Role of uniform horizontal magnetic field on convective flow

0.58 MB·English

by  Pinaki Pal

Role of uniform horizontal magnetic field on convective flow
Pinaki Pal1 and Krishna Kumar2
1Department of Mathematics, National Institute of Technology, Durgapur-713 209, India
2Department of Physics and Meteorology, Indian Institute of Technology, Kharagpur-721 302, India
TheeffectofuniformmagneticfieldappliedalongafixedhorizontaldirectioninRayleigh-B´enard
convection in low-Prandtl-number fluids has been studied using a low dimensional model. The
modelshowstheonsetofconvection(primaryinstability)intheformoftwodimensionalstationary
rolls in the absence of magnetic field, when the Rayleigh number R is raised above a critical value
Rc. The flow becomes three dimensional at slightly higher values of Rayleigh number via wavy
instability. These wavy rolls become chaotic for slightly higher values of R in low-Prandtl-number
(Pr) fluids. A uniform magnetic field along horizontal plane strongly affects all kindsof convective
flows observed at higher values of R in its absence. As the magnetic field is raised above certain
value, it orients the convective rolls in its own direction. Although the horizontal magnetic field
does not change the threshold for the primary instability, it affects the threshold for secondary
(wavy)instability. Itinhibitstheonsetofwavyinstability. ThecriticalRayleighnumberRo(Q,Pr)
at the onset of wavy instability, which depends on Chandrasekhar’s number Q and Pr, increases
3 monotonically withQforafixedvalueofPr. ThedimensionlessnumberRo(Q,Pr)/(RcQPr)scales
with Q as Q−1. A stronger magnetic field suppresses chaos and makes the flow two dimensional
1
with roll pattern aligned along its direction.
0
2
PACSnumbers: 47.20.Bp,47.20.-k,47.52.+j,47.35.Tv,47.65.-d
n
a
J
I. INTRODUCTION of modes, often makes the understanding of the basic

6 physicsofconvectiveflowdifficult. Here,low-dimensional
2
models play a very important roll. Low-dimensional
Convective flows in low-Prandtl-number fluids in the
models are useful for modeling largescale flows. It takes
] presence of magnetic field have been studied for many
S much less computer time and yet provide valuable infor-
yearsduetoitsimportanceingeophysicalandastrophys-
P mation of the dynamics of the fluid flow.
icalproblems[1,2]. Generallythisflowiscalledmagneto
.
n convection. There are also industrial applications of this
Inthis paper we study the effectof uniformhorizontal
i kind of flow in crystal growth [3] and in fusion reactor
l magnetic field applied along a fixed horizontal direction
n asheatexchanger[4]. Therehasbeenextensivetheoreti-
(y-axis)usingalowdimensionalmodel. Intheabsenceof
[
cal and numerical studies of thermal convectionin fluids
magnetic field the model shows the onset of convection
1 in presence of external magnetic field [1, 5–11]. These in the form of steady two dimensional rolls (2-D). The
v studies reveal the stabilizing effect of the magnetic field
2-D rolls become three dimensional (3-D) via wavy roll
0 onconvectiveflow. Inseveralexperiments[12–14]ithave
instability [20], when the Rayleigh number R is raised
2 beenshownthatmagneticfieldaffectstheconvectiveflow
slightly above critical value R . The wavy rolls become
2 c
strongly.
6 chaotic, if R is raised about 2% above Rc for low val-
1. Fauve et. al. [15, 16] studied the effect of horizontal ues of Prandtl number Pr. These rolls may be aligned
magnetic field, both in the longitudinal and the trans- either along x-axis or along y-axis. Two sets of rolls
0
verse directions, on wavy roll instability in a Rayleigh- competewitheachotherwithfurtherincreaseinR. The
3
1 B`enard experiment with mercury. They found that the horizontal magnetic field does not shift the threshold of
: horizontal magnetic field inhibited the oscillatory insta- the primary instability (stationary convection). How-
v
bility and made the convection two dimensional. Libch- ever, it delays the onset of the secondary instability in
i
X aber et. al. [17] and Hof et. al. [18] considered the ef- the form of wavy rolls. If Chandrasekhar’s number Q,
r fectofexternalhorizontalmagneticfieldonlow-Prandtl- which is proportional to the square of the applied hor-
a numberthermalconvectionandfoundthatthe magnetic izontal magnetic field B , is raised above certain value,
0
field delayed the onset of wavy instability. So magnetic the rolls are orientedalongthe direction ofthe magnetic
fieldtypicallyinhibitstheoscillatoryinstabilitybutthere field in the horizontal plane. The dimensionless num-
aresituationswhenmagneticfieldcanstimulateinstabil- ber R (Q,P )/(R QP ) scales with Q as Q−1 at the on-
o r c r
ities [19]. Therefore, a better understanding is necessary set of wavy instability. If the intensity of the magnetic
for this kind of flow. field is sufficient, it suppresses chaotic flow and makes
Numerical simulations complement experiments be- the convection periodic for relatively higher values of R.
