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Ferrothermohaline convection
N
ELSEVIER
Journalof magnetism ~ i ~ and magnetic materials
Journal of Magnetism and Magnetic Materials 176 (1997) 321-330
Ferrothermohaline convection G. Vaidyanathan a'*, R. Sekar b, A.
Ramanathan
c
aDepartment of Physics, Pondicherry Engineering College, Pondicherr), 605014, India b Department of Mathematics, Pondicherry Engineering College, Pondicher~ 605014, India c Department of Mathematics, Mahatma Gandhi Government Arts College, Mahe, U. T ofPondicher 0" 673311, India Received 9 February 1996; received in revised form 14 May 1997
Abstract
The ferroconvective instability of two-component fluid heated from below and salted from above has been analysed. This is a familiar example of ferrofluids in which the ferric components are dissolved into organic solvants acting as carrier. Moreover, the salt is magnetic which modifies the established magnetic field. The effect of salinity has been included in magnetisation and density of the ferrofluid. A linear stability analysis has been carried out to study the onset of ferroconvection both by stationary as well as oscillatory modes. It is found that the salinity of ferrofluid enables the fluid to get destabilised more when it is salted from above. Numerical and graphical results are presented. )2) 1997 Elsevier Science B.V. All rights reserved.
Keywords." Magnetic fluid; Convection; Linear stability
1. Introduction
The importance of ferrofluids was realised soon after the method of formation of ferrofluids during mid-sixties. This is because it had a very large potential application in various fields. However, due to the availability of colloidal magnetic fluids (ferrofluids), many other uses of these fascinating liquids have been identified which are governed with the remote positioning of magnetic field controlling the magnetic fluid. Due to these wide range of application of ferrofluids, its commercial usage includes vacuum feed throughs for semi-conductor manufacturing and related uses [1], pressure seals for compressors and blowers [2]. It is also used in liquid-cooled loudspeakers which involves small bulk quantities of the ferrofluid to conduct heat away from the speaker coils [-3]. This innovation increases the amplifying power of the coil, and hence, it leads to the loudspeaker to produce high-fidility sound. In order to bring the drug to a target site in human body, a magnetic field applied acted on a drop of ferrofluid injected in human body [-4]. The novel zero-leakage rotating-shaft seals are used in computer disk drives [5]. However, there are many more applications in various fields [-6].
* Corresponding author. 0304-8853~/ 97~/ $17.00 ~, 1997 Elsevier Science B.V. All rights reserved PII $ 0 3 0 4 - 8 8 5 3 ( 9 7 ) 0 0 4 6 8 - X
G. Vaiclvanathan el al. / Journal qf Magnetism and Magnetic Materials" 176 (1997) 321 330
322
Finlayson [7] studied the effect of ferroconvection of a single-component fluid. He explained the concept of thermomechanical interaction in ferrofluids. Das Gupta and Gupta [8] have shown the stabilising effect of rotation on setting up of convective instability in ferrofluids. Recently, Vaidyanathan et al. [9] illustrated the inhibiting effect of onset of convection in ferrofluids in the presence of a porous medium. Sekar et al. [10] and Sekar and Vaidyanathan [11] obtained the condition for rotation to determine the nature of instability that may set in ferrofluids. The above authors analysed convective instability in a single-component ferrofluid. Normally, ferrofluids are suspension of magnetic salts in a carrier organic fluids [12] and hence, it is appropriate to study the convective instability in two-component fluids in which the ferric salts are treated as solute and organic carrier as solvents. Hence, the study of convection in the two-component ferrofluids will throw more light on convective instability. This is referred to as a new type of convection known as ferrothermohaline convection studied by Baines and Gill [13]. In the present paper, onset of convection of two-component ferrofluid is studied. The linear stability analysis is carried out to study the onset of ferroconvection both by stationary as well as oscillatory modes.
2. Mathematical formulation A horizontal layer of an incompressible ferromagnetic fluid of thickness 'd' in the presence of transverse applied field, heated from below and salted from above is considered. The temperature and salinity at the bottom and top surfaces z = -T-½dare To +_AT/2 and So-'-AS/2, respectively. Both the boundaries are taken to be free and perfect conductors of heat and solute. The mathematical equations governing the above investigation are as follows. The continuity equation for an incompressible fluid is V.q=0.
(1)
The momentum equation as given by Finlayson [7] is
Dq Po Dt -
V p + PO +
V. (HB) + flVZq.
(2)
The temperature equation for an incompressible ferrofluid is
[ p o C , , , ~ - / ~ o H \(~M) oT..
