Finite bandwidth, long wavelength convection with boundary imperfectons: near resonant wavelength excitation

J. Sci.

VOL. 21 NO. (1998) 171-182

171

FINITE

BANDWIDTH,

LONG WAVELENGTH

CONVECTION

WITH BOUNDARY

IMPERFECTIONS: NEAR-RESONANT

WAVELENGTH

EXCITATION

D.N.RIAHI

Departmentof Theoretical andApplied Mechanics University ofIllinoisatUrbana-Champaign

Urbana,Illinois61801, U.SA. (ReceivedJune12, 1996)

ABSTRACT. Finiteamplitude thermalconvection withcontinuous finite bandwidthof long wavelength

modes in aporous layerbetweentwohorizontal poorlyconductingwalls is studied whenspatially non-uniformtemperatureisprescribedatthe lower wall. The weaklynonlinearproblemissolved byusing

multiple scales and perturbation techniques. The preferred long wavelength flow solutions are determined by a stability analysis. The case of nearresonant wavelength excitation is consideredto determinethenon-modal type of solutions. Itisfound that, undercertain conditions on the form of the

’boundary

imperfections, thepreferredhorizontalstructureofthesolutions is ofthesamespatial formas thatof thetotal or some subset of the imperfection shape function It iscomposed ofa multi-modal pattern withspatialvariations over thefastvariables and withnon-modal amplitudes,whichvaryover the slow variables The preferred solutions have unusual properties and, in particular, exhibit ’kinks’ in certain vertical planes which are parallel to the wave vectors of theboundary imperfections. Along

certain vertical axes, where some of these vertical planes can intersect, the solutions exhibit multtple

’kinks’.

KEY WORDS AND PttRASES: Convection, imperfections, long wavelength, long wavelength convection,near-resonance, boundary imperfections,finite-bandwidth

1991AMSSUBJECTCLASSIFICATION CODES: 76R10, 76R99, 76S05, 80A20, 76R05,76D30.

1. INTRODUCTION

Recently Riahi [1] investigated the problem offinite amplitude

discr,ete-modal

convection in a

horizontal layerwithboundary imperfections dueto spatially modulatedboundarytemperatures For

resonant wavelength excitation case, regular or non-regular multi-modal pattern convection in the discrete-modal domain arefoundtobe abletobecomepreferredin somerangesof the amplitudesof the

boundary temperatures, provided the wave vectors of such discrete-modal patterns are contained in certainsubsetof thewave vectorsrepresenting some portion of the modulatedboundarytemperatures

The presentpaperextendstheabovework tothe more complicatedcase offiniteamplitudeconvection with continuous finite bandwidth of modes in a horizontal layer. In order to make the problem mathematically simpler,we consideraporouslayerwithboundary imperfectionstobe duetospatially

non-uniformtemperatureatthe lower wallonlyandassumethat the horizontal wallsarepoorconductors ofheat. Suchlater assumption providesthe analytical method ofapproachduetoBusseand Riahi[2] andRiahi

[3]

tobeusedinthe presentproblem.

The presentinvestigation of the continuousfinitebandwidth ofconvectionmodes applies the method ofapproach due to Newell and Whitehead

[4].

As was explained by Newell and Whitehead [4], continuous-modalanalysis ofconvectionleadsto a widerclassofsolutions which can describeadequately

172

Reesand Riley [5,6] investigated the effects of one-dimensionalsinusoidalboundary imperfections

onweaklynonlinearth=rmalonv=ction in aporousmdiumanddtermined, inparticular, thenonlin=ar systemfor the flow amplitudes, and the effects oftheboundary modulationson thestability ofdifferent

rollcells, and the evolutionof the unstable rollswerestudi=d Rs [7]investigated the effectof one-dimensionallongwavelengththermal modulations onth=onset ofConvectionin aporousmediumand

predicted, in particular,the preference ofamode in the form ofrectangular cells forcertainrangesof valuesofthe modulation wavenumber. Thepr=sentpaper is,in asense,anextension ofReesandRiley’s work

[5]

tomoregeneral two-dimensional imperfections andtomoregeneral andnewthree-dimensional flows andis an=xampleofanimperf=tbifurcation drivenbyimperf=t heatingand/or _cooling. There havebeen literature associated with anumber of authors whose works areall r=lvantto thepresent

problem andwere reviewed in Riahi ].