causeverylow-Prandtl-numberfluidscannotbeachieved A stronger magnetic field suppresses the oscillatory con-
in the laboratory. Even the fluids like mercury (P = vection even at higher values of R. The flow becomes
r
0.025) do not allow very good visualization in experi- steady and two dimensional in the presence of stronger
ments. However, the simulations are costly in terms of horizontal magnetic field. The results of the model are
computer time. The simulations, due to large number also compared with those observed in experiments [16].
2
In sectionII, we describe the physicalsystemtogether III. LOW DIMENSIONAL MODEL
with boundary conditions. Section III deals with the
derivation of the low-dimensional model. The results of WeapplystandardGalerkintechniquetoderiveasim-
the model and their comparison with experimental and ple model to describe the convection in presence of hor-
numerical simulations are described in section IV. Con- izontal magnetic field. The spatial dependence of the
clusions are given in section V. independent fields are expanded in Fourier series com-
patible with the boundary conditions. The expansions
are truncated to describe the superposition of two mu-
II. HYDROMAGNETIC SYSTEM
tually perpendicular sets of wavy rolls [21] for zero (P )
r
convection[22] and extend the model to consider low P
r
Asaphysicalsystem,athinhorizontallayerofelectri- convection. The expansions for vertical velocity, vertical
cally conducting fluid of thickness d, uniform kinematic vorticity and temperature field are as follows:
viscosity ν, thermal diffusivity κ, magnetic diffusivity λ
andcoefficientofvolumeexpansionαiskeptbetweentwo v3 = [W101(t)cos(kx)+W011(t)cos(ky)]sin(πz)
horizontalplates,andisheateduniformlyfrombelow. A +W (t)cos(kx)cos(ky)sin(2πz)
112
uniform horizontal magnetic field B0 = (0,B0,0) is ap- +W (t)sin(kx)sin(ky)sin(πz) (6)
111
pliedalongafixeddirection(sayalongy-axis). Notethat
ω = +Z (t)cos(kx)+Z cos(ky)
we have taken x and y-axes along horizontal directions 3 100 010
and z-axis along vertical upward direction. We consider +Z111(t)cos(kx)cos(ky)cos(πz) (7)
the flow of liquid metals which have magnetic Prandtl θ = [T (t)cos(kx)+T (t)cos(ky)]sin(πz)
101 011
number, P (= ν) of the order of 10−5 [4]. Therefore
m λ +[T112(t)cos(kx)cos(ky)+T002(t)]sin(2πz)
we set P = 0 for our study. Now we choose units d
m
for length, d2/ν for time, B0ν for the induced magnetic +T111(t)sin(kx)sin(ky)sin(πz) (8)
λ
field and νβd/κ for temperature, where β is the uniform
Thehorizontalcomponentsofvelocityareobtainedusing
temperature gradientbetweenthe plates andgetthe fol-
the solenoidal property of the velocity field. In P 0
m
lowing set of governing dimensionless equations under →
limit, the magnetic fieldfluctuationb is slavedto v, and
Boussinesq approximation:
the components of b are determined using Eq. 4. We
∂ ( 2v ) = 4v +R 2 θ Qeˆ [ (∂ b)] then derive the model by projecting the hydrodynami-
t ∇ 3 ∇eˆ 3 [(∇ωH )−v (3v·∇)×ω]∇,× y (1) cal equations (1-5) on the above mentioned modes. The
3
− ·∇× ·∇ − ·∇ model, given in the Appendix, consists of twelve dimen-
∂ ω = 2ω +[(ω )v (v )ω ]
t 3 3 3 3 sional coupled first-order nonlinear ordinary differential
∇ ·∇ − ·∇
+Qeˆ3 [ (∂yb)], (2) equations for the above Fourier amplitudes.