]dT/Jat
+/toT
(~M) ~,,u--dt dH =
KxV2T + q)"
(3)
The mass flux equation is given by
~tt + q" V
S = g s V2S,
(4)
where, Po, O, q, t, p, I~,H, B, C~.u, T, M, K1, S, K, and q~ are the density, acceleration due to gravity, velocity, time, pressure, dynamic viscosity (constant), magnetic field, magnetic induction, heat capacity at constant volume and magnetic field, temperature, magnetisation, thermal conductivity, salinity, mass diffusivity and viscous dissipation factor containing second-order terms in velocity, respectively.
323
G. Vaidyanathan et al. / Journal of Magnetism and Magnetic Materials 176 (1997) 321-330
Using Maxwell's equation for non-conducting fluids [12] one can assume that the magnetisation is aligned with the magnetic field and depends on the magnitude of the magnetic field, temperature and salinity, so that H
(5)
M = ~ M(H, T, S).
The magnetic equation of state is linearised about the magnetic field Ho, the average temperature To and the average salinity So to become M = Mo + z ( H - Ho) - K ( T -
(6)
To) + K2(S - So).
The density equation of state for a Boussinesq two-component fluid is (7)
p = po[-1 -- ~t(T -- To) + ~s(S -- So)],
where $~t is the thermal expansion coefficient and ~ is the solute analog of at. Basic state is assumed to be quiescent state and the basic state quantities are obtained by substituting, the velocity of quiescent state in the governing Eqs. (1)44) and the solutions of Eqs. (17(7) are obtained following the techniques of [7, 10-12]. Further, for the stability of the above equations one studies using linear theory. Following the analysis of [7, 11] one can use normal mode techniques for all dynamical variables. This can be written as
(s)
f (x, y, z, t) = f (z, t) exp i(kxx + kyy),
where the wave number k is given by, k 2 = k~ + k 2
(9)
The vertical component of the momentum equation can be written as
po~ Uz2 - k : w - -(1- +Z) + P°g°~skZS
~z-KO
]k2
, oK2 sI
(1 + ) / ) ~ z + K 2 S
-pogoqk20+~
poKK2 (82 ) p ° K K 2 flsk20 + - fisk2S + !~ -- k 2
(1 + Z)
(1 + Z)
~
w.
]
k2
(10)
The modified Fourier heat conduction equation is
poN-.oKToat
az =/¢' ~ z : - k2 0+
pc/~,
p°K2T°fi' +
(l+z)
w,
(I+z)
J
(11)
where pC = poC~,,n + poKHo. The salinity equation is
8~-+/~sw=K~ Uz:-k 2 S.
(12)
Using the analysis similar to Finlayson [-7] one gets ,82q~ (1 + Z)~z2
k2(
l+~
Mo) 80 8S ° q5 - K~zz + K2~zz = 0 ,
(13)
G. Vaidyanathan et al. /Journal of Magnetism and Magnetic" Materials 176 (1997) 321 330
324
w h e r e the n o n - d i m e n s i o n a l t e r m s used are
vt
w*
t* = ~ ,
wd = ~-'
a a=kd,
D-
~z*'
T * - K1 aR1/2 p C fltvd O,
(1 + z ) K l a R 1/2 KpCfitvd2 05,
05,
z z*
d'
K aRJ/2 S * - - - S . pCfl~vd
(14)
T h e n o n - d i m e n s i o n a l f o r m of g o v e r n i n g e q u a t i o n s c a n be w r i t t e n as
~t*
(D 2 - a2)w * = MlaR1/2D05 * - (1 + M1)aR1/2T * + M1MsaR1/2D05 * + (1 + M4)aR~/2S *
ST* P,~- - M2P, ~
-- M 1 M s a R 1 / 2 T * + M 4 M 5 laR~/2S* + (D 2 - a2)2w *,
(15)
(DO*) = (D 2 - a2)T * + [1 - M2 -- M 2 M s ] aR1/2w *,
(16)
~S* Pr ~ - = z( D2 - a2) S* - M 5 M 6 laR~/2w*
(17)
D205 * - M3a205 * - D T * + - - ~c
(18)
and
DS* = O,
where the d i m e n s i o n l e s s p a r a m e t e r s used are
M1
poK2i, (1 + Z)Po~tg
M3 -
(1 + (Mo/Ho)) (1 -]- Z) ' K 2fl s
Ms -
Kilt '
lxoK2itTo (1 + z ) p C '
M2
t~oK2 il~ (1 + Z ) P o ~ g
M4
K, M6 = ~ ,
R
pC P" - K I '
pCoqiltgd 4 vK~
Rs -
poCoc~fisgd 4 Ksv
Ks z = --K1 pC.