Th= g=n=ral problemunder onsid=rationCanhavepra=ialvalues inthaton=mightwanttomake

the tmprature ofaboundarynon-uniform ifth transport processes ar=enhanced [1 or ifthe flow

struCtureould b ontrolll.

The presentpaperonsid=rscontinuous finite bandwidthof three-dimensional modes ofconvection andarbitrarynon-uniformt=mpratureboundary onditioninthe form of suchcontinuousmodsatthe lower wall. We hav foundanumber ofimerstingresults. Inparti=ular, we found stable nvlope

solutions whoseflow patternsCanhavquite unusualbehavior. Forexample,d=pendingontheform of the boundary imp=rf=tions, one suCh pattern coupletsin different parts ofspace may be $0 outof phase, andthe solutions can =xhibit ’kinks’ inthehorizontal structure

2. GOVERNING SYSTEM

Weconsider an infinitehorizontalporouslayer of average depthd filled with fluidand bounded by

twoinfinitehalf spaceswith thethermal conductivitywhich isassumedtobe smallincomparisonwith thatoftheporousmedium. Inthesteady static state,a constant heatfluxtraverses the systemsuchthat

the meantemperatures

Tt

and

T

areattainedatthelowerandupperboundariesof thefluid. Introduce a cartesian system, withthe originatthe centerplane of the layer andwiththe z-coordinate in the vertical directionanti-paralleltothe gravityvector. Weshallexaminethe effects of lower boundary temperature perturbationvariationsatafixedvalueof

AT

T

T

and represent the magnitude ofsuch variation relative to

AT

by

.

Itisassumed that

<

o(1).

Wedefine atemperaturerelative tothe conduction

stateby

T,(x,y,z,t)

-

z

+

T(x,y,z,t)

+

T,,,

(21)

where x and y are the horizontal variables and is the time variable It is convenient to use non-dimensionalvariables in whichlengths, velocities,timeand temperatureTarescaled respectively by d,

A/(dp,,c),d2p,,c/)

and

AT/R.

Here

R

gKATdp,,c/(,)

is the Rayleigh number, is the

coefficientof thermal expansion, g is acceleration duetogravity,

K

isthe

Darcy

permeabilitycoefficient,

po is areference

(constant)

fluid density, c isthe specific heatat constantpressure, ,kisthethermal

conductivity of theporousmedium(fluid-solid mixture)and#isthekinematicviscosity. Then,withthe usualBoussinesq approximation that densityvariationsare taken into accountonlyin thebuoyancyterm, the Darcy-Boussinesq-Oberbeck equations for momentum, continuity and heat in the limit ofinfinite

prandtl-Darcy number

[8]

are obtained, which are given in Riahi

[3].

These are equations for 0

(dimensionless T),u(velocityvector)and

P

(modifieddeviationofpressurefrom its staticvalue). Thegoverning equationsaresimplified by using the representation

LONG WAVELENGTH CONVECTION WITH BOUNDARY IMPERFECTIONS 7

forthe divergence free u_ [3] Here _z is a unitvector in thevertical direction. Taking the vertical component ofdouble curl ofthemomentum equation andusing (2 2) in the heat equation yield the

followingequations:

Ag_

(V

2

+

0)

0, (2.3a)

-

0-

RA2

f-V0, (2.3b)

where

A2

isthe horizontalLaplacian These equations mustthen be solved subject totheboundary

conditions

1

=0

at z=

+/-2,

O_O

rlT[O

6Rh(x y)]

_{at z}

Oz

O0 1

Oz

+

rlT20=O

at z=

,

(2 4a)

1

2’

(2 4b)

(2 4c)

where

h(x,

y) is a given spatiallynon-uniform functionofxand y. The formulation oftheboundary

conditionsfor 0follows from thoseduetoKellyand Pal[9]and Riahi[1 forisothermal boundariesand

thosedueto

Sparrow

etal.

[10],

BusseandRiahi

[2]

andRiahi

[3]

for poorly conducting boundaries

’For

further details regarding boundary condition alternative to

(2.4b),

the reader is referred to Riahi

1,11]. Theparameter

72

givenin(24b,c)is a Biotnumber,which isassumed tobesmall

(7

<<

1)

in the present paper. For problemstreated in Riahi [1,11], 3’ oo. Additional parameter r/ introduced in (2 4b,c)isrelatedtothehorizontal wavenumber for the classicallinearproblem(Busseand Riahi[2],

Riahi

[3])

bythe relation

c r/7, (2.5)

anditspresence in(2.4b,c)isneededtocovertheclassical linear discrete modalresults[3].