·∇×
P [∂ θ + (v )θ]=v + 2θ, (3)
r t 3
·∇ ∇
2b = ∂ v, (4)
y IV. RESULTS AND DISCUSSIONS
∇ −
v = b=0, (5)
∇· ∇·
where v (x,y,z,t) (v ,v ,v ) is the velocity field, b Weintegratethemodelusingtheode45solverofMAT-
1 2 3
(x,y,z,t) (b ,b ,b≡)the induced magneticfield due to LAB to investigate the effect of magnetic field. We set
1 2 3
convection≡,θ(x,y,z,t)thedeviationintemperaturefield k = kc = π/√2. The model is integrated in the absence
from the steady conduction profile, and ω (ω ,ω ,ω ) of magnetic field using random initial conditions for dif-
1 2 3
visthevorticityfieldinthefluid. Th≡edimension- ferent values of the reduced Rayleighnumber r=R/Rc.
l≡es∇s p×arameters are: Rayleigh number R = αβgd4/νκ, Once the system attains steady state, magnetic field is
thermal Prandtl number P =ν/κ and Chandrasekhar’s switched on by setting Q nonzero. We start our discus-
r
number Q = B02d2, where g is the acceleration due to sionbypresentingtheresultsofthemodelintheabsence
ρ0λν of magnetic field in the next subsection.
gravity and ρ is the reference density of the fluid. eˆ
0 3
is a unit vector directed vertically upward and 2 =
∇H
∂ +∂ is the horizontal Laplacian. We assume the
xx yy A. Convection in the absence of magnetic field
horizontal boundaries to be stress-free, thermally and
electrically good conductors. The boundary conditions
In the absence of magnetic field, the model shows the
are then given by
onsetofconvectionintheformofsteadytwodimensional
rolls as the value of r becomes slightly above unity. As
∂v1 = ∂v2 =v =θ =0,∂b1 = ∂b2 =b =0 r is increased above ro(Q,Pr) = R(o)(Q,Pr)/Rc, where
3 3
∂z ∂z ∂z ∂z R(o)(Q,P ) is the threshold for oscillatory (secondary)
r
at z = 0,1. In what follows we also use one parameter instability in the presence of horizontal magnetic field,
r called reduced Rayleigh number, being the ratio of R the convective flow becomes three dimensional via oscil-
andcriticalRayleighnumberR =27π4/4,thethreshold latoryinstability[20]. Inthemodel,thesecondaryinsta-
c
Rayleigh number for onset of stationary convection. bility appears as wavy rolls [23] as r is raised above r .
o
3
With little further increase in r, we find chaotically os- displayedthe behaviour ofthe relativedistance fromthe
cillating wavy rolls either oriented along y-axis or along thresholdofoscillatoryinstability(R(o) R(o))/P infig-
x-axis depending upon the choice of initial conditions. Q − 0 r
ure 2(a) and the angular frequency ω at the onset of
o
These wavy rolls are observed for 1.0011 < r < 1.19 for
oscillatory instability in figure 2(b) as a function of Q.
Pr = .025. For r ≥ 1.19, the model shows chaotic com- HereR(o) andR(o) arethethresholdvaluesofoscillatory
petition of rolls and squares [21–23]. In the next subsec- Q 0
(secondary) convection in the presence of the external
tionswepresenttheresultsofthemodelontheeffectsof
magnetic field (Q = 0) and in the absence of external
magnetic field on the onset of wavy roll instability and 6
magnetic field(Q = 0) respectively. The results of our
chaotic flows.
modelisingoodagreementwiththenumerical[5]andex-
perimental[16]resultsshowingthepowerlaw. Thequan-
tity (R(o) R(o))/P scales with Q as Qα with α= 1.1,
B. Inhibition of oscillatory instability while tQhe −expe0rimenrts and direct numerical simulations
gave α = 1.2. The frequency of oscillation at the onset
The convection generated magnetic field adds a term of oscillatory instability is linearly proportional to the
Q ∆−1 ∂2v only in the dimensionless Navier-Stokes Chandrasekhar’s number Q.
e−quation,(cid:16)w∂hye2r(cid:17)e ∆−1 denotes the inverse of Laplacian.
