exact solutions for thermal problems

7/27/2019 Exact Solutions for Thermal Problems

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Review of Applied Physics (RAP) Volume 1 Issue 1, December 2012 www.seipub.org/rap

1

ExactSolutionsforThermalProblemsBuoyancy,Marangoni,VibrationalandMagneticFieldControlledFlows

Marcello

Lappa*1

Telespazio

ViaGianturco31,80046,Napoli,Italy*[email protected]

Abstract

In general, the thermalconvection (NavierStokes and

energy)equationsarenonlinearpartialdifferentialequations

that inmostcasesrequire theuseofcomplexalgorithms in

combination with opportune discretization techniques for

obtainingreliablenumericalsolutions.Therearesomecases,

however,in

which

such

equations

admit

analytical

solutions.

Suchexact solutionshaveenjoyedawidespreaduse in the

literatureasbasic states fordetermining the linear stability

limits in some idealized situations. This review article

provides a synthetic review of such analytical expressions

for a variety of situations of interest inmaterials science

(especiallycrystalgrowthandrelateddisciplines),including

thermogravitational (buoyancy), thermocapillary

(Marangoni), thermovibrational convection as well as

mixed cases and flow controlled via static and uniform

magneticfields.

Keywords

ThermalConvection;NavierStokesEquations;AnalyticSolutions

Introduction

Finding solutions to the NavierStokes equations is

extremelychallenging.Infact,onlyahandfulofexact

solutions are known. For the cases of interest in the

presentarticle(buoyancy,Marangoni,vibrationaland

magnetic convection), in particular, these exact

solutions exist when the considered system is

infinitelyextendedalongthedirectionoftheimposed

temperature gradient or the orthogonal direction; ingeneral, such solutions are regarded as reasonable

approximations in the steady state of the flow

occurring in thecoreofrealconfigurationswhichare

sufficiently elongated (the core is the region

sufficiently away from the end regions, where the

fluidturnsaround,tobeconsiderednotinfluencedby

such edge effects). Even though limited to cases of

great simpicity, these solutions have proven able to

yielddirectlyorindirectlyinsightsandunderstanding

thatwouldhavebeendifficulttoobtainotherwise.

Hereafter, first theattention is focusedon thecaseof

systems subject to horizontal heating (or to

temperature gradient along the direction in which

theyareassumedtobeinfinitelyextended),thenother

possiblevariantsandconfigurationsareconsideredin

thesectiononinclinedsystemsandsubsequentones.

Analytic Solutions for Thermogravitational

and Marangoni Flows

y

xCold Hot

u

g

d

FIG.1SKETCHOFLAYEROFINFINITEEXTENTSUBJECTTO

HORIZONTALHEATING

In the following three subsections (focusing on the

Hadley flow, Marangoni flow and related hybrid

states) the horizontalboundaries are assumed tobelocated at y=1/2 and 1/2, respectively. The velocity

components alongy and z are zero (v=w=0) and the

componentalongxissolelyafunctionofy,i.e.u=u(y)

(whichmeanstheassumptionof planeparallelflow

is considered).Temperaturedependsonxandy (i.e.

these solutions are essentially twodimensional).

Rayleigh and Marangoni numbers are defined as

Ra=GrPr=gTd4/ and Ma= RePr=Td2/,respectively(whereisarateofuniformtemperatureincreasealongthexaxis,disthedistancebetweenthe

boundaries, is the thermal diffusivity, is thekinematic viscosity, T is the thermal expansioncoefficient,Prtheratiobetweenand,GrandRetheGrashof and Reynolds numbers, respectively).

Referring velocity and temperature to the scales /dandd,respectivelyandscalingalldistancesond,thegoverning equations in nondimensional form

(incompressibleform)read

0 V (1)

g

iTRaVVVp

t

VPrPr 2

(2)

TTVt

T 2

(3)


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www.seipub.org/rap Review of Applied Physics (RAP) Volume 1 Issue 1, December 2012

2

where ig is the unit vector along the direction of

gravity and the Boussinesq approximation hasbeen

used for the buoyancy production term in the

momentumequation.

In the presence of a free liquid/gas interface such

equations must be considered together with thesocalled Marangoni boundary condition, which

neglectingviscousstressinthegas(herethedynamic

viscosityofthegassurroundingthefreeliquidsurface

willbe assumed tobe negligiblewith respect to the

viscosityoftheconsideredliquid)reads

TMan

VS

S

(4)

where n is thedirection locallyperpendicular to the

consideredelementaryportionofthefreeinterface,S

the derivative tangential to the interface andVs the

surfacevelocityvector.