3. ANALYSIS

The caseofnear resonantwavelengthexcitationcorrespondstothe criticalregime whereR

Rc

and,

o()

[,91.

Here

Re

isthecritical value of

R

belowwhichthereis no motion andEisthe

amplitude ofconvection. Weconsiderthefollowing doubleseriesexpansions for

,

O,

and

R

inpowers

of 7and

6

(3 1)

Because of the nature of the present thermal boundary conditions, it turns out that many ofthe coefficients vanish andonlysystemsto orders61/3

"/61/3 7261/3 762/3

and

726

needtobe analyzedin order to determine the nonlinearproperties ofthe system inthe doublelimitof small7and small6[2,3

Upon inserting

(3.1)

into

(2.3)-(2

4)and disregardingthe quadratic terms, we find thelinearproblem

whose system isgivenin Riahi[3],andit isthe classical linearsystemwhose discretemodal solution was

critical two-dimensional modein theform of rolls alongthe y-axis Inthe presentproblem, weneedto

formulatethe casefor poorly conducting boundaries where the critical modeattheonsetof convection, based onthe discrete modal analysis, is knowntobe three-dimensional mode in the form ofsquare

cells [2,31.

Hence,

weneedtoallowan

o(6a/a)

bandof modesinthezdirection and an

0(61/3)

band of modes in the y direction centered around the critical three-dimensional mode inthe form ofsquares whose wavenumber vector is along the line y z in the horizontalplane. Thus, the general linear solutionof such finite bandwidth modes canbewritten as

N ,(0)

t

+7b

1)

+0(72)

fl(z)

Wn(z,y)An(zs,

y.,ts), ,:-N

(3.)

N

0

)

+70

1)

+0(72)

gx(Z)

W(z,)A.(zs,u,,t,),

where

A,

are functionsoftheslowvariablesformulatedas

=2

=’)’(5, =’76,

L

(5, (3 3)

thefunction

W,

has therepresentation

andsatisfiestherelation

W.

exp(i.K,

r_),

(3.4a)

A2W,

c2W,.

(34b)

Here r_ is the horizontal position vector,

x/c:’-,

c is the horizontal wave number of the flow

structure, N is a positive integer, and the horizontal wave vectors

.K,

(Kx,K)

of the flow

structuresatisfytheproperties

_K

._z

o,

I_KI-

c,

_K_

_K.

(3 5)

The amplitude functions

A,

satisfythe condition

A

A_, (3 6)

wherethe asterisk indicates thecomplex conjugate The expressions for

fl

(z)

and gx

(z)

aregivenin Riahi

[3],

and thedetailsof the results for

Ro

and itsminimum/attainedat c

cc

canalsobe foundin Riahi[3]. Here

Ro

=-

R(o

)

+

7R(o

)

+

o(7

)

_and

cc

r/,7’, where r/= and

2[

+

s-/v,]

+

o(d)

Intheorder

76]

equation(2.3b)yields

t920

1)

where

02

2

(3.7)

The solvability condition for this equation is obtainedby multiplying

(3.7)

with

W,7

and averaging over thewholelayer Duetosymmetric property ofthelayerand(2.4),this conditionyields

I

)

o.

FINITE BANDWIDTH, LONG WAVELENGTH CONVECTION WITH BOUNDARY IMPERFECTIONS 17 5

Inordertodeterminethe solutionfor(3 7),theboundarycondition 1 (1)

/92

=0 at z=

+

_(39a)

has been used for the horizontally averaged component

of/91)

because the horizontal mean ofthe

boundary temperatureisgivenas anexternal parameter of theproblem Theremainingcomponent

of/91)

satisfiedtheboundarycondition

0-

=0 at z=

-t-2,

(39b)

where anover-barin(3.9)denoteshorizontalaverage[2] The system(3 7)and(3 9)then yield

N l,P=N N

9(

D292(z)

IArl

AtAr,

D2f2

(z,

lp)WlWp

+

g3(z)E

Ds,.,A,W,.,,

_{(3 10a)}

m=-N I,P=-N n=-N

where

lz,

lz2

7 0

(

0

g3(z)

5

+

,

D

0--’

Ds

=_2i

g-x

+

Kv

$1

=-

(K-I" K.

)

/2

(3Ob)

and the expressions for the functions

f2

and#2 are givenin Riahi

[3].

Incontrast to thediscretemodal analysiscase[1],there is a need to determine the solution

1)

for(2 3a)inthe order

782/3.

