0

dT+ 8

3πA2

0 = 0 andd%0

dT= 0 (4.60)

with solution%0 = 0 and

A0(T ) = 1

1+ 43π T

. (4.61)

This approximation happens to be quite good. Comparison with a numerically obtained “exact”solution shows a relative error in the amplitude of less then 2· 10−4 for ε = 0.1.

10:56 21 Mar 2002 43 version: 21-03-2002


Example II.34 Sound propagation in a slowly varying duct. Consider a hard-walled circularcylindrical duct with a slowly varying diameter described in polar coordinates(x, r, θ) as

r = R(εx) (4.62)

with ε a dimensionless small parameter. In this duct we have an acoustic medium with constant meanpressure and sound speedc0. Sound waves of circular frequencyω are described by Helmholtz’sequation

∇2 p + k2 p = 0 (4.63)

wherep is the acoustioc pressure perturbation,k = ω/c0, and the boundary condition of a vanishingnormal gradient at the wall yields

∂p

∂r− εR′(εx)

∂p

∂x= 0 atr = R(εx). (4.64)

For constantR and constantk the general solution can be built up from modes of the following type

p = AJm(αmµr) e−imθ−ikmµx , (4.65)

αmµ = j ′mµ/R, k2mµ = k2− α2

mµ, Re(kmµ) ≥ 0, Im(kmµ) ≤ 0,

(whereJm denotes them-th order Bessel function) and we assume for the present problem, followingthe previous section, that there are solutions close to these modes. We introduce the slow variable

X = εx

so thatR = R(X), and we seek a solution of slowly varying modal type:

p = A(X, r; ε) e−imθ e−i ε−1∫ X0 γ (ξ ;ε) dξ (4.66)

Since

∂2 p

∂x2 =(−γ 2A − 2iεγ

∂ A

∂X− iεγ ′A + ε2 ∂

2A

∂X2

)exp

(· · ·)

we have for (4.63)[−γ 2A − 2iεγ

∂ A

∂X− iεγ ′A + ε2 ∂

2A

∂X2 +∂2A

∂r2 +1

r

∂ A

∂r− m2

r2 A + k2A

]exp

(· · ·)= 0.

After suppressing the exponential factor, this is up to orderO(ε)

L(A) = iεA

∂

∂X

(γ A2), ∂ A

∂r+ iεR′γ A = 0 at r = R(X), (4.67)

where we introduced for short the Bessel-type operator

L(A) = ∂2A

∂r2 +1

r

∂ A

∂r+(

k2− γ 2 − m2

r2

)A

and rewrote the right-hand side in a form convenient later. Expand

A(X, r; ε) = A0(X, r)+ εA1(X, r)+O(ε2),

γ (X; ε) = γ0(X)+O(ε2),

10:56 21 Mar 2002 44 version: 21-03-2002


substitute in (4.67), and collect like powers ofε, to obtain

O(1) : L(A0) = 0,∂ A0

∂r= 0 atr = R(X), (4.68)

O(ε) : L(A1) = iA0

∂

∂Xγ0A2

0,∂ A1

∂r= −i R′γ0A0 atr = R(X). (4.69)

Since variableX plays no other rôle in (4.68) than that of a parameter, we have forA0 the “almost-mode”

A0(X, r) = P0(X)Jm(α(X)r),

α(X) = j ′mµ/R(X), (4.70)

γ 20 (X) = k2− α2(X), Re(γ0) ≥ 0, Im(γ0) ≤ 0,

The amplitudeP0 is still undetermined, and follows from a solvability condition forA1. As before,amplitudeP0 is determined at the level ofA1, without A1 necessarily being known.

Multiply left- and right-hand side of (4.69) withr A0 and integrate tor from 0 to R(X). For theleft-hand side we utilize the self-adjointness ofL.

∫ R

0r A0L(A1) dr =

∫ R

0r A0L(A1)− r A1L(A0) dr

=[

r A0∂ A1

∂r− r A1

∂ A0

∂r

]R

0= −iγ0RR′A2

0.

For the right-hand side we apply Leibnitz’s rule

∫ R

0

∂

∂X

(iγ0A2

0

)r dr = d

dX

∫ R

0iγ0A2

0r dr − iγ0RR′A20.