(19)
atz=-T-½.
(20)
2.1. Exact solution f o r f r e e boundaries T h e b o u n d a r y c o n d i t i o n s are
w* = D2w * = T* = D05* = S* = O T h e exact s o l u t i o n s satisfying w* = A e ~t* c o s ~z, D05" = O e ~* cos r~z,
Eq. (20) are
T* = B e ~t* cos rcz, 05* = D_ e~t, sin r~z, 7~
S* = C e at* cos gz, (21)
w h e r e A, B, C a n d D are c o n s t a n t s . I n the a b o v e s o l u t i o n the lowest m o d e of sin(nrcz), for n = 1 is a s s u m e d as s o l u t i o n . T h e s o l u t i o n c a n be o d d or even m o d e s c o m p a t i b l e with b o u n d a r y c o n d i t i o n s . In the p r e s e n t case of c h o o s i n g reference at the c e n t r e e n a b l e s to c h o o s e lowest e v e n m o d e , n a m e l y cos(~z) for all d y n a m i c a l
G. Vaidvanathan et al. / Journal of Magnetism and Magnetic Materials 176 (1997) 321 330
325
variables. Substitution of Eq. (21) in Eqs. (15)-(18) leads to A(/r2 + a2)[-(/r2 + a2) + 0-1 -- B[(1 + m l ) + m l m s ] a R 1/2
+ C[(1 + M4) + M 4 M ~ 1]aR~/2 + DMI(1 + M s ) a R 1/2 = 0, AaR1/2(1 - M2 - M 2 M s )
--
B[(1r 2 + a 2) +
(22)
Pral + D M z P r o = 0,
Am6aRls/2 + C[z(~ z + a 2) + Pra] = 0, -- BTrZRls/2 + D( g2 + a 2 M 3J~RX/2 s
+
(23) (24)
CMsM61R1/2,j~2 =
(25)
0.
For the existence of non-trivial eigenfunctions the determinant of the coefficients of A, B, C and D in Eqs. (22)-(25) must vanish. Following the techniques and analysis of Refs. [7-91 on Eqs. (22)-(25) leads to (26)
Ua 3 + Va 2 + W a + X =O.
Eq. (26) helps one to obtain eigenvalue R for which solution exists. If oscillatory instability exists, a is taken to be equal to i0-1 if and only if al = 0. By using the technique similar to [7 9] the Rayleigh number for stationary mode has been obtained using (~2 q_ a2)3 _ a2Rs(1 + M4 + M 4 M ~ 1) R = aZ(1 + M1 + M 1 M s ) - TrZMla2(1 + Ms)(1 + (Ms/r))/(~z 2 + aZM3)"
(27)
When the salinity Rayleigh number is taken as zero, this tends to the critical Rayleigh number obtained in Ref. [7] for single component ferrofluid. When M1 = 0 the classical Rayleigh problem for bouyancy-induced convection is obtained [a 2 = ~2/2 and Re = 277r41 [14]. When all the magnetic parameters M t - M 6 vanish, this reduces to double-diffusive convection [13]. For M1 very large one gets the results for the magnetic mechanism, and the critical thermal Rayleigh number for stationary mode is obtained using N = RM1 =
((7~2 + a2) 3 -- aZRs(1 + m ~ + m 4 m ~ 1))(I~[2 q- aZM3) a2(1 + Ms)[a2M3 _ TcZM3.c - 11
(28)
Following the analysis and techniques of Refs. [7-9] the critical Rayleigh number for oscillatory mode has been calculated using (29)
R = ZZ4~4 + ( X Z 2 + Z Z 3 ) a 2 + X Z ~ . Dr
when M1 is very large one gets the thermal Rayleigh number as N = RM1
2122(74 -- ( X l l Z 4 q- 2 1 1 2 3 ) 0 -2 q- X l l Z 1
Dr,
'
(30)
where Dr = X 2 + a2Z 2,
2 2 Dr, = X21 + 0-1Zll,
X = (~2 + a2)a2(1 + M I
+
0-21
X2Y1
+ X1Y3
= X 1 Y 2 + X 3 Y 1'
M1Ms)(/1; 2 + a 2 M 3 ) ' c -- aeTrZMl(1 +
X l 1 = (/-c2 -}- aZ)a2(1 + Ms)[(~ 2 + a2M3)z + 7r2z + ~2M5], Z = a2(1 + M , + M1Ms)Pr(~ 2 + aZM3) -- Tt2(1 + M s ) z P r M l a 2,
