Numerical analysis of a new mixed-formulation for eigenvalue convection-diffusion problems

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Numerical analysis of a new mixed-formulation for

eigenvalue convection-diffusion problems

Charles Pierre, Franck Plouraboué

To cite this version:

Charles Pierre, Franck Plouraboué. Numerical analysis of a new mixed-formulation for eigenvalue

convection-diffusion problems. SIAM Journal on Applied Mathematics, Society for Industrial and

Applied Mathematics, 2009, 70 (3), pp.658-676. �hal-00260546�

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eigenvalue onve tion-diusion problems

C. Pierre F. Plouraboué

Re eived: date / A epted: date

Abstra t

Amixedformulationisproposedandanalyzedmathemati allyfor oupled onve tion-diusion in heterogeneous medias. Transfer in solid parts driven by pure diusion is oupled with onve tion/diusion transfer in uid parts. Thisstudy is arriedout for translationnaly invariant geometries (general in-nite ylinders) and unidire tional ows. This formulation brings to the fore a new onve tion/diusion operator, the properties of whi h are mathemati- allystudied : its symmetryisrst shown using asuitable s alarprodu t. It isproved to be self-adjoint with ompa t resolvent on asimple Hilbertspa e. Its spe trum is hara terized as being omposed of a double set of eigenv al-ues: one onverging towards

−∞

and the other towards

+

∞

, thus resulting in a non-se torial operator. The de omposition of the onve tion-diusion problem into a generalized eigenvalue problem permits the redu tion of the originalthree-dimensional probleminto atwo-dimensional one. Despitebeing nonse torial,a omplete solutionontheinnite ylinder,asso iatedto astep hange of the wall temperature at the origin, is exhibited with the help of the operator's two sets of eigenvalues/eigenfun tions. On the omputational point of view, a mixed variational formulation is naturally asso iated to the eigenvalue problem. Numeri alillustrations are provided for axi-symmetri al situations,the onvergen e ofwhi hisfoundto be onsistentwiththe numer-i aldis retization.

Keywords Conve tion-diusion

·

variational formulation

·

Hilbert spa e

·

mixedformulation

1 Introdu tion

Conve tion-diusion problems are of importan e in many elds of appli ations in thermal, hemi alorbiomedi alengineerings ien es. Morespe i ally,heatormass diusion oupled with unidire tional onve tion is present in many types of equip-mentssu hasheatpipes,heatex hangers(shell,tubeorplate), hromatographsand rea torsandmassex hangersinmi ro- hannelarti ialdevi es,ando ursinreal bi-ologi altissues. This framework overs both parallelor ounter ow ongurations. A lassi al strategy for des ribing the temperature eld

T

of tube like ongura-tions inthe appliedliterature is generallyto assumethe followingseparate-variable

(3)

de omposition:

T (x, y, z) =

X

λ∈Λ

c

λ

T

λ

(x, y)e

λz

,

(1)

where

z

is the longitudinal oordinate along whi h the ow is aligned and

x

,

y

are transverse oordinates. The usual subsequent steps [7℄ are then to sear h for the eigenvalues/eigenfun tions

λ

/

T

λ

and nally ompute the amplitude oe ients

c

λ

.

This strategy, althoughwell established, raises several important questions. To our knowledge, there is no omplete theoreti al foundation for de omposition (1). This la k of theoreti al framework, despite ommonly used terminology, does not permit

Λ

and

T

λ

to be dened via an eigenvalue problem, all the more so a sym-metri alone. On the one hand, this is a fundamental problemfor the denition of

Λ

's topologyand lo ation;though itis always assumed tobereal and dis rete. On the other hand, this is a pra ti al issue for the numeri al omputation of

T

λ

and oe ients

c

λ

for whi h nodire torthogonal properties are availablefrom asimple, s alar-produ t-based,denition.

Fora lear understanding of these points,one needs to look ba k to the origins. GraetzandNusselt[9,17℄studiedasimpliedversionoftheproblem: auidowing inasingledu tathighPe letnumberPe(whi histheratioof onve tiontodiusion time s ales), when longitudinal diusion is negligible ompared to radial diusion. The du tis assumed to beeither a ir ular ylinder ormade of two parallelinnite plates. Su h asymmetri ongurationa tually leadsto simpliedone-dimensional problems. In the ase of a ylindri al du t, with the radial oordinate denoted by

r

, one omes ba k tothe Graetzproblem

1 r

∂

r

(r∂

r

T ) =

Pe

v(r)∂

z

T,

withaPoiseuilleparaboli velo ityprole

v(r)

. Inthissimpliedframework, sear h-ingfor a separate variablesolution

T (r, z) = f (r)g(z)

, rstly leads after a straight-forward al ulation to

g(z) = e

λ

Pe

z

and se ondly leads to the eigenvalue problem dening

f (r)

nowdenoted

T

λ

(r) := f (r)

:

T (r, z) = T

λ

(r)e

λ

Pe

z

,

1 r

∂

r

(r∂

r

T

λ

) = λvT

λ

,

(2)

whi h allows the denition of

λ

/

T

λ

as eigenvalues/eigenfun tions. Problem (2) is, moreover,symmetri negative,selfadjointwith ompa tresolvent,justifying de om-position(1)where

Λ

appearsasadis retesubsetof

R

−

. Moreover,the oe ients

c

λ

anbeeasily omputed usings alarprodu tsthanks tothe symmetryof thesystem

c

λ

=

Z

T

0 (r)T

λ

(r)rdr,

where

T

0

isthe inlet ondition at

z = 0

.

(4)

an be performed in a simple way. Indeed, many studies have explored possible extensions to that framework. Among these extensions, two are of parti ular im-portan e: the extended Graetz problem where the longitudinal diusion term is no longer negle ted, and the onjugated Graetz problem in whi h oupling with a solidwallwherediusion o ursis onsidered. Welistbellowthe di ulties metby previous ontributors when onsideringthese extensions.

Lookingfor aseparate variable solution

T (r, z) = f (r)g(z)

nolongerprovidesin a straightforward way a denition of the type

g(z) = e

λz

. However assuming su h a usual de omposition

T (r, z) = T

λ

(r)e

λz

, one do not get anymore an eigenvalue problem. Pre isely,in the ase of the onjugated Graetz problem,the new problem tobesolved for

T

λ

reads:

₁

r

∂

r

(r∂

r

T

λ

) = λ

Pe

vT

λ

uid part

1 r

∂

r

(r∂

r

T

λ

) = −λ

2 _T

λ

solid part

+

oupling ondition onthe uid/solid interfa e

,

where the quadrati term

λ

2

is a ounting for the axial diusion along

z

. In su h a form, one an see that this problem is not an eigenvalue problem on the whole uid+solid domain.

Addingaxialdiusionnowpermits informationba k-owinthe

z < 0

dire tion, not only along the ow with

z > 0

. Therefore both positive and negative eigen-values

λ

are physi allyexpe ted: the previoussymmetri -negativestru ture ofthe Graetz problem is no longer relevant here. However, until Papoutsakis work [18℄ detailed below, no attention had been paid to this important point. Early papers ontheextended/ onjugated Graetzproblem[25, 1, 15,5, 6,13,26,27,16℄assumed anegativespe trum (that ould, atleast inprin iple,be omplex)anda omplete set of eigenfun tions by plugging a Graetz-problem-like series solution into the diusion onve tion equation.