This term represents some kind of anisotropic viscos-
103 P = 0.025 100
imtyag[1n5e]t.icTfiheelrdefdorireeicttiinonh.ibAitssvaelroecsiutylt,vathrieatmioangsnaeltoincgfitehlde Pr102 Prr = 0.1 (a) ωy ()080 (b)
iesbnenohrhtvihabendiotcsnetshtPahtrhteeatonhsndtesaeQtotino.soneFfatirogyos.fc21i-lo(Dlasac)tciolcolrlanyetvaoeirrnclyysttioasinhnbsoitrlwiaetbgsyiihmliaotenyw.ddIttgehpreieseasnottdbalys-- (o)(o)(R − R )/Q0111000−011 nset frequenc 246000
O
bility boundary of the 2-D convection regime pushed to-
10−2 0
wards higher values of r due to the application of uni- 10−2 10−1 100 101 0 25 50 75 100
Q Q
form magnetic field in horizontal plane. This result
corroborates the experimental [15, 16] as well as nu- FIG. 2: (a)Scaling of (RQ(o) −R0(o))/Pr with Q for different
merical [9, 10] results. The threshold for wavy insta- values of Pr. The solid line is parallel to the best fit, and its
bility depends on Prandtl number Pr as well as Chan- slopeis1.1±0.01. (b)Dimensionlessfrequencyωoattheonset
drasekhar’s number Q. The combination of dimension- ofoscillatoryinstabilityasafunctionofQforPr =0.025. The
lessnumbersro(Q,Pr)/(QPr)showspowerlawbehaviour best fit (solid line) shows thelinear increase of ωo with Q.
(see Fig.1(b)), and scales with Q as Q−1. We have also
4
(a) Q=5
P)r3 Q=2
Q, Q=0 C. Orientation of convective rolls
r(o2
1 Figure 3 shows the effect of horizontal magnetic field
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
P on the orientation of the convective rolls. Fourier modes
r
104 (b) P = 0.1 W101(bluecurve)andW011(greencurve)oscillatechaot-
QP)r Prr = 0.025 iucraell3y(ain)sthhoewasbcsheanocteicowf tahveyeroxltlesranlaolnmgxag-anxeitsicfofirerld=. 1F.i0g5-
P)/(r102 intheabsenceofexternalmagneticfield(Q=0). Theset
Q, ofwavyrollsinitiallyalongx-axisorientthemselvesalong
(o
r the direction of the magnetic field (see, figure 3(b)) for
100 Q=5. ForslightlyhighervaluesofthereducedRayleigh
10−2 10−1 100 101
Q number(r =1.2),themodelshowscompetitionbetween
mutually perpendicular sets of wavy rolls (Fig. 3(c)) in
FIG.1: (a)Variationofthethresholdro(Q,Pr)forwavyinsta- the absence of external magnetic field (Q = 0) in stead
bility as a function of Prandtl numberPr for different values of one set of wavy rolls (Fig. 3(a)). However, the set
ofChandrasekhar’snumberQ. (b)Scalingofthecombination of wavy rolls along x-axis disappears and only the rolls
ro(Q,Pr)/(QPr) with Q for Pr =0.025 (stars) and Pr =0.1 along y-axis survive (see Fig. 3(d)) for Q = 6. We con-
(circles). The quantity ro(Q,Pr)/(QPr) scales as Q−1. The clude that the external magnetic field orients the wavy
solid line shows the same power law behaviour. rolls along its own direction.
4
100(a) r = 1.05, Q = 0 100(b) r = 1.05, Q = 5 100 (a) Q = 0 58 (b) Q = 24
−55000 −55000 W, W101011 −55000 W101 57
−1000 10 20 30 40 50 −1000 10 20 30 40 50 −1000 10 20 30 40 50 560 0.25 0.5
W011100(c) r = 1.2, Q = 0 100(d) r = 1.2, Q = 6 90 (c) Q = 38 97.34 (d) Q = 45
W, 101 500 500 W10189.5 W10197.339
−50 −50
−1000 2 4 6 8 10 −1000 2 4 6 8 10 890 0.25 0.5 97.3380 0.25 0.5
Time Time Time Time
FIG. 3: Time series of W101 (blue curve) and W011 (green FIG. 4: Time series of W101 and W011 at r =1.05 and Pr =
curve) for P = 0.025 showing (a) chaotic wavy rolls along 0.025 for four different values of Q: (a) Chaotic wavy rolls
x-axis (r = 1.05, Q = 0) and (b) chaotic wavy rolls along along y-axis (for Q = 0) are represented by the variation
y-axis(r=1.05, Q=5), (c) chaotic competition of mutually of the Fourier mode W101 (blue). The mode W011 (green)
perpendicular wavy rolls (r = 1.2, Q = 0), and (d) chaotic remains zero as the rolls are already oriented along y−axis,
wavy rolls along y-axis(r=1.2, Q=6). (b)periodicwavyrolls(Q=24),(c)periodicwavyrollswith
smaller amplitude of oscillation (Q=38), and (d) stationary
rolls along y-axis(Q=45).