As a result

∫ R

0rγ0A2

0 dr =[

12γ0P2

0

(r2− m2

α2

)Jm(αr)2

]R

0

= 12γ0P2

0 R2(1− m2

j ′mµ2

)Jm( j ′mµ)

2 = constant

or:

P0(X) = constant

R(X)√γ0(X)

. (4.71)

Example II.35 Ray acoustics in a temperature gradient. When a sound wave propagates in freespace through a medium that varies on a much larger scale than the typical wave length (typically:temperature gradients, or wind with shear), the same ideas of multiple scales may be applied. Incontrast to the duct, where the wave is confined by the duct walls, the waves may now freely refractand follow curved paths. These paths are called rays.

Consider an infinite 3D medium with varying temperature (typical length scaleL) but otherwise witha constant mean pressure, so that we have equation

∇·( 1

k2∇ p)+ p = 0 (4.72)

10:56 21 Mar 2002 45 version: 21-03-2002


wherek = k(εx) = ω/c0(εx), for a time harmonic sound fieldp ∝ eiωt . The small parameterεrelates the typical wave lengthλ ∼ 2πc0/ω to L, so ε ∼ λ/L. Assuming the field to be locallyplane we try an approximate solution having the form of a plane wave but with slowly varying (real)amplitudeA and phaseτ

p(x) = A(X; ε) e−iτ (X;ε)/ε (4.73)

whereX = εx the slow variable. The surfaces

τ (X) = εωt (4.74)

describe the propagating wave front. Note that the vector field∇τ is normal to the surfacesτ =constant. Define the operator

∇ =( ∂

∂X,

∂

∂Y,

∂

∂Z

)

so that∇ = ε∇. Substitute (4.73) in (4.72):

∇ p :=(ε∇A − i A∇τ

)e−iτ/ε, (4.75a)

∇2 p :=(ε2∇2

A − 2iε∇A·∇τ − iεA∇2τ − A|∇τ |2

)e−iτ/ε, (4.75b)

to obtain

(k2− |∇τ |2)A − iεk2

A∇·( A2∇τ

k2

)+ ε2k2∇·( 1

k2∇A)= 0. (4.76)

Expand

A(X; ε) = A0(X)+ εA1(X)+O(ε2)

τ (X; ε) = τ0(X)+O(ε2)

and collect like powers in (4.76). We find to leading orderτ0 andA0:

|∇τ0|2 = k2 (4.77)

∇·( A20∇τ0

k2

)= 0. (4.78)

Equation (4.77) is theeikonal equation, which determines the wave fronts and the ray paths. Equation(4.78) is called thetransport equation and describes theconservation of wave action, which is hereequivalent toconservation of energy [36]. It relates the amplitude variation to diverging or convergingrays.

The eikonal equation is a nonlinear first order partial differential equation, of hyperbolic type, whichcan always be reduced to an ordinary differential equation along characteristics ([36, p.65]).

10:56 21 Mar 2002 46 version: 21-03-2002


Exercises

1. Derive asymptotic solutions (forε → 0) of the equation

εx3− x + 2= 0.

2. Derive step by step, by iteratively re-scalingx(ε) = γ0(ε)X (ε) and balancing, re-scalingX (ε) =γ1(ε)Y (ε) and balancing, etc., that a third order asymptotic solution (forε → 0) of the equation

log(εx)+ x = a

is given by

x(ε) = logε−1+ log(logε−1)+ a + o(1).

3. Derive Webster’s equation for sound of long wave length propagating in slowly varying horns, bythe method of slender approximation. The reduced wave equation for pressure perturbationsp andwavenumberk is given by

∇2p + k2 p = 0,

within a duct given in cylindrical polar coordinates by

S = r − R(X, θ) = 0, X = εx, ε is small.

The wave number isO(ε), so we scalek = εκ. The duct wall is hard, so we have the boundarycondition

∇ p·∇S = 0 at S = 0.

4. Consider thevan der Pol equation for variabley = y(t; ε) in t and small parameterε:

y′′ + y − ε(1− y2)y′ = 0.