Stillintheseearlyworks,aspointedoutbyMi helsenetal. [16℄,thedi ultiesof determiningboththenon-orthogonaleigenfun tionsandtheexpansion oe ients

c

λ

appear riti al. From a omputational point of view the strategy used by Hsu et al. [13, 26, 27℄ using the Gram-S hmidt re-orthogonalization pro edure has a high ost, espe iallywhenapproa hing the entran eregionwherealarge numberof eigenvalues is ne essary for a orre t representation of the solution.

The domain denition and inlet ondition also raise new questions and di- ulties. In early papers, the ow domain is set as the positive real axis and the assumption of uniform uid temperature at the inlet has been widely used. As pointed out in [18, 28℄, e.g., when axial diusion is permitted, the uniform inlet onditionis invalidsin ethe temperaturewouldbealteredby upstream ondu tion beforerea hing the inletlo ation.

The most important progress in over oming these di ulties has been made by Papoutsakis and Ramakrishna in a series of innovative papers [19, 20, 18℄. [18℄ proposes a new formulation of the extended Graetz problem, adding a se ond un-known temperatureux, leadingtoasymmetri eigenvalue problem. Thisapproa h thus answers the problem regarding the spe trum lo ation (real eigenvalues only) and provides a ni e formalism for the amplitude oe ient

c

λ

omputation. This approa h has been su essfully used in a series of re ent papers by Weigand et al.

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amongwhi h we non- omprehensively quote [11, 12℄. In our opinion, threeimportantissues are stillpending:

1- Papoutsakis etal. formulationonly overs symmetri al ongurations su h as ir ulardu ts or re tangular hannels,

2- The extensionto the onjugated Graetz problemproposed in [19, 20℄ (adding solid parts with diusion)remains heavy and ompli ated,

3- From atheoreti al point of view, only the symmetry of the problem has been proved: neither its self-adjointness nor the resolvent ompa tness have been proved. In parti ular the ompleteness of the eigenvalue set

T

λ

has not been shown yet, weakening the legitima yof the proposed de omposition (1). The aimof this paperistoaddressthese issuesinavery generaltube ongura-tion(we assumenosymmetry ofthe tubese tion)forany generalunidire tional ve-lo ityprole(forexampleallowingnon-Newtonianvelo ityproles). Amajorresult we wish to a hieve here is to derive a formulation of the initial three-dimensional problem into a two-dimensional one, whose numeri al dis retization is obviously mu h easier. The physi al and geometri al frameworks are des ribed in se tion 2. Se tion3develops a theoreti alinvestigationof Equation(1)de omposition forthe temperature solution. Subse tion 3.1 introdu es a reformulation of the problem whi h allows the sear h for separate variable solutions and leads to an eigenvalue problem. In subse tion 3.2 the fun tional properties of the eigenvalue problem op-erator are established. It isproved tobe symmetri and moreover self adjointwith a ompa t resolventona basi Hilbert spa e. Atthe end ofthis theoreti al se tion, these results are used in 3.3 to display a full de omposition of a temperature eld for whi h fareld onditions are substituted for aninappropriateinlet ondition at

z = 0

. This de omposition appears e ient from a omputational point of view sin e it only exhibits the eigenvalues/eigenve tors of the problem as well as eas-ily omputable oe ients using simple s alar produ ts. In se tion 4, it is shown that the eigenvalue problemisnaturally equivalenttoa mixed variationalproblem, thus providing a simple omputational framework to solve the eigenvalue problem in terms of mixed nite element methods. The remaining part of this se tion is devoted to the analysis of the numeri al onvergen e of the method. We restri t ourselvestosymmetri ongurationswhereanalyti alsolutionsare available allow-ingan a priori error estimate of the solution. In this lastse tion we notably study the previously dis ussed extended Graetz and onjugated Graetz problems.

2 Physi al statement

2.1 Geometry, general assumptions and notations

Thedomain onsidered hereisaninnite ylinder

Z = Ω × R

havinga ross se tion

Ω ⊂ R

2

(assumptions on

Ω

are stated below). The oordinatesystem relative to

Ω

willbe denoted by

(x, y)

and the axial oordinateby

z ∈ R

.

The domain ross-se tion

Ω

is assumed bounded and its boundary

∂Ω

is taken tobesmooth(

C

1

(6)

x

y

z

Ω

1 Ω

2 Ω

3 v

1 = 0

v

2 v

3 n

n

_2,1

Figure1: Domain ross-se tion

Ω

(left)and wholedomain

Z = Ω × R

(right)

a olle tionof open sub-domains

Ω

i

(

1 ≤ i ≤ N

) with smooth boundaries, disjoint (

Ω

i

∩ Ω

j

= ∅

if

i 6= j

) and su h that

Ω = ∪

i

Ω

i

. The interfa e between

Ω

i

and

Ω

j

(if non-empty) is denoted by

Γ

ij

= Ω

i

∩ Ω

j

, and its unit normal, outward from

Ω

i

towards

Ω

j

, will be denoted by

n

ij

. These assumptions ensure that the semi-norm

R

Ω

|∇u|

2 _dx

is a norm on

H

1

0 (Ω)

equivalent to the

H

1

norm (Poin aré inequality) and also that

H

1

0 (Ω)

and

H

1 _(Ω

i

)

have ompa t embedding into

L

2 _(Ω)

and

L

2 _(Ω

i

)

respe tively(see e..g [8,3℄).

The ow in the uid part is assumed to be established and laminar, so that the velo ity

v

= v(x, y)e

z

is along the

z

dire tion and is a fun tion of

(x, y)

only. The velo ity prole

v

is only assumed bounded:

v ∈ L

∞

_(Ω)

, though itis physi ally ontinuousinallappli ations. Solidsub-domains

Ω

i

aretakenintoa ountbysetting

v

|Ω

i

= 0

.

v > 0

(resp.

v < 0

) on

Ω

i

naturally means that

Ω

i

is a uid sub-domain where the owis in the

z > 0

(resp.

z < 0

) dire tion.

The ondu tivity

k

is isotropi but heterogeneous. Pre isely,

k

is a bounded, positive and pie ewise onstant fun tion onstanton every

Ω

i

:

0 < α ≤ k(x) ≤ β < +∞

a.e. in

Ω ,

k

i

:= k

|Ω

i

∈ R .

(3)

T

i

:= T

|Ω

i

indi ates the restri tion of the fun tion

T

to the sub-domain

Ω

i

. Conventionally here, the dierential operators div,

∇

are onsidered on

R

2

only: div

p

= ∂

x

p

1 + ∂

y

p

2

and

∇f = (∂

x

f, ∂

y

f )

, for a ve tor eld

p

and as alar fun tion

f

respe tively.

2.2 Energy equation

On the innite ylinder

Z = Ω × R

. The dimensionlessenergy equation is div

(k∇T ) + k∂

2 z

T =

Pe

v∂

z

T ,

(4)

where Pe is the dimensionless Pé let number. On the ylinder boundary

∂Z

, on-stanttemperatures are imposed, with a step hange atthe entry

z = 0

:

(

T

|∂Z

= 1

if

z < 0

T

|∂Z

= 0

if

z > 0

.

(5)

Relevantlimit onditions as

z → ±∞

therefore are:

T (·, z) →

z→−∞

1 ,

T (·, z) →

z→+∞

0 .

(7)

Coupling onditionsatthesub-domaininterfa esalsoarerequired,physi ally stand-ing for the ontinuity of the temperature ( on entration) and of the normal heat (mass) ux, they read:

T

i

= T

j

and

k

i

∇T

i

· n

ij

= k

j

∇T

j

· n

ij

on

Γ

ij

,

(7)

whenever the interfa e

Γ

ij

is non-empty,the dot produ t naturallystanding forthe s alar produ tin

R

2

.