Forthecaseconsideredhere(steadyflowwithu=u(y)

andv=w=0,asshowninFig.1)eqs.(1)(3)reduceto:

2

2

Pry

u

x

p

(5)

RaTy

pPr

(6)

2

2

2

2

y

T

x

T

x

Tu

(7)

whileeq.(4)canberewrittenas:

x

TMa

y

u

(8)

Assumingsolutionsinthegeneralform(withMa=0or

Ra=0 for pure buoyancy or Marangoni flows,

respectively):

0

0

)()( 21 ygMaygRa

w

v

u

V (9)

)()( 21 yfMayfRaxT (10)

1, 2, g1 and g2 can be determined as polynomialexpressions with constant coefficients satisfying

specificequationsobtainedbysubstitutingeqs.(9)and

(10) into eq. (7) and into the equation resulting from

crossdifferentiationofeqs.(5)and(6)(i.e.3

3

d u TRa

dy x

).

Suchsystemofequationsread:

2

1

2

1dy

fd

g (11a)

2

2

2

2dy

fdg (11b)

013

2

3

3

1

3

dy

gdMa

dy

gdRa (12)

tobe

supplemented

with

the

proper

boundary

conditions:

Kinematicconditions:

solid boundary: u=0 g1(y)=g2(y)=0 (13)

free surface: eq. (8) 01 dy

dg and 12 dy

dg (14)

Thermalconditions:

Adiabatic boundary: 0y

T 021 dy

df

dy

df (15)

Conducting boundary: T=x1=2=0 (16)

Inaddition,itmustbetakenintoaccountthatsincethe

considered parallel flow is intended tomodel a slot

withdistantendwalls,continuityrequiresthatthenet

fluxoffluidatanycrosssectionoftheslotbezero),i.e.:

02

1

21

udy 02

1

21

2

21

21

1

dygdyg (17)

Beforegoing furtherwiththedescriptionofsolutions

satisfying the systemofequations (1117), it isworth

noting that some insights into their expected

polynomial order can be given immediately on the

basisofeq. (12).According to suchequation, in fact,

d3g1/dy3=1forMa=0,whiled3g2/dy3=0forRa=0,which

leads to thegeneral conclusion that thepolynominal

expression for uwillbe of the third order for pure

buoyancy flow and of the second order for pure

Marangoni flow,while (on thebasisof eqs. (11)) the

respective temperature profiles are of the fifth and

forthordersiny.

ThermogravitationalConvection:

The

Hadley

flow

In linewith the considerations above, in the case of

liquidconfinedbetween twohorizontal infinitewalls

with perpendicular gravity, eqs. (5)(7) admit as an

exactsolutionthefollowingvelocityprofile:

46

3 yy

Rau (18)

generallyknownastheHadleyflow(Hadley,1735)as

it was originally used as a model of atmospheric

circulation(see,

e.g.,

Lappa,

2012)

between

the

poles

andequator (froma technologicalpointofview, this


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3

solutionisalsostronglyrelevanttothemanufactureof

bulksemiconductorcrystals).

The corresponding temperature profile changes

according to the type of boundary conditions

considered.Foradiabaticwallsitreads:

16

5

6

5

120

24yyy

RaxT (19a)

whereasforconductingboundariesitbecomes:

48

7

6

5

120

24yyy

RaxT (19b)

ThepolynomialexpressionsandgforsuchsolutionsareplottedinFigs.2.

-0.010 -0.005 0.000 0.005 0.010

Velocity

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

y

a)

-0.0008 -0.0004 0.0000 0.0004 0.0008Temperature

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

y

b)

FIGS. 2 EXACT SOLUTION FOR THE CASE OF

THERMOGRAVITATIONALCONVECTION IN INFINITELAYER

WITH TOP AND BOTTOM SOLID WALLS: (a) VELOCITY

PROFILEg(y);(b)TEMPERATUREPROFILE(y)FORADIABATIC(SOLID)ANDCONDUCTING(DASHED)BOUNDARIES.

MarangoniFlow

Intheabsenceofgravityandreplacingtheuppersolid

wall with a liquidgas interface supporting the

development of surfacetensiondriven (Marangoni)

convection,theexactsolutionreads(Birikh,1966):

4

13

4

2yy

Mau (20)

Foradiabatic interfaceand insulatedbottomwall the

temperaturedistributioncanbeexpressedas:

165

23

2323

48234 yyyyMaxT (21a)

thatforconductingboundariesmustbereplacedwith:

16

3

2

1

2

323

48

234yyyy

MaxT (21b)

-0.30 -0.20 -0.10 0.00 0.10Velocity

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

y

a)

-0.01 0.00 0.01 0.02

Temperature

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

y

b)FIGS. 3 EXACT SOLUTION FOR THE CASE OF

THERMOCAPILLARYCONVECTIONININFINITELAYERWITH

BOTTOM SOLID WALL AND UPPER FREE SURFACE: (a)

VELOCITYPROFILEg(y); (b)TEMPERATUREPROFILE(y)FORADIABATIC (SOLID) AND CONDUCTING (DASHED)

BOUNDARIES.