D. Suppression of chaos
effect of increasing magnetic field on different patterns
As mentioned above, in the absence of magnetic field
are shown in three columns of figure 5. As Q increases,
the model shows three types of chaotic patterns: chaotic
the wavy rolls orient themselves along the direction of
wavy rolls along y-axis, chaotic wavy rolls along x-axis
the horizontal magnetic field. The chaos is suppressed
and a competition of mutually perpendicular sets of
for relatively higher values of Q and the flow becomes
chaotic wavy rolls depending upon the choice of initial
steady and two dimensional for further higher values of
conditionsandthevalueofr. Theexternaluniformhori-
Q . The route to chaotic flow from stationary convec-
zontalmagnetic field of moderate strengthapplied along
tive motion is displayed in figure 6. As Chandrasekhar’s
y-axis orients the rolls along its own direction, if it is
number Q is decreased from higher to lower value, the
not already oriented along the magnetic field (see, fig-
system shows a series of bifurcations. For much higher
ure. 3). Once the roll-patterns are oriented along the
values of Q the convection is stationary in the form of
field direction, the effect of further increase of magnetic
2-D rolls along the direction of the magnetic field. With
field is similar to its effect on a roll-patterns already ori-
further decrease in Q, there is a Hopf bifurcation from
ented along y-axis. Figure 4(a) displays the time series
stationarystate. Theconvectiveflowshowstimeperiodic
of the Fourier modes W (blue) and W (green) in
101 011
behaviour (figure 6a-b). As Q is lowered below certain
the absence of magnetic field (Q = 0). This represents
value of Q, one more independent frequency develops in
chaotic wavy rolls along y-axis. If the external magnetic
the flow. Consequently, the convection becomes quasi-
field along y-axis is gradually increased to a large value,
periodicintime(figure6c-d). FurtherdecreaseofQleads
chaotic flow is suppressed and a periodic flow develops.
to chaoticmotion(figure 6e-f). The route to chaosis via
The time series of the Fourier mode W corresponding
101
temporal quasi-periodic convection. The chaotic motion
to the periodic flow is displayed in figure 4(b). Further
becomes more complicated with further decrease in Q
increase in the field strength makes the amplitude of os-
(see figure 6g-h).
cillationofthemodeW smaller(see,figure4(c)). The
101
flow becomes stationary and two-dimensional (2-D) as The amplitude andfrequency ofoscillatoryconvection
field strength is increased more. Figure 4(d) shows the due to Hopf bifurcation, as Q is lowered from higher to
time series of W corresponding to 2-D stationary so- lower values for a fixed value of r, are displayed in fig-
101
lution. The mechanism of the suppression of chaos and ure 7. The dimensionless frequency decreases linearly
reorientation of wavy rolls in the direction of the exter- withthedecreaseinQ(figure7a). Thesquareoftheam-
nal magnetic field (y-axis) is displayed very clearly in plitude of the oscillatory motion increases linearly with
figure 5. It displays the isotherms at z = 0.5 computed the decreaseinQ nearthe onsetofHopfbifurcation(fig-
from the model for different values r and Q. The top ure 7b). However,the square of the amplitude starts in-
rowshowsthe variousconvectivepatternsintheabsence creasingslowerthanlinearwithdecreaseinQawayfrom
of magnetic field (Q = 0): (a) chaotic square patterns theonsetofoscillatorybehaviour. Ultimately,itsgrowth
(r = 1.2), (b) chaotic wavy rolls along y-axis (r = 1.05) is stopped and it starts decreasing with decreasing Q.