3 Mathemati al analysis

3.1 Problem reformulation

Equation (4)is reformulated intoa system of tworst order dierentialequations:

∂

z

T =

Pe

v k

−1

_{T − k}

−1

div

(p)

(8)

∂

z

p

= k∇T ,

(9)

where

T

still denotes the dimensionless temperature (or on entration), the addi-tional unknown

p

denotes a ve tor valued fun tionon

Ω

.

Introdu ing the following unbounded operator

A : D(A) ⊂ H 7→ H

on an Hilbert spa e

H

andwithdomain

D(A)

(whosedenitionsfollow),system(8)takestheform of anODE onthe innitedimensional spa e

H

with unknown

Φ(z) ∈ H

:

d

dz

Φ(z) = AΦ(z) , Φ(z) =

T (z)

p(z)

, A =

Pe

vk

−1

_−k

−1

div

(·)

k∇·

0 .

(10)

The spa e

H

is dened as the Hilbert spa es produ t

H = L

2 _{(Ω) × (L}

2 _(Ω))

2

, where

(L

2 _(Ω))

2

is the spa e of square integrableve tor valued fun tionson

Ω

.

H

is equipped with the followings alar produ t,

(Ψ

1 , Ψ

2 )

H

=

T

1 p

₁

,

T

2 p

₂

H

=

Z

Ω

T

1 T

2 kdx +

Z

Ω

p

₁

· p

₂

k

−1

dx .

(11)

Note that this s alar produ t on

H

is equivalent to the anoni al one (taking

k =

1

) by using assumption (3). It has been modied to ensure the symmetry of the operator

A

.

RelativetoahomogeneousDiri hletboundary ondition,the domain

D(A)

is given as

D(A) := H

1

0 (Ω) × H(

div

, Ω)

,where

H(

div

, Ω) = {p ∈ (L

2 _(Ω))

2 _,

div

(p) ∈ L

2 _(Ω)}

in the distribution sense. We shall refer to[4℄ for the basi properties of the spa e. Su ha denition of

D(A)

ensures that

A : D(A) ⊂ H 7→ H

in (10) iswelldened. Proposition 3.1. Theoperator

A

is dense and symmetri :

(8)

Proof. The density of

A

dire tly follows from its denition. Denoting

Ψ

j

=

T

j

p

j

,

j = 1, 2

, using the Green formulaand the fa tthat

T

j

∈ H

1

0 (Ω)

yields:

(AΨ

1 , Ψ

2 )

H

=

Z

Ω

Pe

vT

1 T

2 dx −

Z

Ω

div

(p

1 ) T

2 dx +

Z

Ω

∇T

1 · p

2 dx

=

Z

Ω

Pe

vT

2 T

1 dx +

Z

Ω

p

1 · ∇T

2 dx −

Z

Ω

T

1

div

(p

2 ) dx

= (Ψ

1 , AΨ

2 )

H

.

3.2 Spe tral analysis of

A

In this se tion, the main theoreti al result of our study is proved. Weshow that

A

is self adjoint and that (0 ex epted), its spe trum is made of eigenvalues of nite order only, the orresponding eigenfun tions forming a Hilbert ( omplete) base of

(

Ker

A)

⊥

₌

Ran

A

. We observe that denoting by

Ψ

n

= (T

n

, p

n

)

the omponentsof the

n

th

eigen-fun tion(

AΨ

n

= λ

n

Ψ

n

),and introdu ing

T (x, y, z) = e

λ

n

z

T

n

(x, y)

, we have: div

(k∇T

n

) + λ

2 n

kT

n

= λ

n

Pe

vT

n

and div

(k∇T ) + k∂

2 z

T =

Pe

v∂

z

T ,

and

T

isa solutionofthe originalenergy equation (4). In identally,we alsore over theso- alled eigen-values/fun tionsofthepreviouslyquotedliterature[1,2,6,5,13, 26, 27,15, 25℄. This theorem therefore brings full legitima yto the de ompositions routinelyfound inthe literature.

Theorem 3.2.

A : D(A) ⊂ H 7→ H

is self-adjoint and has a ompa t resolvent. Weintrodu etheKernelof

A

,Ker

A = {(0, p), p ∈ H

0 (

div

, Ω)}

, where

H

0 (

div

, Ω) =

{p ∈ H(

div

, Ω),

div

p

= 0}

. Then there exists a Hilbert base

(Ψ

n

)

n∈N

of Ran

A =

(

Ker

A)

⊥

omposedof eigen-fun tions:

Ψ

n

∈ D(A)

,

AΨ

n

= λ

n

Ψ

n

,

kΨ

n

k

H

= 1

. The oordinates of

Ψ

n

are denoted

Ψ

n

= (T

n

, p

n

) = (T

n

, k∇T

n

/λ

n

)

. We therefore have

D(A) =

(

Ψ ∈ H ,

X

n

|λ

n

(Ψ, Ψ

n

)

H

|

2 < +∞

)

, AΨ =

X

n

λ

n

(Ψ, Ψ

n

)

H

Ψ

n

,

for all

Ψ ∈ D(A)

.

Moreover this base an be split into two parts

Ψ

+

i

i∈N

and

Ψ

−

i

i∈N

su h that:

0 > λ

+

₁

≥ · · · ≥ λ

+

_j

≥ · · · → −∞ ,

0 < λ

−

₁

≤ · · · ≤ λ

−

_j

≤ · · · → +∞ ,

(13)

The orrespondingeigen-fun tionsare denoted

Ψ

±

n

. Eigen-values(fun tions), a ord-ing to this de omposition, are respe tively alled upstream (+) and downstream (-) .

(9)

Lemma 3.3. For any

f ∈ L

2 _(Ω)

. there exists a unique

T ∈ H

1

0 (Ω)

satisfying div

(k∇T ) = f

in the distribution sense. That solution also satises on ea h sub-domain

Ω

i

:

T

i

∈ H

2 _(Ω

i

)

, div

(k∇T ) = f

in

L

2 _(Ω

i

)

(strong sense) and

kT

i

k

H

2 (Ω

i

)

≤

Ckf k

L

2 _(Ω)

(

C

independent on

f

). Moreover

T

satises on every interfa e

Γ

i,j

the oupling onditions (7)in the tra e sense.

Proof.

A

is dense and symmetri . Sin e

vk

−1

_{∈ L}

∞

_(Ω)

,

A

is also a ontinuous perturbation of the symmetri operator

A

0 : D(A) ⊂ H 7→ H

dened as

A

0 =

0 −k

−1

div

(·)

k∇·

0

. Using the Kato-Relish theorem (see e.g. [23℄ p. 163), the self-adjointnessof

A

0

impliesthe self-adjointnessof

A

. Toprovethe self-adjointness of

A

0

, one shows that

A

0 + i

has range

H

(see e.g. [22℄).

Letus x

(f, q) ∈ H

. We sear h for

T ∈ H

1

0 (Ω)

su h that:

∀ ϕ ∈ H

₀

1 (Ω) :

Z

Ω

T ϕkdx +

Z

Ω

k∇T · ∇ϕdx =

Z

Ω

∇ϕ · qdx −

Z

Ω

iϕf kdx .

Ontherightone learlyhas a ontinuouslinearformon

H

1

0 (Ω)

,whereasthe leftside exhibits a symmetri ,positive, ontinuous and oer ivebilinear produ t on

H

1

0 (Ω)

. As a result, the Lax-Milgram theorem applies (see e.g. [8℄) ensuring the existen e and uniqueness of su h a

T

. Let us dene

ip = q − k∇T ∈ (L

2 _(Ω))

2

. From the above equality we obtain:

∀ ϕ ∈ C

_c

∞

(Ω) :

Z

Ω

ip · ∇ϕdx =

Z

Ω

k(if + T )ϕdx .