ThepolynomialexpressionsandgforsuchsolutionsareplottedinFigs.3.

HybridBuoyancyMarangoniStates

As shown by eqs. (5)(7), under the considered

conditions(u0,v=w=0),thenonlinearconvectivetermof the momentum equation becomes zero and the


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4

energy equation reduces to u=2T/y2, i.e. thegoverning equations are linear; as a consequence,

morecomplexsolutionscanbebuiltassuperposition

(addition)ofothersimpleexistingsolutions.

Along these lines, in thepresence of vertical gravity

andaliquid

gas

interface

supporting

the

development

ofsurfacetensiondriven (Marangoni)convection, the

velocity profile can be obtained as the sum of two

terms, the first term corresponding to the pure

buoyancydriven flow and the second term

representing the contribution of a pure

thermocapillarydrivenflow,whichprovidesasimple

theoreticalexplanation for thegeneral formgivenby

eqs.(9)and(10).

Theresultingshearflowissetupbyacombinedeffect

of buoyancy and viscous surface stress due to the

temperaturedependence of surface tension.The firstcontributionrelated topurebuoyancyflow,however,

exhibitssomedifferenceswithrespecttoeq.(18)asthe

upper solidwall consideredearliermustbe replaced

with a stressfreeboundary (see Fig. 4); the velocity

profilereads:

4

1338

48

23 yyyRa

u (22)

Hence, the resulting profile formixed gravitational

Marangoniconvectionhastheform:

413

441338

48223 yyMayyyRau

(23)

Theassociatedtemperaturedistributionis:

2

13

16

195

2

51058

2048

1 2345 yyyyyyRa

xT

2

1

16

5

2

3

2

323 234 yyyyyMa (24)

Thecases

of

adiabatic

and

conducting

horizontal

surfacescanbeobtainedbysettinginthisrelation=0and=1,respectively.

Equation(24)canbeextendedtothemoregeneralcase

inwhichthebottomisconductingandaBiotnumber

is introduced todescribeheat transferat the top free

surface by assuming =Bi/(1+Bi). The solutions forsymmetrical(topandbottom)conditionscanagainbe

recoveredaslimitingcaseswhenBi0andBi:WhenBi, in fact,1andeq. (24)reduces to theconductingcase;whenBi0 (0), it reduces to the

insulating boundary conditions for the temperatureprofile.

-0.030 -0.020 -0.010 0.000 0.010 0.020

Velocity

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

y

a)

-0.001 0.000 0.001 0.002 0.003

Temperature

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

y

b)FIGS. 4 EXACT SOLUTION FOR THE CASE OF PURE

THERMOGRAVITATIONALCONVECTIONIN INFINITELAYER

WITH BOTTOM SOLID WALL AND UPPER STRESSFREE

SURFACE: a) VELOCITY PROFILE g(y); b) TEMPERATURE

PROFILE (y) FOR ADIABATIC (SOLID) AND CONDUCTING(DASHED)BOUNDARIES.

GeneralProperties

The present section is devoted to illustrate somefundamentalpropertiesofthesolutions introduced in

thepreceding

sections

with

respect

to

the

well

known

Rayleighs condition, which so much attention has

enjoyed in the literature as a means for gaining

insights intothefluiddynamicinstabilitiesofparallel

flows in the limitasPr0 (Rayleigh,1880;Lin,1944;Rosenbluth and Simon, 1964; Drazin and Howard,

1966).

Letusrecallthatsuchatheoremreads:Inashearflowa

necessary conditionfor instability is that theremust be a

pointofinflectioninthevelocityprofileu=u(y),i.e.apoint

whered2u/dy2=0.

Thesecondderivativeofsolution(18)gives:


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5

yRady

ud

2

2

(25)

that means the profile has an inflection point at

midheight(y=0)andsatisfiestheRayleighsnecessary

condition.

Forpurebuoyancy flowwithupper free surface, i.e.

solution(22),thesecondderivativereads:

648482

2

yRa

dy

ud (26)

that gives the inflection point at a sligtly different

position(y=1/8).

Theinflectionpoint,however,isnolongerpresentfor

thecaseofpureMarangoniflowsince:

02

32

2

Mady

ud (27)

thatmeansMarangoniflowsolutionsofthetypegiven

by eq. (20) do not satisfy the Rayleighs necessary

condition.

Themost interesting case in this regard is, perhaps,

givenby themixed state representedby eq. (23). In

suchacase:

WyRa

dy

ud72648

482

2

(28)

thatmakesthe locationoftheinflectionpointa linear

functionofthenondimensionalparameterW=Ma/Ra:

8

112

Wy (29)

-0.12 -0.08 -0.04 0.00 0.04

Velocity

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

y

W=0.25

W=0.1 W=0

Inflection point

FIG. 5 VELOCITY PROFILE g1(y)+Wg2(y) FOR THE CASE OF

MIXED THERMOGRAVITATIONALTHERMOCAPILLARY

CONVECTION IN INFINITE LAYER WITH BOTTOM SOLID

WALLANDUPPERSTRESSFREESURFACE:THE INFLECTION

POINTDISAPPEARSFORW=0.25.