Ms)(z + Ms),
G. Vaidyanathan et al. / Journal of Magnetism and Magnetic Materials" 176 (1997) 321 - 330
326 Z1 =
_ (~2 _1_ a2)2(,/.~2 _}_ a 2 M 3 ) [ . r ( g 2 q_ a2)2 q_ a2(1 + M 4 +
M4Ms 1 ) R s M 6 ] ,
Z 2 = (~z2 + a2)(~ 2 + a2M3)P~[(rc 2 + a 2) + (1 + r ) ] , Z 3 = a2(1 q- M1 + M , M s ) P r ( g 2 + a 2 M 3 )
-
-
(,1~2 __ a e ) 3 [ ( g 2 + aeM3){Pr(1 + "r) + z } ] ,
Z 4 = ( r d + a2)(rc 2 + a 2 M 3 ) P 2, Zll
= a2(1 + M s ) P r [ - - ( r c
2 + a 2 M 3 ) + ~2(1 + M s ) ] .
(31)
Table l Stationary mode of instability of two-components ferrofluids (R)~
M3
M5 = 0.1, M 4 = 0.1, Ratio of mass transport to heat transport z = 0.05
r = 0.09
~ = 0.1
a~
N~
a~
Nc
ac
Nc
ac
N~
l 5 t0 15 20 25
6.73 4.33 3.71 3.43 3.24 3.08
11404.95 4266.68 3481.12 3201.48 3053.18 2959.93
5.98 3.91 3.39 3.12 3.00 2.87
7968.60 3200.94 2643.79 2444.20 2338.61 2272.24
5.50 3.67 3.20 2.96 2.83 2.74
6250.61 2629.60 2190.22 2032.45 1949.16 1897.11
5.32 3.57 3.12 2.92 2.78 2.69
5682.42 2433.55 2033.73 1890.18 1814.46 1767.23
l 5 10 15 20 25
6.25 3.71 3.16 2.92 2.78 2.69
7882.70 2012.28 1500.47 1333.22 1248.77 1197.50
5.52 3.43 2.96 2.78 2.65 2.60
5505.94 1622.59 1252.90 1130.19 1068.03 1030.22
5.05 3.24 2.83 2.69 2.60 2.55
4361.36 1419.15 1121.65 1021.75 971.02 940.25
4.90 3.16 2.78 2.65 2.55 2.50
3990.88 1350.45 1076.90 984.59 937.73 909.17
1 5 10 15 20 25
6.06 3.46 2.96 2.74 2.65 2.55
6880.21 1377.37 965.27 839.31 778.23 741.92
5.34 3.20 2.78 2.65 2.55 2.50
4808.78 1190.31 885.66 789.15 741.56 713.12
4.90 3.08 2.74 2.60 2.50 2.45
3830.61 1093.79 843.25 762.15 721.73 697.43
4.74 3.04 2.69 2.55 2.50 2.45
3517.69 1061.32 828.77 752.90 714.85 691.97
100
1 5 10 15 20 25
5.85 3.00 2.60 2.45 2.40 2.35
5780.99 647.14 519.42 490.53 490.53 374.40
5.15 2.9 2.6 2.5 2.4 2.4
4048.20 621.56 502.06 439.05 409.31 351.87
4.72 2.87 2.60 2.50 2.40 2.40
3256.33 550.29 457.17 398.29 369.73 322.74
4.56 2.87 2.60 2.50 2.40 2.40
3007.98 458.61 415.04 317.98 290.02 233.37
500
1 5 10 15 20 25
5.48 5.48 5.48 5.48 5.48 5.48
48.84 15.40 13.87 13.41 13.19 13.06
4.74 4.74 4.74 4.74 4.74 4.74
29.87 9.63 8.62 8.31 8.17 8.08
4.18 4.18 4.18 4.18 4.18 4.18
22.59 6.85 6.07 5.83 5.71 5.65
3.94 3.94 3.94 3.94 3.94 3.94
18.17 4.67 4.57 4.38 4.29 4.24
-- 500
--
~ = 0.07
100
G. Vaidyanathan et al. / Journal (?.['Magnetism and Magnetic Materials' 176 (1997) 321-330
327
There are many types of ferrofluids formed by changing ferric oxides and carrier organic fluids. In the present analysis, the range of values pertaining to ferric oxide, kerosene and other organic carriers are chosen. With the same ferric oxide, the different carriers like alcohol, hydrocarbon, ester, halocarbon, silicone could be chosen. Depending on this, the parametric values of ferrofluid are found to vary within these limits. These values are used for analysis in the paper. Suggestion from Finlayson 1-7] and Das Gupta and Gupta 1-8] have also been taken for variation of these parametric values. In the above analysis M1 takes the value 1000 and M takes values from 0.1 to 0.5, Pr = 0.01. M is allowed to vary from 0.1 to 0.5 and r varies from 0.01 to 0.1.