This equality shows that, in the distribution sense, div

(p) ∈ L

2 _(Ω)

and we have

p

∈ H(

div

, Ω)

. Thus

Ψ = (T, p) ∈ D(A)

and one has

(A

0 + i)Ψ = (f, q)

,soproving the self adjointness of

A

0

and

A

.

To prove that

A

has a ompa t resolvent, we introdu e the pseudo inverse of

A

,

A

−1

_:

Ran

A 7→ (

Ker

A)

⊥

_{∩ D(A) =}

Ran

A ∩ D(A)

and we prove that

A

−1

is ompa t.

For this let us onsider a bounded sequen e

(f

n

, q

n

) ∈

Ran

A

. There is a unique

(T

n

, p

n

) ∈

Ran

A∩D(A)

satisfying

A(T

n

, p

n

) = (f

n

, q

n

)

.

(T

n

)

thensatises

k∇T

n

=

q

_n

and therefore forms a bounded sequen e in

H

1

0 (Ω)

. The ompa t embedding

H

1

0 (Ω) 7→ L

2 (Ω)

thus impliesthat

(T

n

)

isrelatively ompa t in

L

2 _(Ω)

. We now introdu e

ϕ

n

∈ H

1

0 (Ω)

the unique variational solution to div

(k∇ϕ

n

) =

Pe

vT

n

− kf

n

. Let us prove that

p

n

= k∇ϕ

n

. Sin e

A(T

n

, k∇ϕ

n

) = (f

n

, q

n

)

, we haveto he k that

(T

n

, k∇ϕ

n

) ∈ (

Ker

A)

⊥

:

∀ p ∈ H

0 (

div

, Ω) :

T

n

k∇ϕ

n

,

0 p

H

=

Z

Ω

∇ϕ

n

· pdx = −

Z

Ω

ϕ

n

div

(p)dx = 0

Lemma3.3 thenapplies and ensures that

ϕ

n|Ω

i

∈ H

2 _(Ω

i

)

and that,

(

Pe

vT

n

− kf

n

)

, being bounded in

L

2 _(Ω)

,

(ϕ

n|Ω

i

)

is bounded in

H

2 _(Ω

i

)

. Therefore both omponents of

(∇ϕ

n|Ω

i

)

are bounded in

H

1 _(Ω

i

)

, thus implying that both omponents of

(p

n|Ω

(10)

alsoarebounded in

H

1 _(Ω

i

)

. The ompa tembedding

H

1 _(Ω

i

) ⊂ L

2 (Ω

i

)

thenensures that

(p

n

)

is relatively ompa t in

L

2 _(Ω)

. Consequently,

A

−1

is ompa t and self adjoint on the separable spa e Ran

A

. Therefore there exists a Hilbert base

(Ψ

n

)

n∈N

for Ran

A

made of eigen-fun tions:

Ψ

n

∈ D(A)

,

Aψ

n

= λ

n

Ψ

n

.

A

−1

being ompa t, 0 is the only limit point for sub-sequen es of

(1/λ

n

)

and thus

{−∞, +∞}

are the only two possible limit points for sub-sequen es of

(λ

n

)

. It is easily seen that, whatever the value of

α ∈ R

,

A + α

is bounded neither below nor above. The spe trum is therefore alsoneither bounded below nor above. Thus

{−∞, +∞}

are both limitpointsfor the spe trum, implying de omposition(13).

3.3 Solution derivation

The results of the previous se tion are used here to derive the solution

Φ(z) =

(T (z),

p

(z))

to(8)-(10) su h that

T

satises the boundary,limit and interfa e on-ditions in (5)-(6) and (7). We point out that the boundary ondition (5) implies that, for

z < 0

, one does not have

Φ(z) ∈ D(A)

. For this to be taken into a ount, we shall onsider the (maximal) extension

A

to operator

A

:

• D(A) = H

1 _{(Ω) × H(}

div

, Ω)

,

• A : D(A) 7→ H

has the same algebrai expression as

A

in (10).

Unlike

A

,

A

isnot symmetri :

(AΨ

1 , Ψ

2 )

H

= (Ψ

1 , AΨ

2 )

H

+

Z

∂Ω

T

1 p

2 · nds −

Z

∂Ω

T

2 p

1 · nds ,

(14)

for allpairs of fun tions in

D(A)

, with the usual notations.

Denition 3.4. We shall dene a solution to (8)-(10) with onditions (5),(6) and (7)as a fun tion

Φ : z ∈ R 7→ Φ(z) = (T (z),

p

(z)) ∈ H

su h that:

• Φ ∈ C (R, H)

( ontinuity on

R

),

• Φ ∈ C

1 _{(R − {0}, H)}

( ontinuous Fre het dierentiability on

R − {0}

),

• ∀z ∈ R − {0}

,

Φ(z) ∈ D(A)

and

d

dz

Φ(z) = AΦ(z)

,

and su h that

T

satises the limit ondition (6) as

z → ±∞

in

H

's norm and the boundary, interfa e onditions (5)-(7) for all

z 6= 0

in the tra e sense.

That formalism being stated:

Proposition 3.5. There exists a unique solution

Φ

to (8)-(10) with onditions (5),(6) and (7). Dening the onstants

(α

n

)

,

α

n

:=

1 λ

2 n

Z

∂Ω

k∇T

n

· nds =

1 λ

n

Z

∂Ω

p

n

· nds,

(15)

(11)

this solution is givenas follows:

Φ(z) =











−

X

n

α

n

Ψ

n

+

X

n

α

−

_n

e

λ

−

n

z

Ψ

−

_n

z ≤ 0

−

X

n

α

_n

+

e

λ

+

n

z

Ψ

+

_n

z ≥ 0

(16)

The expression an moreoverbe simplifyed and the temperature eld isgiven by:

T (z) =











1 +

X

n

α

−

n

e

λ

−

n

z

T

−

n

z ≤ 0

−

X

n

α

+

_n

e

λ

+

n

z

T

_n

+

z ≥ 0

(17)

Sin e

A

is not se toral (is not the innitesimal generator of an analyti semi-group,seee.g. [10℄),somepre autionshavetobetakenindemonstratingthe propo-sition. Adetailed proof follows.

Proof. Using the Hilbert base

(Ψ

n

)

of

(

Ker

A)

⊥

, the solution

Φ

is sought in the form

Φ(z) =

P

n

(Φ(z), Ψ

n

)

H

Ψ

n

. All oe ients must therefore satisfy the ODE

d

dz

(Φ(z), Ψ

n

)

H

= (AΦ(z), Ψ

n

)

H

. Then using (14), the boundary ondition (5) and the equality

k∇T

n

= λ

n

p

n

, we nd that

d

dz

(Φ, Ψ

n

)(z) = (Φ, AΨ

n

)(z) + ω(z)

Z

∂Ω

p

_n

· nds = λ

n

(Φ, Ψ

n

)(z) + λ

n

α

n

ω(z) ,

where

ω(z) = 0

when

z > 0

and

ω(z) = 1

otherwise. Looking for a bounded and ontinuous solutionto this ODE on

R

givesus a unique solution,a ording to

λ

n

's sign (

λ

+

n

< 0

and

λ

−

n

> 0

):

(Φ, Ψ

−

_n

)(z) =

(

α

−

n

e

λ

−

n

z

− 1

z < 0

0 z > 0

,

(Φ, Ψ

+

n

)(z) =

−α

+

n

z < 0

−α

+

n

e

λ

+

n

z

z > 0

.