The inflection point disappears when y>1/2, i.e. forW>1/4. Accordingly, these solutions satisfy the

Rayleighs necessary condition only if 0


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6

6), anddepending on the type of thermal conditions

alongthewalls.

In the following these expressions are given first for

the case of adiabatic walls (eqs. (32)(35)); then the

configuration with conducting boundaries is

considered(eqs.(38)(41)).

Heating from below (0


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7

u

Hot

Coldy

x

g

FIG. 7SKETCHOFALAYEROF INFINITEEXTENT INCLINED

WITH RESPECT TO THE HORIZONTAL DIRECTION WITH

HEATING APPLIED THROUGH THE BOTTOM WALL

(TEMPERATURE GRADIENT IMPOSED ALONG THE y

DIRECTION).

Equations (32)(35) and (38)(41) provide analytical

solutions if the imposedT isparallel to thewallsasshown inFig.6;thermogravitationalconvection inan

inclined layer,however,alsoadmitsexactsolutions if

the imposed T is primarily perpendicular to the

walls i.e. ifsuch temperaturegradientactsacross the

thicknessofthelayer(i.e.layerwithupperandlower

wallskeptatuniformdifferenttemperaturesasshown

inFig.7).Suchasolutionreads:

4

)sin(6

3 yy

Rau (42)

withRa=gTd3/.

Thecorrespondingdistributionoftemperaturealongyis governedby diffusion only and is approximately

uniformthroughouttheplaneofthelayer.

Velocity(u)tendstozero(asexpected)inthe limitas

0(insuchacase,infact,itisawellknownfactthatconvection arises only if a given threshold of the

Rayleighnumberisexceeded).

y

x

Cold Hot

v g

FIG. 8 SKETCH OF A TRANSVERSELY HEATED VERTICAL

CAVITYINTHELIMITASTHEVERTICALSCALEOFMOTION

TENDSTOINFINITY.

Itisalsoworthnoticingthatfor=90onerecoverstheidealizedcaseofa transverselyheatedverticalcavity

(in the limit as the vertical scale ofmotion tends to

infinity,seeFig.8):

46

3 xx

Rav (43)

this profile is generally referred to as conduction

regimesolution(owingtotheassociatedtemperature

profilealongx,thatasmentionedbeforeislinear).

Convection with Vibrations

PureThermovibrational Flow

Letusrecallthatthermovibrationalconvectioncanbe

regarded as a variant of the standard

thermogravitational convection forwhich the steady

Earth gravity acceleration is replaced by an

accelerationoscillatingintimewithagivenfrequency

(Lappa,2004,2010).

Disturbances induced in a fluid by a sinusoidaldisplacement of the related container along a given

direction( n istherelatedunitvector)

nts t)bsin( (44)

wherebistheamplitudeand=2f(fisthefrequency)induceanacceleration:

t)sin(

gtg (45)

where 2 g b n

;whichmeansvibratingasystemwith

frequencyfand

displacement

amplitude

b

corresponds to a sinusoidal gravitymodulationwith

the same frequency and acceleration amplitude b2and vice versa (accordingly, hereafter the terms

gravity modulation , periodic acceleration ,

container vibration and gjitter will be used as

synonyms).Thisalsomeans that2 3

Tb Td

Ra

canbe

regardedasavariantoftheclassicalRayleighnumber

with the steady acceleration being replaced by the

amplitudeoftheconsideredperiodicacceleration.

Ingeneral,

the

treatment

of

the

problem

is

possible

in

terms of three independent nondimensional

parameters only, where the first is the wellknown

Prandtl number (Pr) and the others are the

nondimensional frequency () anddisplacement (),definedas:

2d (46)

d

Tb T

(47)

where,obviously2=PrRa

In the specific case of sufficiently small amplitudes

(1)ofthe


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9/14


9

AsillustratedindetailbyBirikh(1990),thisapproach

leadstothefollowingmathematicalexpressions:

)cos()sin()cosh()sinh(

))cos()sinh()2sin()2cosh(

16)(

3

1

yyRyu


)sin()cosh()2cos()2sinh(

3

yy (62)


)cos()sinh()2cos()2sinh(2)tan(

2

1)(

yyyyf

)cos()sin()cosh()sinh()sin()cosh()2sin()2cosh(

yy (63)

forthecasewithadiabaticwalls

)(sin)(cosh)(cos)(sinh

)cos()sinh()2sin()2cosh(

16)(

22222

1

yyRyu


)sin()cosh()2cos()2sinh(

22222

yy (64)


))cos()sinh()2cos()2sinh(2)tan(

2

1)(

2222

yyyyf


)sin()cosh()2sin()2cosh(2222

yy (65)

forthecasewithconductingwalls

where

4/12 )(cos22

1 vRa (66)

ModulatedBuoyant

Flows

There have alsobeen studies on the effect of time

modulated gravity (or vibrations) on systems with

basic buoyant convection induced by vertical static

(steady and uniform) gravity and horizontal

temperaturegradients

The simplest model for this kind of flows is the

canonical layer of infinite horizontal extent already

examinedbeforeforpurebuoyancy(theHadleyflow)

andvibrationalflow(purethermovibrationalflow).

