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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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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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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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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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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), 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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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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AsillustratedindetailbyBirikh(1990),thisapproach
leadstothefollowingmathematicalexpressions:
)cos()sin()cosh()sinh(
))cos()sinh()2sin()2cosh(
16)(
3
1
yyRyu
)cos()sin()cosh()sinh(
)sin()cosh()2cos()2sinh(
3
yy (62)
)cos()sin()cosh()sinh(
)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)(cos)(sinh
)sin()cosh()2cos()2sinh(
22222
yy (64)
)(sin)(cosh)(cos)(sinh
))cos()sinh()2cos()2sinh(2)tan(
2
1)(
2222
yyyyf
)(sin)(cosh)(cos)(sinh
)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()sin()cosh()sinh(
)cos()sinh()2sin()2cosh(
16)(
3
1
yyRyu
)cos()sin()cosh()sinh()sin()cosh()2cos()2sinh(
3
yy (68)
)cos()sin()cosh()sinh(
)cos()sinh()2cos()2sinh(2
2
1)(
2
1
yyy
R
Ryf
)cos()sin()cosh()sinh()sin()cosh()2sin()2cosh(
yy (69)
forthecasewithadiabaticwalls
)(sin)(cosh)(cos)(sinh
))cos()sinh()2sin()2cosh(
16)(
22222
1
yyRyu
)(sin)(cosh)(cos)(sinh
)sin()cosh()2cos()2sinh(22222
yy (70)
)(sin)(cosh)(cos)(sinh
)cos()sinh()2cos()2sinh(2
2
1)(
22222
1
yyy
R
Ryf
)(sin)(cosh)(cos)(sinh
)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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-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
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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
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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.
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www.seipub.org/rap Review of Applied Physics (RAP) Volume 1 Issue 1, December 2012
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.