Construct by using the Lindstedt-Poincaré method (“method of strained coordinates”) a first orderapproximation of a periodic solution.

5. Following example II.34, derive a multiple scales solution of sound waves in a slowly varying ductwhile also the sound speed is a slowly varying function ofx . The pertaining equation is therefore(4.72).

6. Consider a generalisation of example II.35.

Acoustic perturbations in pressurep′, densityρ ′, velocityv′ and entropys′ of an inviscid mean flow,given by densityρ0, velocity v0, pressurep0, sound speedc0, entropys0 and constantsCP , CV andγ , are given by the linear equations

∂ρ ′

∂t+ v0·∇ρ ′ + v′ ·∇ρ0+ ρ0∇·v′ + ρ ′∇·v0 = 0

ρ0

(∂v′

∂t+ v0·∇v′ + v′ ·∇v0

)+ ρ ′v0·∇v0 = −∇ p′

∂s′

∂t+ v0·∇s′ + v′ ·∇s0 = 0.

s′ = CV

p0p′ − CP

ρ0ρ ′, where

CP

CV

p0

ρ0= γ

p0

ρ0= c2

0

10:56 21 Mar 2002 47 version: 21-03-2002


Assuming the perturbations to be harmonic in time, such that the typical wave lengthc0/ω is shortcompared to the typical length scaleL of the mean flow variations, we introduce the ray approxima-tion

p′, ρ ′, v′, s′ = P(X), R(X), V (X), S(X)× eiωt−iτ (X;ε)/ε

whereω " c0/L. Introduce a suitable small parameterε ∼ c0/ωL, and deriveto leading orderthe equations for the ray amplitudesP, R, V , S and the phaseτ . Eliminate∇τ to obtain an eikonalequation similar to (4.77).

10:56 21 Mar 2002 48 version: 15-03-2001

III Appendix

1 Dimensionless numbers

Archimedes Ar g�ρL3/ρν2 particles, drops or bubblesArrhenius Arr E/RT chemical reactionsBiot Bi hL/κ heat transferBiot Bi h D L/D mass transferBodenstein Bo V L/Dax mass transfer with axial dispersionBond Bo ρgL2/σ gravity against surface tensionDean De (V L/ν)(L/2r)1/2 flow in curved channelsEckert Ec V 2/CP�T kinetic energy against enthalpy differenceEuler Eu �p/ρV 2 pressure resistanceFourier Fo αt/L2 heat conductionFourier Fo Dt/L2 diffusionFroude Fr V 2/gL multiphase flowGrashof Gr β�T gL3/ν3 natural convectionHelmholtz He f L/c acoustic wave numberKapitza Ka gη4/ρσ 3 film flowKnudsen Kn λ/L low density flowLewis Le α/D combined heat and mass transferMach M V/c compressible flowNusselt Nu hL/κ convective heat transferPéclet Pe V L/α forced convection heat transferPéclet Pe V L/D forced convection mass transferPrandtl Pr ν/α = CPη/κ convective heat transferRayleigh Ra β�T gL3/αν natural convection heat transferReynolds Re V L/ν fluid flowSchmidt Sc ν/D convective mass transferSherwood Sh h D L/D convective mass transferStanton St h/ρCP V forced convection heat transferStanton St h D/V forced convection mass transferStrouhal Sr f L/V hydrodynamic wave numberWeber We ρV 2L/σ film flow, bubble formation, droplet breakup

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1. DIMENSIONLESS NUMBERS

Nomenclature

CP specific heat J/kg K T,�T temperature KD diffusion coefficient m2/s t time sDax axial dispersion coefficient m2/s V velocity m/sE activation energy J/mol c sound speed m/sf frequency 1/s κ thermal conductivity W/m Kg gravitational acceleration m/s2 α = κ/ρCP thermal diffusivity m2/sh heat transfer coefficient W/m2 K β coef. of thermal expansion K−1

h D mass transfer coefficient m/s λ molecular mean free path mL length m η dynamic viscosity Pa sp,�p pressure Pa ν kinematic viscosity m2/sR universal gas constant J/mol Kρ,�ρ density kg/m3

r radius of curvature m σ surface tension N/m

10:56 21 Mar 2002 50 version: 15-03-2001

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