Byanalogywiththeanalogousproblemtreatedforthe

pure thermovibrational case, let us start the related

discussionbyobservingthatundertheeffectofhigh

frequency vibrations (in the framework of the

averaged formulation discussed before) also this

system admits solutions corresponding to quasi

equilibrium, i.e. states inwhich themean velocity is

zero. These states are made possible by a perfect

balanceofthestaticcomponentofthebodyforceand

pressuregradientestablishedintheliquid.Therelated

mathematical (necessary) conditions of existence for

the specific case considered here, i.e. vertical steady

gravity andhorizontal temperature gradients o xT i ,

reduceto:

0)sin()cos( RaRav (67)

Whenrequisites formechanicalquasiequilibriumare

not satisfied timeaverage fluid motion arises. Such

convective states are twodimensional and admitanalytic solution in the form of planeparallel flow.

Following thegeneral concepts introducedbefore for

pure thermovibrational flows, these solutions canbe

representedas


)cos()sinh()2sin()2cosh(

16)(

3

1

yyRyu

)cos()sin()cosh()sinh()sin()cosh()2cos()2sinh(

3

yy (68)


)cos()sinh()2cos()2sinh(2

2

1)(

2

1

yyy

R

Ryf

)cos()sin()cosh()sinh()sin()cosh()2sin()2cosh(

yy (69)

forthecasewithadiabaticwalls


))cos()sinh()2sin()2cosh(

16)(

22222

1

yyRyu


)sin()cosh()2cos()2sinh(22222

yy (70)


)cos()sinh()2cos()2sinh(2

2

1)(

22222

1

yyy

R

Ryf


)sin()cosh()2sin()2cosh(2222

yy (71)

forthecasewithconductingwalls(seeFig.10)

where

4/1222

1R (72)

RaRaR v )sin()cos(1 (73)

)(cos

2

2 vRaR (74)

-0.010 -0.005 0.000 0.005 0.010

Temperature

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

y

Ra =1000v

Ra =0v

a)


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10

-0.0002 0.0000 0.0002

Temperature

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

y

Ra =1000v

Ra =0v

b)FIG. 10 EXACT SOLUTION FOR THE CASE OF MIXED

THERMOGRAVITATIONALTHERMOVIBRATIONAL

(AVERAGE) CONVECTION IN INFINITE LAYER WITH TOP

AND BOTTOM SOLID CONDUCTINGWALLS (=0, Ra=1): (a)VELOCITYPROFILEu(y);(b)TEMPERATUREPROFILE(y).

MixedMarangoni/ThermovibrationalConvection

Likebuoyancyflow,alsothethermalMarangoniflow

canbestronglyaffectedbyimposedvibrations.Sucha

case is considered in this section (surfacetension

inducedflowinteractingwithvibrationaleffectsinthe

absenceofastaticaccelerationcomponent, i.e.Ma0,RaorRav0andRa=0).

Interesting results along these lines deserving

attention are due to Suresh andHomsy (2001),who

considered the infinite parallel Marangoni flow

subjected to gravitational modulation at low

frequencies (where the averaged model is not

applicable) for the case in which finitefrequency

vibrationsareperpendiculartothe layer(=90),(Fig.11; they assumed as base unmodulated Marangoni

flow thepopularreturn flow solutiongivenbyeq.

(20) and employed a quasisteady approach, in the

limitofverylowforcingfrequency).

FIG.11SKETCHOFFLUIDLAYEROFINFINITEEXTENTWITH

UPPER FREE SURFACE SUBJECTED TO VIBRATIONS s t bsin( t)n

WITHARBITRARYDIRECTION.