Itis

I,P=N N N

1)

E

AtApf2(z,,p)WtWp +

g2(z)

E

]Aml2

+

f3(z)E

DsmAmWm,

I,P=-N rn=-N m=-N

t:-P

(3 10c)

where

f3(z)

6z6+

4-z

53

z2

+

3840"

47

Intheorder

-8

equations

(2.3)-(2

4)yield the followingsystemfor

(3 10d)

0/9(0

02/9

2) Ot, Oz

(3 lla)

,1

(3 lb)

1

(3 11c)

The function hgivenin(3.1 lb) actuallyisassumedtohavethe following arbitrary representation

N(b))

where

(3.12a)

W(

b)

exp(i_K

).

r.),

_(3.12b)

L

are functionsof

x

andy,N(b) _{is apositiveinteger whichmay tendtoinfinity,andthe horizontal} wavenumbervectors

K

b)

/’

rc() rc()

’

forthe boundaryimperfectionssatisfy the properties

\

...

,w)

D N RIAHI

L

L_.

Weshall assume that

c(

b)

ac

Multiplying(3.1 la) by

W,,

averagingover thewholelayerand using(31lb,c)yields

(3 14)

where anangular bracketindicates anaverageoverthefluidlayer.

Using(3 15),doingsomescalingsont,

Lm

and

A,

(forallpossiblem)andapplyingthe conditions

giveninRiahi [3] forthe non-zero average product, weendupwiththefollowing simplifiedform for

(315)

s--

1)-K,-x+K,"y

A= -A.

IA[2+

lain

N()

+

L(W)),

(n= -N,-..,-1,1,..-,N).

(316)

-N()

Theabove system is a collection of 2N paifl differemifl equations for the 2N

uo

nions

A(n= -N,...,

1,1,---,N)

To

distinish

the physicfllyreliable solution

(s)

ongfll the study solutions of(316), the

stability of

A(m=-N,...,-1,1,-..,N)

th

respe

to disturbces B(x,y,t) are

investigated The system ofequations for the time dependent dismrbces th addition ofatime

dependence ofthefoexp(at)egiven by

=.

(A.A=B

+

A.AB=

+

B.IAI

)

N

+

(A.A=B

+

A.AB=

+

B.A[

)

O,

(n

N,...,

1,1,...,N),

( 17)

where

B. satis

conditions of the fo (3.6) for a

"

Tng

complex conjugate of(3,17)

d replacing the subscript n by -n, it then c be seen

er

sgme remangement that

B.

exp(at.) B:.exp(a’) wMehimplies thataisreM Itisclefrom(3.16)-(3 17)thatthe

bound

imperfection

ects

thesteadysolutionsdirectlyas a source t in(3.16),wMle theimperfection affects thedisturbcesindirectlytough thestudysolutions

4. SOLUTIONS

Weconsiderthe system

(3.16)

for thecaseswhere

A.

t(.

+

v.)

+

i.[cosh(.

+

V.)]-’

+

,

(a.a)

where

d

U.

d

V.

ethereMd

ima#n pros

of

A.,

resptively Such assumption

(4.1)

is

suested

by theemensionof the simpleone-dimensionMenvelopesolutionduetoNewell d tehead

[4]

in their studies offitebddth,fitplitud rollsconvection in alayerth isothermboundmes

The coefficients d

b.

given in (4.1a) e rl constts We e

assung

here that (4.1) is

FINITE BANDWIDTH, LONG WAVELENGTH CONVECTION WITH BOUNDARYIMPERFECTIONS 177

L, =gm

tanh(z,

+

y’,)

+

ih,,[cosh(x,

+

yn)]-1 (42)

where gra andh,,arereal constants. Thejustificationfor suchassumptionwas confirmedby using(4 1)-(4 2)into(316)whichledtothefollowingalgebraic system.

N

-..2

Pl)an

an

2a2n

"q’-

(1

+

emn)

a2m

grn(WW2)),

_{(4 3a)}

m=-N m=-N(b)

N

^2

m=-N -N()

(4 3b)

N

---2

2(1

0, (4 3c)

N

--.2

b’,

2(1

+

a,

-b’,)

+

(1

+

,’,’,)(b2m

a)

=0,

(n= -N,-..,

-1,1,

.,

N).

(43d)

This is a systemof8Nequations for

4(N

+

N

(b))

unknowncoefficients ara, bin,gm and

hm.