and (c) chaotic wavy rolls along x-axis (r = 1.05). The This may be the influence of quasi-periodic bifurcation
5
20 1010
(a) (b) Q=24
f
0 Q=24 100
−20 10−10
56 56.5 57 57.5 58
20 1010
(c) (d) Q=22.8
0 Q=22.8 100 f2 f1
−20 10−10
53.5 54 54.5 55 55.5
100 1010
(e) Q=22.4 (f) Q=22.4
0 100
Z010−100 SD(f)10−10
0 50 100P
100 1010
(g) Q=20 (h) Q=20
0 100
−100 10−10
−100 −50 0 50 100 0 10 20 30
W101 Dimensionless frequency ( f )
FIG. 5: Isotherms at z = 0.5 for different values of r and
Q. (a) The left column displays isotherms for r = 1.2, and FIG. 6: Quasi-periodic route to chaos as a function of Q for
for three values of Q. The chaotic competition of rolls and Pr =0.025andr=1.05. Theleftcolumndisplaysprojection
squares (top row) for Q = 0, time-periodic wavy rolls (mid- of the phase space trajectories on W −Z plane for four
101 010
dle row) along y-axis for Q = 60, and stationary 2-D rolls differentvaluesofQ. Therightcolumn showspowerspectral
(bottom row) for Q = 80. (b) The middle column displays density (PSD)of theFourier mode W as a function of the
101
isothermsforr=1.05andthreedifferentvaluesofQ. Chaotic dimensionless frequency f for thecorresponding values of Q.
wavyrolls (top) alongy-axisforQ=0becometime-periodic Time-periodicbehaviouroftheconvectiveflow(a)forQ=24
wavy rolls (middle) for Q= 24 and finally spatially periodic and one peak in PSD corresponding to theperiodic flow (b).
stationary 2-D rolls (bottom) along y-axis for Q = 45. (c) Thetemporalquasi-periodicbehaviouroftheflow(c)forQ=
Isotherms for r = 1.05 but with initial conditions different 22.8 and two peaks (d) confirming the quasi-periodic flow.
thanusedin(b)aredisplayedinrightcolumn. Chaoticwavy Chaotic flow (e) for Q = 22.4 and its PSD (f). The chaotic
rollsalongx-axis(top)orientthemselvesalongy-axisastime- flowbecomes more complex for Q=20 (g) with flatteningof
periodic wavy rolls (middle) for Q = 38, and finally become peaks in PSD (h).
2-D stationary rolls along y-axis for Q=45.
fioNpfrfgroueWucmssorseeenentlv8tthhe.eiacnnTvtumieothmhoneabedltheesimralomsoacepdssotlaemoravlot.petnfrueugadtngescedittmdhioieiNnnlauNtroshiufsteyesRlstdweailnyirttuelhecmnitgtubhhmneeurbnmbuaeertmehrtaboihcvbeaeirtolaouRsinnrismeieondt-f mensionless frequency (f)111111123456 (a) 2Oscillation amplitude) 1112468024000000000000000000000 (b)
ulations [6]. In addition, the results of the model also Di10 ( 0
23 28 33 38 43 23 28 33 38 43
show the convergence of Nusselt numbers for higher val- Q Q
ues of R as observed in simulation [6] for various values
FIG. 7: (a) Nondimensional frequency f of thewavy convec-
of Q. For intermediate values of R, the Nusselt number
tion with Q at a fixed value of thereduced Rayleigh number
obtainedfromthemodelshowsdifferenceswiththesame
r=1.05. Thebestfit(solidline)showsthelineardependence
obtained in DNS. This may be due to severe truncation
off withQ. (b)Squareoftheamplitudeoftheperiodicwavy
of the Fourier expansion of various fields in the model.
convection as a function of Q at r=1.05.
V. CONCLUSIONS
inhibits the onset of wavy rolls, which appear at the
secondary instability. The stationary straight rolls at
In this paper we presented a low-dimensional model primary instability as well as wavy rolls at secondary
to study the effect of uniform horizontal magnetic field instability orient themselves along the direction of the
on Rayleigh-B´enard convection in low-Prandtl-number magnetic field, if it is sufficiently strong. The threshold
fluids. The model is capable of capturing convective value of the Rayleigh number at the onset of wavy in-
patterns in the form of two sets of mutually perpendic- stability shows power law behaviour. Stronger magnetic
ular straight (2-D) rolls as well wavy (3-D) rolls. The field causes the suppression of chaotic flow. The results
external field affects the flow patterns significantly. It obtained from this simple model have good qualitative
6
101 Pr = 0.025 (a) 101 Pr = 0.1 (b) agreement with those observed in experiments and
simulations.