Following the general concepts introduced earlier,

such solution (see Figs. 12) canbe expressed as the

superposition of two components, i.e. the steady

Marangoni componentproportional toMa (Td2/)andaperiodicvibrationalcomponentproportional to

Ra(b

2

Td4

/=2

/Pr,where

is

the

rate

of

uniformtemperatureincreasealongthexaxis):

)exp()()( tiygRayMagu BM (75)

)exp()()( tiyfRayMafxT BM (76)

where, as already reported in the case of pure

Marangoniflow,foradiabatichorizontalwalls:

4

13

4

1)( 2 yyygM (77)

16

5

2

3

2

323

48

1)( 234 yyyyyfM (78)

and,as

analytically

determined

by

Suresh

and

Homsy

(2001):

2321)exp()exp()(

mcmmcmmcygB

(79)

and: )exp()exp()exp()( 321 mdndndyfB

22

54 /)exp( nmdmd forPr1 (80a)

)exp()()( 31 mddyfB 4

542 /)exp()( mdmdd forPr=1 (80b)

with

2

1y (81a)

2/1

Pr

i

m (81b)

2/1 in (81c)

))sinh()cosh((2

1)12)(exp(4

2

1mmmm

mmmc

(82a)

))sinh()cosh((2

1)12)(exp(4

2

2mmmm

mmmc

(82b)

)( 123 ccmc (82c)

)( 214 ccc (82d)

ifPr1

)exp()exp()(

)sinh(2

112122

2

1 mcnccmn

m

nnd

222

)exp(1)exp(

nm

nmc (83a)

)exp()exp()(

)sinh(2

112122

2

2 mcnccmn

m

nnd

222

)exp(1)exp(nm

nmc (83b)


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11

22

13

mn

mcd

(83c)

22

24

mn

mcd

(83d)

2

35

n

cd (83e)

ifPr=1

)exp(()sinh(2)sinh(4

1111 mcmcm

mmd

42

))exp(1(2))exp(

m

mmc (84a)

)exp(()sinh(2)sinh(4

1122 mcmcm

mmd

42

))exp(1(2))exp(

m

mmc (84b)

21

3

c

d (84c)

22

4

cd (84d)

2

35

m

cd (84e)

withthehorizontalboundarieslocatedaty=1/2.

a)

b)FIGS. 12 EXACT SOLUTION FOR THE CASE OF MIXED

MARANGONITHERMOVIBRATIONAL

CONVECTION

IN

INFINITELAYERWITHADIABATICBOUNDARIES(Pr=1,Ma=1,

=90,Ra=5,=1).

Solutions in the Presence of Static and

Uniform Magnetic FieldsInmanymaterial processing techniques, amagnetic

fieldiscommonlyusedtocontroltheliquidflows.Its

action can lead to the braking of the flow (i.e. the

reductionoftherateofconvectivetransport)ortothe

dampingofpossibleoscillatoryconvectiveinstabilities.

Forthesereasons,anumberoftheoreticalstudieshave

appearedduringrecentyearsconcerning theeffectof

magnetic fields on thebasic flowmotion in several

geometrical models of semiconductor growth

techniques.

Thepresent section illustrates theeffectofaconstant

magnetic field (static and uniform) on the

fundamental solutions for gravitational and

Marangoniconvection

introduced

in

earlier

sections.

PhysicalPrinciplesandGoverningEquations

Themotionofanelectricallyconductingmeltundera

magneticfieldinduceselectriccurrents.Lorentzforces,

resulting from the interaction between the electric

currentsandthemagneticfield,affecttheflow.

In the following, as stated in the title of the present

subsection, the physics of the problem and the

introductionofthecorrespondingmodelequationsare

considered for the simple and representative case of

constantmagneticfields.

Auniformmagnetic fieldwithmagnetic fluxdensity

(alsoreferredtoasmagneticinduction)Bogeneratesa

damping Lorentz force through the aforementioned

electric currents induced by the motion across the

magnetic field that in dimensional form can be

expressed asL fF J B where fJ is the electrical

current density. Like other body forces (e.g., the

buoyancy forces), an account for this force can be

yielded simply adding a relevant term to the

momentumequation.

Following this approach, scaling the magnetic flux

density with Bo and the electric current densityfJ

with eVrefBo, themomentum equation including theLorentzforce,canbewritten innondimensional form

(assuming as usualVref=/d) and in the absence ofphasetransitionsas:

Bofg

iJHaiRaTVVVpt

V 22 PrPrPr

(85)

whereHaisthesocalledHartmannnumber


12/14


12

2/1

eo dBHa

(86)

eistheelectricalconductivityandiBotheunitvectorinthedirectionofBo.

Thenondimensional

electric

current

density

fJcan

be

expressedviatheOhmslawforamovingfluidas:

Bof iVEJ (87)

whereEistheelectricfieldnormalizedbyVrefBo.Since

(see Lappa, 2010), in many cases of technological

interest) the unsteady induced field is negligible, in

particular, the electric field can be written as the

gradientofanelectricpotential:

eE (88)

Theconservationoftheelectriccurrentdensitygives:

0f

J (89)

which combined with eq. (88) leads to a Poisson

equationfortheelectricpotential:

ViiV BoBoe 2 (90)

Finally,themomentumequationcanberewrittenas:

BoBoe

g

iiVHa

iRaTVVVpt

V

2

2

Pr

PrPr

(91)

BuoyantConvection

with

aMagnetic

Field

Thesimplestmodelinsuchacontextistheflowinan

infinite planar liquid metal layer confined between

two horizontal solid walls driven by a horizontal

temperature gradient (the canonical infinite Hadley

flowwhosegeneralpropertieshavebeen treated ina

previoussectionintheabsenceofmagneticfields).