Generally,

’solution for N

>

N(b) _{is not possible, unless}

a

_{b, 0 for rn}

>

_{N(). Non-trivial solutions}

(a,

g:

o,

bra

-7/:

0)

arealways possible forN

<

N() _{Ofcourse we areassumingthe casesof significant}

boundary imperfections,so thatthelasttermsintherighthand sides of(43a,b)are non-zero

Itshould be noted that onecould,ingeneral,consideranysolutionoftheformA, F,

(x,,

y,)for (3.16),for given functions

F,,,

where the functions

L

aretobesochosentosatisfy(3 16) However,

detailed investigation of stability of such solution requires a knowledge of particular forms of

F,,

although,as weshallsee later in thissection, all such typeofsolutions canbecome stablefor sufficiently large

[R’)

and

R’)<

0, provided the horizontal averages of functions involvingthebaseflow and disturbancequantitiesand/ortheir first orhigherderivatives remain finite. The main reasonfor assuming (4 1)isthe ratherunusualnon-modalproperties ofthe real parts

U’,

of suchsolutions Where

U,

isweak

namelyatx, y, 0,V, isstrongand vice versa For large

Ix,],

the solutions(3.2)fory’, oareof

scale

27r/a,:

except thatthesenseof rotationofthecorrespondingpatterncoupletsdueto isreversed ontheoppositesidesofthe

xs

0 axis.

Next,

weinvestigate stability of thesolutionsof the form(4.1)withrespecttodisturbances

B’,

of the general form

B,.,

U,.,

+

iV,,, (4 4)

whereU’, and

V,,

arethe real andimaginarypartsof

B,,

respectively. Using(4.1)and(4.4)and(3 17) leadusthe following systems for

U,

and

V,,:

a-R

)

+32",+V,

--xn

+--yn

V’,+2U’,VnV,.,+

(1-t-2’,)

(4.5a)

2-,VraUra

+

2V,V,Vra

+

+

Vm

,,

0,

(n

N,...,

1,1,...,N).

(4.5b)

178 D N. RIAHI

/I)

<

0, providedthe horizontalaverageswhich involve

n,

,

n,

n

and their first orhigherorder derivatives remain finite This resultholds for generalbase flow solutions whichincludethose considered in(4 1), providedthat theboundaryimperfectionsaresignificantsothat thelasttermsin thefight hand

sidesof(4 3a,b)arenon-zero. Forthe casesof insignificant boundary imperfections, where noneofthe wave vectors

_Kn

arealong anyof the wavevectors

_K/,

then the lasttermsinthefighthand sidesof (4.3a,b)vanish Itis thennotdifficulttoshow from

(4.3)-(4

5)that the baseflowsolutions with kinks

,

a,

tanh(x,

+

/,)

and

,

0for all n, are unstable and

pl)

2 and a 1 follow Forthe cases ofsignificant boundary imperfections, such solutions can be stable as the following examples

indicate

Letusconsider nowthe following specific examples for significant boundary imperfectionscases in ordertoillustratethe inter-relations between theboundary imperfectionsandtheresultingpreferred flow

patterns and to demonstrate specific conditions on

pl)

under which absolute stability ofdifferent solutions with kinds arepossible.

EXAMPLE

N() _{1. Considerthe caseN 1first. Supposeg 0and}

hi #

0 Then(4 3) implies that botha,and b,arenon-zeroand

pl)

2(1

+

h).

Assumethat the disturbancequantities

U

and

V,

ingeneral, are functionsofslow variables. Multiply(4.5a) (forn 1)byU, (45b)(for

n 1)byV1, addthetworesulting systemsandaverageoverthe slow variables Theresultingsystem thenyieldthe result that nostablesolution ispossible forsufficientlylargeh,whilestablesolutionmay be possibleforsufficiently small

h

though such possibility cannotbeprovedrigorously Itisimplied fromtheseresultsthatsmaller

P)

casesarefavoredoverlarger

pl)

cases. Supposenow that g :/:0 and

h

0. Then(4.3)implies thatnon-zero valuefor

b

ispossible for

pl)

>

1 since

R

1)

2a

1 and

a

>

1,while

P)

canbe small for

bl

o. For

bl

o,wefind

12

1,

Vt

1) 2 .ql/l, 2-1 al. (4 6)

Hence,

z

I is preferred for 9

<

0, while I is preferred for 1

>

0 Forming again the

integralsystemfortheaveraged amplitude ofthedisturbances,wefindthat

b

0 solutionfor(4.5)is

stablefor

P

<

1. Applythe samemethod ofapproachastheone describedaboveforthe₁ 0 andh 0case,we find

a

=l+b],

P])

=2a-gt/a],

bz

=ah/(gt-at), a_ -at, b_t -bt, (4.7)

so

on ,y

I l’l

<

0. ,oo

preferredfor91

<

0,whiletheoppositeistrueforal

>

0. Ascanbe seenfrom(4.6)-(4.7),the solution

for the caseg

:#

0and/zz 0ispreferredovertheonefornon-zerogl and h since itleadstosmaller

P)

for givengzand

h.