This gives us de omposition (16) and the uniqueness of the solution. Let us now prove that

Φ

dened by (16) isa solutionwith the sense in3.4.

Consider the (unique) fun tion

ϕ

∞

∈ H

1

0 (Ω)

su h that div

(k∇ϕ

∞

) =

Pe

v

. We introdu e

Φ

∞

=

1 k∇ϕ

∞

∈ H

,afun tionthat learlysatises

Φ

∞

∈ D(A)

,

AΦ

∞

=

0

and

Φ

∞

∈ (

Ker

A)

⊥

. Letusprove that

Φ

∞

= −

P

n

α

n

Ψ

n

(thusexplaininghowto gofrom (16) to(17)). Sin e

λ

n

p

n

= k∇T

n

:

(Φ

∞

, Ψ

n

)

_H

=

Z

Ω

T

n

kdx +

1 λ

n

Z

Ω

k∇ϕ

∞

· k∇T

n

k

−1

dx =

Z

Ω

T

n

kdx −

1 λ

n

Z

Ω

Pe

vT

n

dx,

and using the equality

λ

n

kT

n

=

Pe

vT

n

−

1 λ

n

div

(k∇T

n

)

, weobtain

(Φ

∞

, Ψ

n

)

H

= −

1 λ

2 n

Z

Ω

k∇T

n

· nds = − α

n

.

(12)

Thus

−

P

n

α

n

Ψ

n

= Φ

∞

∈ H

, and it follows that

Φ

±

∞

= −

P

n

α

±

n

Ψ

n

∈ H

and

Φ

∞

=

Φ

−

∞

+ Φ

+

∞

. We use the fa t that

Φ ∈ D(A)

if and only if

P

n

|λ

n

(Φ, Ψ

n

)

H

|

2 < +∞

. Sin e

λ

+

n

→

n

−∞

(resp.

λ

−

n

→

n

+∞

), it is straightforward to he k that the two fun tions,

f (z) =

X

n

α

−

n

Ψ

−

n

e

λ

−

n

z

, g(z) =

X

n

α

+

n

Ψ

+

n

e

λ

+

n

z

,

satisfy:

• f ∈ C ((−∞, 0], H)

,

g ∈ C ([0, +∞), H)

( ontinuity),

• f ∈ C

1 _{((−∞, 0), H)}

,

g ∈ C

1 _{((0, +∞), H)}

( ontinuous Fre het dierentiabil-ity),

•

for

z < 0

(resp.

z > 0

),

f (z) ∈ D(A)

(resp.

g(z) ∈ D(A)

)and

d

dz

f (z) = Af (z)

(resp.

d

dz

g(z) = Ag(z)

).

The fun tion

Φ

in (16) an be rewritten as

Φ(z) = Φ

∞

+ f (z)

,

z ≤ 0

and

Φ(z) =

−g(z)

,

z ≥ 0

(whi h fun tions a tually mat h at

z = 0

using

Φ

∞

= Φ

−

∞

+ Φ

+

∞

). It is therefore ontinuous on

R

, Fre het dierentiable on

R − {0}

,

Φ(z) ∈ D(A)

and

d

dz

Φ(z) = AΦ(z)

for

z ∈ R − {0}

sin e

AΦ

∞

= 0

. It is also lear that

T (z)

satises the limit ondition (6) and the boundary ondition (5)for

z 6= 0

.

Itremainstobeprovedthatitalsosatisestheinterfa e onditions(7)for

z 6= 0

. For this, letus onsider the previously introdu edfun tion

f

whose omponents willbe denoted as

f (z) = (t(z), p(z))

. Sin e

λ

−

n

→

_n

+∞

,it is easyto he k that, for

z < 0

,

Af (z) ∈ D(A)

. Therefore

k∇t(z) ∈ H(

div

, Ω)

whi h implies that div

(k∇t)(z) ∈

L

2 _(Ω)

for

z < 0

. Applying 3.3, it follows that

t(z)

satisesthe interfa e onditions (7). Thesameresultappliesto

g(z)

for

z > 0

and, asaresult,to

T (z)

for

z 6= 0

.

4 Mixed variational formulation and approximation

4.1 Mixed variational formulation

Let us onsider the following variational problem: nd

(λ, T, p) ∈ R × L

2 _{(Ω) ×}

H(

div

, Ω)

su hthat,

∀(u, q) ∈ L

2 _{(Ω) × H(}

div

, Ω)

,

Z

Ω

Pe

vT udx −

Z

Ω

u

div

(p)dx = λ

Z

Ω

T ukdx

(18)

−

Z

Ω

T

div

(q)dx = λ

Z

Ω

p

· qk

−1

_{dx .}

(19)

It is lear that whenever

Ψ

n

is an eigen-fun tion as given in theorem 3.2, then

(λ

n

, T

n

, p

n

)

satises the variational problem above. Conversely if

(λ, T, p)

satises (18)-(19) for all

(u, q) ∈ L

2 _{(Ω) × H(}

(13)

H

1

0 (Ω)

(using the dense embeddingof

H(

div

, Ω)

into

H

1/2

_(∂Ω)

′

, see [4℄). Therefore

Ψ = (T, p) ∈ D(A)

and satises

AΨ = λΨ

. Thus

Ψ = Ψ

n

for some

n

and solving (18)-(19) is equivalent tonding allthe eigen-values/fun tionsof operator

A

.

4.2 Axi-symmetri al implementation

In order to test this variational formulation, we have derived a one-dimensional version of the problemwhi his interesting inthe ase ofanaxi-symmetri al ong-urations. The motivation is to test the onvergen e of the problem numeri ally on knownsolutions. Thesimplest aseis onve tion-diusioninsideasingle ylinderfor whi h, in the limit of large Pé let number, we should re over the Graetz spe trum [9℄ for the operator

A

. In this se tion we onsider the somewhat more general ase oftwo on entri ylinders,forwhi h

Ω = Ω

1 ∪ Ω

2

,with

Ω

1

aninnerdisklledwith liquidand

Ω

2

an outer solid orona. When the size of the se ond domainto set to zero, the single ylinder problemis found again asa parti ular ase.

Aliquidowsinside

Ω

1

withaunidire tional,longitudinal,dimensionlessvelo ity

v(r)e

z

whi hvariesfromamaximalvalue atthe ylinder enter

r = 0

tozeroatthe boundary with the se ond ylinder pla ed at

r = r

0

. We hoose the dimensionless velo ity to follow the usual Poiseuille ow prole

v(r) = 2

Pe

(r

2

0 − r

2 )

, although any ontinuous prole being zero at the boundary ould be hosen. The velo ity normalization is set so that normalized surfa e averaged velo ity ux is the Pé let number :

1 kΩ

1 k

Z

Ω

1 v(r)dΩ

1 =

Pe Where

kΩ

1 k = πr

2

0

is the inner disk area asso iated with the rst inner ylinder se tion. In orona

Ω

2

the velo ity is taken to be zero; no onve tion o urs in this se ond domain. Continuity of ux and temperature (7) are applied at the domain frontier

∂Ω

2 ∩ ∂Ω

2

with uniform ondu tivity

k = 1

. The radial dimensionless distan e is hosen so that

r = 1

orresponds to the outer boundary of the se ond ylinder

∂Ω

2 − ∂Ω

1 ∩ ∂Ω

2

wherea homogeneousDiri hlet boundary ondition (5)is hosen.