Notably, for electrically insulating horizontal walls

such a flow admits analytical expression even in the

cases inwhich it issubjected toamagneticfieldwith

directionvertical

or

parallel

to

the

applied

temperaturegradient,orevendirectedspanwisetothe

basicflow(seeFig.13).

FIG.13SKETCHOFLIQUIDLAYERUNDERTHEEFFECTOFA

HORIZONTALTEMPERATUREGRADIENTANDAMAGNETIC

FIELD WITH VERTICAL, STREAMWISE OR SPANWISE

DIRECTION.

In thecaseofvertical field, theanalyticsolution (see,

e.g.,Kaddecheetal.,2003)reads:

yHaHay

Ha

Ra

u )2/sinh(2

)sinh(2 (92)

y

y

Ha

Hay

HaHa

RaxT

6)2/sinh(

)sinh(

2

1 3

22 (93)

with thehorizontalboundaries locatedaty=1/2andtheparametergivenby:

)2/sinh(2

)2/cosh(

8

1

HaHa

Ha (94a)

foradiabaticboundariesand

2

1

24

1

Ha (94b)

forconductingboundaries.

If a horizontalmagnetic field parallel to the applied

temperature gradient (magnetic field applied along

thexdirection) isconsidered, there isnodirecteffect

of the field on the parallel flow in the layer as the

velocity is parallel to the field direction (BoV i 0 ),

which (taking intoaccounteqs. (87)(90)) leads to the

vanishingoftheproductiontermineq.(91).Thebasic

flow is,hence, the flowwithoutmagnetic fieldgivenbyeq.(18).

For a magnetic field still horizontal but directed

spanwise to the basic flow (i.e. a magnetic field

appliedalong thezdirection), thebasicparallel flow

inthelayerisstilltheflowwithoutmagneticfield.

A justification for such a behavior is not

straightforward as onewould assume, and deserves

somespecificadditionalinsights.

First of all, it has to be noted that as long as we

consider the basic flow as being horizontally

homogeneous, theelectric field inducedby the liquid

motion isuniform anddirectedperpendicular to the

planeofthelayer.Zerocirculationofthisfield(related

tothefundamentalirrotationalnatureofelectricfields

when steadymagnetic fields are considered) implies

thatnoelectriccurrentsclosingwithintheliquidlayer

canbe induced,buttheelectric impermeabilityofthe

horizontalboundariesalsoprecludesthepossibilityof

any electric current passing normally through the

layer; thereby, insulating horizontal walls lead to a

separation of electric charges over the depth of the

layer, so giving rise to an electrostatic field which


13/14


13

cancelsthatinducedbytheliquidmotion.Asaresult,

there isnoelectric current inducedbyahorizontally

uniformbasic flow ina coplanarmagnetic fieldand,

therefore, there isnodirect influenceof themagnetic

fieldonthebasicflow.

MarangoniFlowwithaMagneticField

Notably, like theHadley flow discussedbefore, also

the infinite surfacetensiondriven flow admits

analytical expression under the effect of amagnetic

field.

Inparticular,ifthemagneticfieldisparalleltothefree

surface, i.e. satisfies Biy=0 (hereafter referred to ascoplanar field), then, following the same arguments

alreadyprovidedfortheinfiniteHadleyflow,itcanbe

concluded that thefieldhasnoinfluenceonthebasic

flow,and consequently theparallel flow remains the

sameaswithoutit(thisisexactlythereturnflowgiven

byeq.(20)).

By contrast, if themagnetic field isperpendicular to

the free surface the analytic solution (neglecting

exponentially small terms for strong magnetic field

Ha>>1)becomes(Priedeetal.,1995):

HayHa

Mau2

1exp

1

22 12

1exp1Ha

HayHa

(95)

HayHa

y

Ha

yMaxT

2

1exp

1

2

132

HayHaHa

y

2

1exp

143

(96)

where, as usual, the horizontal boundaries are

assumed tobe locatedaty=1/2andMa=Td2/ (being the rateofuniform temperature increasealong

the x axis and d the depth of the layer), Ha=Bod

(e/)1/2.

REFERENCES

Birikh R.V., (1966), Thermocapillary convection in a

horizontal fluid layer ,J.Appl.Mech.Tech.Phys.7:43

49.

BirikhR.V., (1990),Vibrationalconvectioninaplanelayer

with a longitudinal temperature gradient, Fluid

Dynamics, 25(4): 500503. Translated from Izvestiya

AkademiiNaukSSSR,MekhanikaZhidkosti iGaza,No.