We now consider the case of arbitraryN

>

1 fortheparticularvaluesofgl and

hz

where(4.6)

holds forN 1, that is gz 0and/z 0. Using

(4.3a)

forn 1,itfollows thatal

:

0 since g 0 Theequations(4.3a,c)then yield

P)

2 g--L

+

2b

+

E

(1

+

b,,)b,,,

al:/:0. (4.8) Comparing

(4.8)

to

(4.6),

itfollowsthat the solution forN 1is

preferred

overthe one for

N

>

Iif at

least one

b

0 for oneparticularvalueofn

(n

1,-..,

N)

since

pl)

for N 1is less than theone

forN

>

1 insuch situation. Ifallthecoefficientsb, forn 1,-..,Narezero, then

(4.3)

impliesthat

suchcase ispossible onlyif all thecoefficients

a

forn 2,---,Nare zero. But,thisresult thenimplies thatno non-trivialsolutionforN

>

1case can bepreferred. Theresultsdiscussedabove thusindicate

thatthepreferredhorizontal platformfunctionH(x,V;x,$t),foreither

b

or0,forthecase,where the

BANDWIDTH, LONG WAVELENGTH CONVECTION WITH BOUNDARY IMPERFECTIONS

(4 9a)

isthefollowing function

H

WnAn

2al tanh(xl

+

Yl)cos(.K1" r’_),

a

1, (49b)

n=-a

wherethe wavevector

_K

isalongthewavevector

_K

b)and

c

b) a

ac.

Itisseen from(4 9)that the preferred horizontal structure of the solutions for N(b) _{1 case is a copy of the boundary}

imperfection shape. Itis along wavelengthpatterncomposed ofamodal rollwithperiodicityoverthe

fastvariables

(z,

y)andanon-modal amplitudewhich varies overthe slow variables

(z

1,

Yl)

andexhibits

kinks locallyinthehorizontalplane. Thefunction H can exhibitunusualbehaviorduetothefunction

tanh(zx

+

Yl).

For example, the sign ofthis function for

(z

+

Vx)

oo is opposite to that for

(z

+

y) -c, and

tanh(z

+

x)

5:0 as

(z

+

y) 5:0 Thus, inthelocal regionswhere

(Zl

+

1)

isclosetozero,

H

canbe 180degreesoutof phase fromone locationtoanotherone within suchlocal regions.

EXAMPLE 2 N1) _{---2. For significant imperfection case, we assume _K1}

.K

b) _and

.K

_K

b) ConsiderfirstN 1. Using(4.3),wefindtheresults(4 6)-(4.7)and,thus,theresultsof

,the

type forN() _{Icase(example 1) follow,sothattwosolutions satisfying(46)oneforwhich(4.9)}

holds and another oneforwhich(4

9)

with

_K

and

.K

) replacedrespectively by

.K

and

.K

b) holds Following the results for N 1, we consider the case 9, 0 and h, 0 for arbitrary

N(m

1,2,

.,

N).

Forthe case where N 2, (4.3) yield the following results for the solutions

correspondingtothe

smallest/)

%,=

2+9,

5r=O, -m=--a, b_,=-b, rn=l,2 (410a)

Pt

1)

2 gl/a 2 g2/a,’2. (4 lOb)

Theseresultsimplythatthere is nosolution, unless

g

g.

(4 11)

Weshall assume thatthe givenconstants gl andg satisfy(4.11). Here

b21

isthecosine oftheangle

between the wave vectors

_K

/and

_K

).