100 100 100 30
20
Nu − 110−1 2030 Nu − 110−1 010
10
10−2 10−2
0
10−130−2 100 102 104 10−130−2 100 102 104
R − R R − R
c c
FIG. 8: Plot of convective heat flux (Nu−1) as a function
of relative Rayleigh number(R−Rc) for Pr =0.025 (a) and PPthanksM.K.Verma,IITKanpur,India,foruseful
Pr =0.1 (b) for several valuesof Q. discussions.
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1 9π4 π
(1982). U˙ = (9π2I+2QA)U+ rT+ V BU
2
[3] D.T.J. Hurle, R.W. Series, Handbook of crystal growth −6 4 4
edited by D.T.J. Hurle, (North Holland, Amsterdam, 1
+ (πV C 2X)Z,
1994). 6π 1 −
[4] A. Gailitis, O. Lielausis, E. Platacis, G. Gerbeth, F. 1 27
Stephani,F.Rossendorf,Rev.Mod.Phys.74,973(2002). V˙1 = (8π2+Q)V1+ π4rS1 CU Z,
−4 8 − ·
[5] F.H.Busse,R.M.Clever,J.Mech.Theo.Appl.(France),
1 27 3π
2, 495 (1983). V˙ = (50π2+Q)V + π4rS U U ,
2 2 2 1 2
[6] F.H.Busse,R.M.Clever,Phys.Rev.A40,1954(1989). −10 20 − 5
[7] R.M.Clever,F.H.Busse,J.FluidMech.201,507(1989). 1 π2 π
Z˙ = (π2I+2QD)Z+ V CU+ XU,
[8] P.Pal, K. Kumar, Indian J. Phys. 81, 1215 (2007). −2 8 1 8
[9] P. Sulem, C. Sulem, P.L. Sulem, O. Thual, Prog. Astro. 1
Aeronaut.100 125(1985). X˙ = (8π2+Q)X+πU Z,
−4 ·
[10] M.Meneguzzi, C.Sulem,P.L.Sulem,O.Thual,J.Fluid
Mech.182 169 (1987). T˙ = 3π2T+ 1 U+ 1S Z+ π(S B+4IY)U,
[11] O.M. Podvigina, Phys.Rev. E. 81, 056322 (2010). −2P P 2 1 4 2
r r
[12] Y.Nakagawa, Proc. R.Soc. A 240, 108 (1957). 2π2 1
[13] Y.Nakagawa, Proc. R.Soc. A 249 138 (1959). S˙ = S + V CZ T+πV Y,
1 1 1 1
− P P − ·
[14] B. Lehnert,N.C. Little, Tellus 9, 97 (1957). r r
[15] S. Fauve, C. Laroche, A. Libchaber, J. Phys. Letts. 5π2 1 π
S˙ = S + V U BT,
(France) 42, L455 (1981). 2 − P 2 P 2− 2 ·
r r
[16] S. Fauve, C. Laroche, A. Libchaber,J. Phys. Letts.
4π2 π π
(France) 45, L101 (1984). Y˙ = Y U T V S ,
1 1
[17] S.Fauve,C.Laroche,A.Libchaber,B.Perrin,Phys.Rev. − Pr − 2 · − 4
Lett.52, 1774 (1984).
[18] B. Hof, A. Juel, T. Mullin, J. Fluid Mech. 545, 193
(2005). where U = (U1,U2)T = (W101,W011)T, (V1,V2) =
[19] K.E. McKell, D.S. Broomhead, R. Jones, D.T.J. Hurle, (W111,W112), Z = (Z1,Z2)T = (Z010,Z100)T, T =
Europhys.Lett. 12, 513518(1990). (T ,T )T = (T ,T )T, (S ,S ) = (T ,T ), X =
1 2 101 011 1 2 111 112
[20] F.H. Busse, J. Fluid Mech. 52, 97 (1972). 0 0 0 1
[21] P.Pal, K. Kumar, Phys.Rev.E65, 047302 (2002). Z111, Y = T002, A = (cid:18)0 1(cid:19), B = (cid:18)1 0 (cid:19), C =
[22] O.Thual, J. Fluid Mech. 240, 229 (1992).
1 0 1 0 1 0
[23] K.Kumar,S.Fauve,O.Thual,J.Phys.II6,945(1996). , D = , and I = . The dot
(cid:18)0 1 (cid:19) (cid:18)0 0(cid:19) (cid:18)0 1(cid:19)
−
()operationimpliesstandardinnerproduct,andthesu-
·
perscript T denotes Transpose of a matrix.