4,pp.1215,JulyAugust,1990.

DelgadoBuscalioniR.andCrespodelArcoE.,(2001),Flow

and heat transfer regimes in inclined differentially

heatedcavities,

Int.

J.

Heat

Mass

Transfer,

44:

1947

1962.

Drazin P. and Howard L.N., (1966), Hydrodynamic

stability of parallel flow of inviscid fluid,Adv.Appl.

Mech.,9:189.

GershuniG.Z.andZhukhovitskiiE.M.,(1979),Freethermal

convection in a vibrational field under conditions of

weightlessness,Sov.Phys.Dokl.,24(11):894896.

Hadley G., (1735), Concerning the cause of the general

tradewinds,Phil.Trans.Roy.Soc.Lond.,29:5862.

KaddecheS.,HenryD.andBenHadidH.,(2003), Magnetic

stabilizationof thebuoyant convectionbetween infinite

horizontalwallswithahorizontaltemperaturegradient ,

J.FluidMech.,480:185216.

LappaM., (2010),ThermalConvection:Patterns,Evolution

andStability,JohnWiley&Sons,Ltd,700pp. ISBN13:

9780470699942, ISBN10: 0470699949, John Wiley &

Sons,Ltd(2010,Chichester,England).

Lappa M., (2004), Fluids, Materials and Microgravity:

Numerical Techniques and Insights into the Physics,

538 pp, ISBN13: 9780080445083, ISBN10: 00804

4508X,ElsevierScience(2004,Oxford,England).

LappaM., (2012),RotatingThermalFlows inNaturaland

IndustrialProcesses,540pp. ISBN13:978111996079

9, ISBN10: 1119960797,JohnWiley& Sons, Ltd (2012,

Chichester,England).

Lin C.C., (1944), On the stability of twodimensional

parallelflows,Proc.NAS,30(10):316324.

Priede J., Thess A., Gerbeth G., (1995), Thermocapillary

instabilitiesin

liquid

metal:

Hartmann

number

versus

Prandtlnumber ,Magnetohydrodynamics,31(4):420428.

Rosenbluth M.N. and Simon A., (1964), Necessary and

sufficient conditions for the stability of plane parallel

inviscidflow,Phys.Fluids,7(4):557558.

Suresh V. and Homsy G.M., (2001), Stability of return

thermocapillary flowsundergravitymodulation ,Phys.

Fluids,13:31553167.

Xu J.J. and Davis S.H., (1983), Liquid Bridges with

thermocapillarity ,Phys.Fluids,26(10):28802886.


14/14


14

Marcello Lappa is Senior Researcher and ActivityCoordinator at Telespazio (formerly, Microgravity

AdvancedResearchandSupportCenter).Hehasabout100

publications(mostofwhichassingleauthor)inthefieldsof

fluidmotion and stabilitybehavior, organic and inorganic

materials sciences and crystal growth, multiphase flows,

solidification,biotechnology andbiomechanics,methodsofnumerical analysis in computational fluid dynamics and

heat/mass transfer, high performance computing (parallel

machines). In2004heauthored thebookFluids,Materials

andMicrogravity:NumericalTechniques and Insights into

thePhysics(2004),ElsevierScience,thatowingtothetopics

treated iscurrentlyusedasa referencebook in the fieldof

microgravity disciplines and related applications. He is

EditorinChief (2005present) of the international scientific

Journal FluidDynamics andMaterials Processing (ISSN

1555256X). In 2010 he authored the book Thermal

Convection: Patterns Evolution and Stability (2010),John

Wiley&

Sons,

in

which

acritical,

focused

and

comparative

study of different types of thermal convection typically

encountered in natural or technological contexts

(thermogravitational, thermocapillary and

thermovibrational)was elaborated (including the effect of

magnetic fields and other means of flow control). He

extended thetreatmentof thermalconvection to thecaseof

rotating fluids in a subsequent book (Rotating Thermal

Flows in Natural and Industrial Processes,JohnWiley &

Sons, 2012), expressly conceived for abroader audience ofstudents (treating not only fluid mechanics, thermal,

mechanicalandmaterialsengineering,butalsometeorology,

oceanography, geophysicsandatmosphericphenomena in

othersolarsystembodies).Inrecentyears,hehastakencare

oftheactivities(atTelespazio)forpreparationandexecution

of space experiments in the field of fluids onboard the

InternationalSpaceStation (ISS) indirectcooperationwith

NASA on behalf of the European Space Agency (ESA).

Related activities include: design of the mission scenario

(MissionOperation ImplementationConcept);development

of all the products required to manage the payload

(procedures,payload

regulations,

joint

operation

interfaces,

payload technical support concept, certification of flight

readiness);coordinationofexperimentscienceteams.

exact solutions for thermal problems

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