Itisseenfrom(4.10)thatthenegativerootfor

a

ispreferred forg,

<

0,whilethe positiverootfor

a

ispreferredforg,,,

>

o

(m

1,

:)

Using

(4.6)

and(4.10),

wefind that the solutionforN 2correspondstosmallervalue

of/1,

for any value of

,

thanthe oneforN 1 case Applyingthe samestability analysisasthe one described inexample 1,wefindthe solu-tion(4.10)isstable

for/1)

<

1. Forthecase/g _1,wefoundthatthe solution(4.6)isstable

for

P)

<

-1. Of course, these stabilityconditions were determined based onupper boundingtype approach,anditispossiblethatthe solutionsfor

N

1 and

N

2 stillremain stable

if/l

is bigger

than -1 and -2, respectively. Forthecasewhere N

>

2 and for significant boundary imperfection

where_/’f,

.Kl(n

1,2),

we findfrom

(4.3)

that no such solution can exist which admits valueof

PI

smaller than thosefor N 1 and N 2 since

a

b,

o(n

3, .-.,

N)

follow. Theresults discussed above indicate that thepreferred solutioncorrespondstoN 2caseif

(4.11)

holds and to

N 1 case if

(4.11)

doesnothold.Inthe later case the results(4.9) follow, while inthe formercase we

have

gl

g,h

(6r/)

-

[gl

tanh(z

+

yl)cos(_K

b).

.r)

+

g2

tanh(x

+

y)cos(.K

1-

r)],

(4.12a)

where

_K

_K(n

b)

(n--

1,

2).

Again, the same results as in the case N(b) _{1 presentedin example}

follow Thepreferredhorizontalstructureofthe solutions is acopy ofthe totalimperfectionshape(case

where(4 I) holds)orcopy ofasubsetoftheimperfection shape(casewhere(4 I)doesnothold) The

longwavelengthpatternexhibitedby (4 12)iscomposedofamodalrectangularpatternwithperiodicity

over thefastvariables and withnon-modal amplitudes,whichvaryoverthe slow variables and exhibit ’kinks’locally

The two examples presented above indicate ageneral theoryfor arbitrary N) _{and forthe case}

where thewavevectorsofthe flow patterncoincidewithasubset ofthe wave vectorsof the boundary imperfection. Such a theory, to be discussed below, is consistent with the results for Nb/ _1,2 presented in the above two examples. For significant boundary imperfection,

_K.,

_K

(for N

())

we m 1,

..,

N()),

and for h, o

(m

1, N

())

and given real constantsg,

(i

1, -,

have withoutlossof generality,thefollowingrelation

g g,,

,...,

(/,

(4 3)

where 1

<_

M(/

_<

_N() _{If(4 13)holdsfor}

M

(b) _N

(),

_{then thepreferredhorizontal structureofthe} solutions is of the samespatial formas thatofthetotal imperfection shapefunction. Itiscomposed ofa multi-modal

(N

N

()

pattern withspatialvariations over thefastvariables

(x,

y)and withnon-modal amplitudes,whichvaryoverthe slowvariables

(x,.,, ,)

(n

1,...,

N).

If(4.13)holds for

M

()

<

_N

(’,

then thepreferredhorizontal structureof the solutionsisof thesamespatial formasthat ofasubsetof

the imperfection shape function. It is composed ofa multi-modal

(N

M

(bl)

patternwith spatial

variations over the fast variables

(x,l)

and with non-modal amplitudes which vary over the slow

,--

M

(/)

Such solutions can exhibit kinks in spatial locations where variables (x,,y,)

(n

1

.,

(x,

+

,)-

o

(n

1,...,M

())

These kinks are in certain vertical planes which are parallel to

significantwavevectorsoftheboundary imperfections Thepreferredsolutions arestable for sufficiently large

[R)[

andP

1)

5. DISCUSSION

Dueto the fact that the present investigationis based onthe assumption that the amplitude of convectionisoforder

6

and 6

<<

1,thepresent results donotchange qualitatively fromthoseforthe

problemwhere thelower boundary’slocation is atz

+

6h(x,

$1). This conclusionactuallyproved by Riahi [11] for the discrete-modal case and appearsto be followed here, as well. The boundary corrugated problem,whose locationisdescribedabove,canincorporate

the,

effects ofroughnesselements of arbitraryshape h,providedN() _{may tendtoinfinity forarbitraryfunctions}

L,

_andthat

a

_maynot

all havethesamevalueas

a.

Thediscussionand resultspresentedinRiahi 11 indicate that the case with

a

>

2a

isexpectedto leadtozerocontribution on variousflowfeaturesand, thus,is irrelevant

for the present problem. However, thecase with

a

< 2a

isexpected tobe relevantfor the non-resonantwavelengthexcitation system which ispresently under investigation bythepresent author, and

theresultswillbe reportexiinthe nearfuture.