Problem (18)-(19) is approximated on a regular one-dimensional mesh dis retizing oordinate

r ∈ [0, 1]

with index

i

on grid

r = i/n

with

i ∈ {1, n}

. We adopt here the lassi al mixed nite element approximation of order 0 of Raviart and Thomas

P

0 × RT

0

(seee.g. [4℄)tothepresentaxi-symmetri al1D formulation. Base elementsforthe s alar

T

aretherefore

P

0

pie ewise onstantfun tionsover thegrid elements, whereas base elements for the 've tor'

p

are the

P

1

ontinuous pie ewise ane fun tions over the grid elements: thus re-establishing the ux ontinuity at the grid points.

The generalized linear eigenvalue problem resulting from this dis retization hoi e is asfollows:

AΨ

n

=

a

b

T

₀

Ψ

n

= λ

n

c 0

0 d

Ψ

n

,

(20)

Where

Ψ

n

is a

2n

omponent ve tor whose rst

n

omponents are the dis rete temperature eld

T

n

= (T

i

)

i∈{1,n}

approximating

T

λ

and the following

n + 1

to

2n

(14)

omponentsdes ribe

p

n

approximatingthe gradienteld

p

λ

= ∂

r

T

λ

/λ

whi his one-dimensionalinthis axi-symetri al ontext. The

n × n

matri es

a

,

b

,

c

and

d

anbe omputed analyti allyand admitthe following oe ients:

a

ij

=

−δ

ij

Pe

2r

0 n

4 (2i − 1)(2i

2 _{− 2i − 2r}

2

0 n

2 + 1)

b

ij

=

−

_n

1 ( δ

ij

i + δ

i−1j

(1 − i) )

c

ij

=

δ

ij

2i−1

_2n

2 d

ij

= −

_12n

1

2 ( δ

ij

8i + δ

i−1j

(2i − 1) + δ

i+1j

(2i + 1) ) ,

(21)

where

(i, j) ∈ {1, n}

2

and

δ

isthe Krone ker symbol.

4.3 Numeri al results and onvergen e

Inthegeneralizedeigenvalueproblem(20),onenotesthatthematri e

A

issymmetri and that the righthand side mass-matrixDiag

(c, d)

issymmetri positive denite. Therefore, problem(20) an benumeri ally solved using the variant of the Lan zos algorithm for generalized eigenvalue problems (see e.g. [24℄). The resulting rst eigenve tors and eigenvalues were omputed using the Fortran library ARPACK and spar e matrixstorage. The resultspresented here orrespond totwoparti ular ongurations:

•

asingle ylinder with a singleradial domain

Ω

1

for whi h

r

0 = 1

and,

•

two on entri ylinders whose radius ratio istwo,so that

r

0 = 1/2

.

We study the numeri al onvergen e of the rst eigenvalues and rst eigenve tors when the Pé let number is varied from lowto high values. We systemati ally om-pared the dis rete numeri alresults with referen e solutions obtained with another iterativemethod explained inthe appendix A.

4.3.1 Single ylinder :

r

0 = 1

In the ase of a single ylinder, for large values of the Pé let number, the upstream part of

A

's spe trum (positive eigenvalues

λ

−

n

asso iated with the

z < 0

region) is di ult to ompute numeri ally for it diverges with Pe [21℄. In ontrast, the downstreampartofthespe trum(negativeeigenvalues

λ

+

n

asso iatedwiththe

z > 0

region) onvergestotheGraetzspe trum,andde aystozeroas

1/

PewhenthePé let number in reases.

Let us rst dis uss the eigenvalue onvergen e. Figure 2 illustrates the relative error

E =

p(λ

n

− λ)

2 _/λ

asso iated with the rst two downstream eigenvalues

λ

+

1

and

λ

+

2

and for the rst upstream one

λ

−

1

. it an be seen in this gure that the onvergen e ofthenumeri alestimationis onsistentwiththe hosen lassi almixed niteelementapproximationspa e

P

0 ×RT

0

,forwhi ha

∼ 1/n

behaviorisexpe ted. Furthermore,thestronginuen eofthe Pé letnumberon onvergen e rate analso be observed. ForsmallPé let number, the spe trumis almostsymmetri al, sothat one expe ts the onvergen e for

λ

+

1

and

λ

−

1

to be very lose, as observed on gure 2a. In ontrast, as the Pé let number in reases, there is a distin t shift in the

(15)

10

1

10

2

10 −3

10 −2

1+

2+

1−

(a)

log

₁₀

E

log

₁₀

n

10

1

10

2

10 −3

10 −2

10 −1

1+

2+

1−

(b)

log

₁₀

E

log

₁₀

n

Figure 2: (a) Relative numeri al error for eigenvalue

λ

+

1

,

λ

+

2

and

λ

−

1

for

P e = 0.1

. The dotted lines orresponds to a

−1

slope asso iated with a

∼ 1/n

behavior. (b) Same onvention as (a) for

P e = 10

.

Sin e

λ

−

1

diverges with Pe , it is more di ult to approximate numeri ally and, then, the relative error asso iated with

λ

−

1

in gure 2b is 30% larger than the one asso iatedwith

λ

+

1

forPe

= 10

. Thisdieren efurtherin reaseswithPé letnumber. We also wish to illustrate the numeri al onvergen e on the eigenfun tion. Figure 3 illustrates the eigenve tor omputation for the temperature and gradient elds asso iated with

λ

+

1

,

λ

+

2

and

λ

−

1

eigenvalues. In the ase of small Pé let numbers, the asymptoti symmetry of the eigenvalue spe trum also implies a symmetry of the eigenve tors,dswhi h is learly visible when omparing the

1+

and

1−

elds in Figure 3. The asso iated leading order eigenfun tion shows a single maximum at

r = 0

,the ylinder enter, and obviouslyde reases tozeroat

r = 1

forthe Diri hlet boundary ondition to befullled. When the asso iatedeigenvalue order in reases, the orresponding eigenfun tion has as many os illations as the eigenvalue order. For example for

λ

+

2

, two riti al points an be seen, a minimum and a maximum, forthe eigenfun tioninFigure3. The superpositionbetween the dis retenumeri al omputation and the exa t solution is also illustrated in Figure 3. One an see thatthe omparisonforthegradientdepi tedinFigure3(b)isroughfor

n = 20

,but nodieren eisvisiblebetween the twofor

n = 320

inFigure3(d). The onvergen e tothe exa t solutionis alsoillustrated in gure4 for

P e = 10

. In this ase the two eigenfun tionsasso iated with

λ

+

1

and

λ

−

1

diermarkedly. The rst one, asso iated with

λ

+

1

, still rea hes a maximum atthe tube enter

r = 0

, whereas the maximum positionofthe se ondone,asso iatedwith

λ

−

1

, isshifted losetothe tubeboundary at

r = 1

. Furthermore, this se ond eigenfun tion de ays to zero atthe tube enter. Thereasonfor thisdistin tbehaviorisnowthe opposite roleof onve tionforthese two temperature proles. For the downstream eigenfun tion asso iated with

λ

+

1

, longitudinal onve tionprevailsoverdiusion. Sin ethis onve tion ismaximumat the tube enter, it di tatesthe shape of the orresponding temperatureprole. For theupstreameigenfun tionasso iatedwith

λ

−

1

,retro-diusionistheonlyme hanism for this temperature to display a ba k-ow exponential de ay. Hen e, sin e the onve tion is maximal at the tube enter, retro-diusion is maximum at the tube

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0

0.5

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 1+

exa

N=20

0

0.5

1 −0.5

0

0.5

1 2+

0

0.5

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 1−

(a)

0

0.5

1 −1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0 1+

exa

N=20

0

0.5

1 −4

−3

−2

−1

0

1

2 2+

0

0.5

1 −1.5

−1

−0.5

0 1−

(b)

0

0.5

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 1+

exa

N=320

0

0.5

1 −0.5

0

0.5

1 2+

0

0.5

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 1−

( )

0

0.5

1 −1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0 1+

exa

N=320

0

0.5

1 −4

−3

−2

−1

0

1

2 2+

0

0.5

1 −1.5

−1

−0.5

0 1−

(d)

Figure 3: (a) Temperature eld

T (i/n) i ∈ {1, n}

for dis retization

n = 20

and Pe

= 0.1

forthersttwodownstreameigenve tors

1+

and

2+

andtherstupstream eigenve tor

1−

. Normalization

T (0) = 1

has been imposed. (b) Temperature gradi-ent

p = ∂

r

T (i/n)/λ

for dis retization

n = 20

and Pe

= 0.1

. ( ) Same onvention as (a)fordis retization

n = 320

. (d)Same onventionas(b)fordis retization

n = 320

.

boundary, where the velo ity vanishes. A boundary layer develops near

r = 1

, the thi kness of whi h de ays to zero as the Pé let number diverges. This boundary layer is responsible for the numeri al di ulties arising in the omputation of the upstream part of the spe trum atlarge Pé let numbers. The slower onvergen e of the eigenve tors

1−

is learly visible in gure 4a and 4b for a rough dis retization of

n = 20

points. Although in this ase, the rst two downstream eigenfun tions,

1+

and

2+

are wellapproximated by the orrespondingeigenve tors,this isnot the asefortheupstreamone

1−

. Nevertheless, forasu ientdis retizationof

n = 320

points,the onvergen e an be satisfa tory asillustrated on gure4 ,d.

We nally wish to illustrate the onvergen e on the eigenve tor by omputing the relative error

E =

p(Ψ

n

− Ψ, Ψ

n

− Ψ)

H

/(Ψ, Ψ)

H

built with the

H

norm (11) for a dis rete eigenve tor

Ψ

n

to onverge to the theoreti al one

Ψ

. Figure 5 shows the onvergen e of the relative error for in reasing point number

n

. As expe ted,

1/n

behavior is observed for both

P e = 0.1

and

P e = 10

, but the error is larger in the latter ase.

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0

0.5

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 1+

exa

N=20

0

0.5

1 −0.5

0

0.5

1 2+

0

0.5

1

0

10

20

30

40

50

60

70 1−

(a)

0

0.5

1 −1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0 1+

exa

N=20

0

0.5

1 −4

−3

−2

−1

0

1

2 2+

0

0.5

1 −600

−500

−400

−300

−200

−100

0

100

200

300 1−

(b)

0

0.5

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 1+

exa

N=320

0

0.5

1 −0.5

0

0.5

1 2+

0

0.5

1

0

10

20

30

40

50

60 1−

( )

0

0.5

1 −1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0 1+

exa

N=320

0

0.5

1 −4

−3

−2

−1

0

1

2 2+

0

0.5

1 −500

−400

−300

−200

−100

0

100

200 1−

(d)

Figure4: Same onventions asgure 3 for Pe

= 10

.

10

100

10 −3

10 −2

10 −1

1+

2+

1−

(a)

log

10 E

log

₁₀

n

10

100

10 −3

10 −2

10 −1

1+

2+

1−

(b)

log

10 E

log

₁₀

n

Figure5: (a)Relativeerror foreigenve torsasso iatedwith eigenvalues

λ

+

1

,

λ

+

2

and

λ

−

₁

for

P e = 0.1

. The dotted lines orresponds to a

−1

slope asso iated with a

∼ 1/n

behavior. (b) Same onvention as(a) for

P e = 10

.

4.3.2 Two on entri ylinders :

r

0 = 1/2

In the ase where two domains are present, it is interesting to test the numeri al implementationoftheuxandtemperature ontinuity(7)between thetwodomains in this formulation. Figure 6 shows some eigenfun tion proles at the same Pé let

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numbers as those previously illustrated for the single ylinder ase, Pe

= 0.1

and Pe

= 10

. It an be observed on this gure that the temperature ontinuity at the

0

0.5

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 1+

exa

n=20

0

0.5

1 −0.5

0

0.5

1 2+

0

0.5

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 1−

(a)

0

0.5

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 1+

exa

n=320

0

0.5

1 −0.5

0

0.5

1 2+

0

0.5

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 1−

(b)

0

0.5

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 1+

exa

n=20

0

0.5

1 −0.5

0

0.5

1 2+

0

0.5

1

0

1

2

3

4

5

6

7 1−

( )

0

0.5

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 1+

exa

n=320

0

0.5

1 −0.5

0

0.5

1 2+

0

0.5

1

0

1

2

3

4

5

6

7 1−

(d)

Figure 6: (a) Temperature eld

T (i/n) i ∈ {1, n}

for dis retization

n = 20

and Pe

= 0.1

for the rst two positive eigenve tors

1+

and

2+

and the rst negative eigenve tor

1−

. Normalization

T (0) = 1

has been imposed. (b) Same onve tion as (a) for for dis retization

n = 320

. ( ) Same onvention as (a)

P e = 10

. (d) Same onvention as(b) for dis retization

n = 320

.

domainborder

r = r

0 = 1/2

isex ellentevenforamodestdis retization

n = 20

. The same observation an bemade onthe gradient eld. The onvergen e to the exa t solutionwhi h an bevisually he ked ongure6 isbetterthanthe onepreviously obtained with the same parameter in gure 4a. This is due to the fa t that there is no boundary layer in the latter ase when two domains are present. The retro-diusion of the upstream eigenve tor

1−

is possible in the se ond annular domain

Ω

2

, so that it is not onned in a small region near the boundary. The resulting temperature gradients are mu h lower and do not diverge with the Pé let number. Hen e,the maximum temperatureobserved forthe

1−

eigenve tor ofgures6 and 6d is indeed lo alized inside the se ond domain at a radial oordinate larger than

1/2

. Obviously, the temperature values asso iated with this maximum are mu h lower thanin the ase of the single ylinder, due tothe smoothing ee t asso iated with permittingretro-diusion inthe se ond domain

Ω

2

.

(19)

ase. The onvergen e rate is onlya littlebetter (not shown).

5 Con lusion

This paperhas presented a new approa hfor omplex three-dimensional ongura-tions of onve tion-diusion in unidire tional ows. We justify a separate variable solutionapproa hby dening the eigenvalue/eigenfun tionde ompositionof an ap-propriatemixed operator. The theoreti alanalysis shows thatthe properties ofthis operatorallowanon-se torialde omposition ofthe solutioninlongitudinally expo-nentiallyde ayingsolutions. This approa h permits full three-dimensionalproblem to be numeri ally restri ted to two-dimensions. Furthermore, a naturally e ient numeri aldis retizationhasbeenproposedusingnite-elements. The relevan eand e ien y of su h a dis retizationhas been analyzedin simple ongurations.

A knowledgements We thank Ingrid Eide and Silvia Risnovska for their numeri al in-putsto this work. We thankGDR Momasfor nan ial support.