An

important demonstration carried outfrom the present investigation isthat theconvectiveflow

canbe admitted, by the boundary imperfections,certainsolutionswhichexhibitkinks in certain vertical

planeswithinthe fluidlayer. Each of theseverticalplanesisparalleltooneactivewave vectorof the boundaryimperfections. Herebyan activewave vector, we meanone whichcoincideswith onewave

vectorof thepreferredflow solution. All suchverticalplanesintersecteach otherato

z-as

Thus along

oz-arAs

therearemultiplekinks inthesolutions. Itispossibleto increasethe complexityin the solutions

FINITEBANDW-IDTH, LONG WAVELENGTH CONVECTION WITH BOUNDARY IMPERFECTIONS verticalplanes,paralleltothe active wavevectors, and theseplanescanintersect eachotheratvertical axes Alongthese vertical axis solutions can exhibit multiple kinks ofvarious degreesS, where S

indicates thenumber ofplanesthat intersect eachotheratone vertical axis

The results presented inthis paper regardingthe preferred solutions, theirstability and the roles

played bytheboundary imperfectionsindicate that theconstants9,,representingthe maximumamplitude of theimperfectioncomponents,controlthestability ofthesolution,that issufficiently high values of

19,

leadtostability Theimperfectionwavevectors

K

control the directions of the wave numbervectors

oftheflow solutions. The spatial forms ofthe amplitudefunctions

L,,

fortheboundary imperfection

leadtounusualbehaviorof thesolutions with kinks. Theimportance of problems ofthetypeconsidered

here,thus, shouldnotbe underestimated Inaddition, to demonstrateexistence andpreferenceofnew andunusual types of solutions, weprovidedawaytocontrolinstabilities and theflowstructureswhich canbe ofsignificanceinflow control applications.

Theproblemstudied heredealswithpoorly conductingboundaries. Thisproblem,as we have shown inthis paper, admitsslowhorizontal variables

:rs

and /s of orders

"/6

duetothefact that for

R

just beyond

Pc,

in the absence of imperfections, three-dimensional solutions in the form of squares are

preferred [2,3] The resulting amplitude system is then a system of non-linear partial differential

equationswhere each equationis secondorderinderivative withrespecttoeither

a:

or/s. Another

equally important problemisoneforthe caseof high conductingboundaries. Thisproblem, asNewell

and Nhitehead [4]demonstrated, hasthe property thatit admits slow horizontal variables

a:

and/s of orders

6

and

6,

respectively, and

:rs

dependenceis moreimportantthan]/s dependence. Thisproperty isdueto thefactthatfor

R

just beyond

Re,

inthe absenceof imperfections, two-dimensionalrollsare

preferred [3,4],whereitisassumedthat :r-axis isalong these rolls Theresulting amplitudesystem will then be a system ofnonlinear partial differential equations where each equation is second order in derivativewith respecttoz, and fourth orderin derivative with respect to/s Althoughtheresults for such a system will bereported elsewhere,it isof interesttonoteherethat such a system can admit non-modal solutions with kinks, different from those discussed in the present paper, and the resulting preferred patternswillbeaffected accordingly.

REFERENCES

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Proc. R. Soc.Lond.A.449(1995),459-478

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RIAHI, N.,

Nonlinear convection in alayerwith ne

,arly

insulatingboundaries,J.

FluidMech. 96(1980),243-256.

[3]

RIAHI, N.,

Nonlinear convection in a porous layer withfinite conducting boundaries, J. Fluid Mech. 129(1983),153-171.

[4] NEWELL,

A.C. and

WHITEHEAD, J.A.,

Finitebandwidth, finiteamplitude convection,J. Fluid Mech.39_(1969),279-303.

[5]

REES,

D.A.S. and

RILEY,

D.S.,Theeffectsofboundary imperfectionsonconvectionin asaturated porous layer: near-resonantwavelengthexcitation,J.FluidMech.199(1989a), 133-153.

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Q.

J.Mech.Appl.Math.43(1990), 189-214.

[8]

JOSEPH,

D.D.,

Stability

of

FluidMotonsIand

II (1976),

Springer.

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R.E. and

PAL, D.,

Thermal convection with spatially periodic boundary conditions resonantwavelengthexcitation,

J.

FludMech.86(1978),433-456.

10]

SPARROW,

E.M.,

GODSTEIN,

R.J.andJONSSON,

V.H.,

Thermal instabilityin a horizontal fluid

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J.

Fluid Mech. 15(1964),

513-528.