Referen es

[1℄ Aleksashenko,V.A.: Conjugatestationaryproblemofheattransferwitha mov-inguid inasemi-innitetubeallowingforvis ous dissipation. J.Eng. Physi s and Thermophysi s 14(1),5558 (1968)

[2℄ Belyaev, N.M., Kordyuk, O.L., Ryadno, A.A.: Conjugate problem of steady heat ex hange in the laminar owof an in ompressibleuid in a at hannel. J. Eng. Physi s and Thermophysi s 30(3),339344 (1976)

[3℄ Brezis, H.: Analyse fon tionnelle, Théorie and appli ations. Masson (1983) [4℄ Brezzi, F., Fortin, M.: Mixed and hybrid nite element methods. Springer

Series inComputationalMathemati s. (1991)

[5℄ Davis,E.J., Gill,W.: Theee tsofaxial ondu tioninthewallonheattransfer with laminarow. Int. J. HeatMassTransfer 13, 459470(1970)

[6℄ Davis, E.J., Venkatesh, S.: The solution of onjugated multiphase heat and mass transfer problems. Chem. Eng. S i34, 775787(1978)

[7℄ Deen, W.: Analysisof transport phenomena. Oxford University press (1998) [8℄ Gilbarg, D., Trudinger, N.: Ellipti partial dierentialequations of se ond

or-der, Grundlehren der Mathematis hen Wissens haften, vol. 224, se ond edn. Springer-Verlag, Berlin(1983)

[9℄ Graetz, L.: Uber die Wärmeleitungsfähigkeit von Flüssigkeiten. Annalen der Physik 261,(7), 337357 (1885)

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[10℄ Henry,D.: Geometri theoryofsemilinearparaboli equations.Springer-Verlag, LNM 840 (1981)

[11℄ Ho, C., Yeh, H., Yang, W.: Double-pass Flow Heat Transfer In A Cir ular Conduit By Inserting A Con entri Tube For Improved Performan e. Chem. Eng. Comm.192(2), 237255(2005)

[12℄ Ho, C.D., Yeh, H.M., Yang, W.Y.: Improvement in performan e on laminar ounterow on entri ir ular heat ex hangers with external reuxes. Int. J. Heatand MassTransfer 45(17), 35593569 (2002)

[13℄ Hsu, C.J.: Theoreti al Solutions for low-Pe let-number thermal entry Region that Transfer in laminar Flow through on entri Annuli. Int. J. Heat Mass Transfer 13, 190724(1970)

[14℄ Ladyzenskaja,O.A., Ural' eva,N.N.: Equationsaux dérivéespartiellesde type elliptique. Monographies Universitaires de Mathématiques, No. 31. Dunod, Paris (1968)

[15℄ Luikov,A.,Aleksashenko,V.,Aleksashenko,A.: Analyti almethodsofsolution of onjugated problems in onve tive heat transfer. Int. J.Heat MassTransfer 14, 10471056 (1971)

[16℄ Mi helsen, M., Villadsen, J.: The Graetz problem with axialheat ondu tion. Int. J. HeatMass Transfer 17, 13911402 (1973)

[17℄ Nusselt, W.: Die abhängigkeitder wärmeübergangszahl von der rohrlänge. Z. Ver. Deut. Ing.54, 1154 1158 (1910)

[18℄ Papoutsakis, E., Ramkrishna, D., Lim, H.C.: The extended Graetz problem with Diri hlet wall boundary onditions. Appl. S i. Res. 36,1334 (1980) [19℄ Papoutsakis, E., Ramkrishna, D., Lim, H.C.: Conjugated Graetz problems.

Pt.1: generalformalismanda lass ofsolid-uidproblems. Chemi al Engineer-ingS ien e 36(8),13811391 (1981)

[20℄ Papoutsakis, E., Ramkrishna, D., Lim, H.C.: Conjugated Graetz problems. Pt.2: uid-uid problems. Chemi al Engineering S ien e 36(8), 13921399 (1981)

[21℄ Pierre,C.,Plouraboue,F.: Stationary onve tiondiusionbetweentwo o-axial ylinders. Int J.Heat Mass Transfer 50(23-24),49014907 (2007)

[22℄ Reed, M., Simon, B.: Methods of modern mathemati alphysi s. I. Fun tional Analysis. A ademi Press, New York (1975)

[23℄ Reed, M., Simon, B.: Methods of modern mathemati al physi s. II. Fourier analysis, self-adjointness. A ademi Press, New York(1975)

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Uni-[25℄ Shapovalov, V.V.: Heat transferin laminarowof anin ompressibleuid ina round tube. J. Eng. Physi s and Thermophysi s 12(5)(1967)

[26℄ Tan,C.W.,Hsu,C.J.: MassTransferofDe ayingProdu tswithAxialDiusion inCylindri al Tubes. Int.J. HeatMass Transfer 13, 18871905 (1970)

[27℄ Tan,C.W.,Hsu,C.J.: LowPe letnumbermasstransferinlaminarowthrough ir ular7. tubes. Int.J. HeatMass Transfer 15, 21872201 (1972)

[28℄ Telles, A., Queiroz, E., Filho, G.: Solutions of the extended Graetz problem. Int. J. HeatMass Transfer 44(2),471483 (2001)

[29℄ Weigand, B.: Anextra t analyti al solutionfor the extended turbulent Graetz problem with Diri hlet wall boundary onditions for pipe and hannel ows. Int. J. HeatMass Transfer 39(8),16251637 (1996)

[30℄ Weigand,B.,Gassner,G.: Theee tofwall ondu tionfortheextendedGraetz problem for laminar and turbulent hannel ows. Int. J. Heat Mass Transfer 50(5-6), 10971105 (2007)

[31℄ Weigand, B., Kanzamar, M., Beer, H.: The extended Graetz problem with pie ewise onstant wallheat uxfor pipeand hannel ows. Int. J. HeatMass Transfer 44(20), 39413952 (2001)

[32℄ Weigand,B.,Wrona,F.: TheextendedGraetzproblemwithpie ewise onstant wallheatuxforlaminarandturbulentows inside on entri . Heatand Mass Transfer 39(4), 14321181(2003)

A Referen e solutions in axi-symmetri al problems

Inthisappendix, wegivesome detailsabouttheanalyti almethodusedin4forthe analysis of the numeri alresults. The method is based ona property of the eigen-fun tions alled

λ−

analy ity: inthe axi-symmetri alframework, any eigenfun tion

T

λ

an bethe expanded inthe form

T

λ

(r) =

X

n∈N

t

n

(r) λ

n

.

(22)

In this des ription the losure fun tions

{t

n

}

n∈N

are independent of the eigenvalue

λ

onsidered and alsoof the onsidered boundary ondition at

r = 1

. They an be omputedusing a simpleiterativepro ess for the omputationof the spe trumand eigenfun tions witha Maple ode.

The onvergen e of the

λ

-analy ity methodhas been establishedfor general axi-symmetri al ongurations. The proof being the topi of a fore oming paper, and forthesakeofsimpli ity,wefo usourattentionhereonthetreatmentofthe Graetz problem. In this ase, the proof for the onvergen e of the

λ

-analy ity method is available in[21℄. The eigenvalues

T

λ

are dened as follows,on the interval

[0, 1]

:

(22)

where

∆

c

stands for the ylindri al part ofthe Lapla eoperator

∆

c

≡ 1/r∂

r

(r∂

r

)

. Eigenfun tions

T

λ

then read (22) where the

t

n

(r)

fulllthe re ursives heme:

t

0 (r) = 1

and:

∆

c

t

n

= v(r)t

n−1

(r) ,

t

n

(0) = 0

for

n ≥ 1 .

We point out that this s heme a tually has a unique solution thanks to the degen-era y of the ODE at