3. Conclusions The thermohaline convection of ferrofluid has been analysed. The Prandtl number is assumed to be 0.01. The magnetisation parameter Mt is taken to be 1000; for a very large value of M 1 , the effect of magnetic
Table 2 Stationary mode of instability of two-components ferrofluids (R)~
M3
Ms = 0.5, M4 = 0.1, Ratio of mass transport to heat transport z = 0.05
c = 0.07
c - 0.09
"c = 0.1
a¢
N¢
at
N~
ac
N~
a~
N~
- 500
5 10 15 20 25
6.36 5.05 4.50 4.15 3.94
6083.67 3432.10 2703.56 2361.09 2160.30
5.27 4.47 4.00 3.71 3.50
4076.70 2445.12 1977.66 1753.72 1621.13
5.05 4.09 3.67 3.43 3.28
3100.47 1941.76 1600,33 1434.88 1336,36
4.87 3.97 3.57 3.32 3.16
2783.02 1773.87 1473.21 1326.95 1239.63
- 100
5 10 15 20 25
6.00 4.64 4.06 3.71 3.50
4584.66 2188.80 1583.23 1314.05 1162.51
5.27 4.12 3.64 3.35 3.20
3059. t9 1601.21 1214.42 1038.19 937.49
4.80 3.81 3.39 3.16 3.00
2339.50 1310.73 1027.44 896.48 820.93
4.61 3.67 3.28 3.08 2.96
2109.39 1214.33 965.27 849.00 781.78
5 10 15 20 25
5.92 4.47 3.87 3.54 3.32
4175.59 1843.29 1273.32 1027.03 891.55
5.17 4.00 3.50 3.24 3.08
2783.28 1371.30 1008.58 847.13 756.60
4.69 3.71 3.28 3.08 2.92
2134.68 1140.73 875.64 755.29 686.78
4.53 3.57 3.20 3.00 2.87
1928.85 1065.39 831.52 724.42 663.23
100
5 10 15 20 25
5.81 4.30 3.67 3.32 3.12
3744.75 1469.45 936.91 717.14 601.26
5.05 3.84 3.35 3.08 2.92
2493.89 1127.59 791.40 647.25 568.68
4.61 3.57 3.16 2.96 2.83
1921.13 963.73 718.17 609.71 549.34
4.42 3.46 3.12 2.92 2.78
1741.12 910.58 693.66 596.75 542.36
500
5 10 15 20 25
4.42 4.42 4.42 4.42 4.42
68.99 40.29 27.21 17.19 15.06
3.64 3.64 3.64 3.64 3.64
19.86 17.75 15.48 14.76 14.41
2.92 2.87 2.87 2.87 2.87
16.85 15.07 11.83 10.71 8.65
3.81 2.45 2.29 2.24 2.24
14.67 12.57 8.91 7.45 6.23
G. Vaidyanathan et al. / Journal of Magnetism and Magnetic Materials 176 (1997) 321 330
328
mechanism is very large, when compared to bouyancy effect [7]. For such fluids, M 2 is assumed to have negligible value and hence taken to be zero [14]. M3 is varied from 1 to 25 because M3 cannot take a value less than one [9]. m 6 is taken to be 0.1. M 4 is the effect on magnetisation due to salinity. This is allowed to vary from 0.1 to 0.5 taking values less than the magnetisation parameter M3. M5 represents the ratio of the salinity effect on magnetic field and pyromagnetic coefficient.This is varied between 0.1 and 0.5. r is the ratio of mass diffusivity to thermal diffusivity which is varied between 0 and 0.1 [10, 14]. The salinity Rayleigh number Rs is varied from - 5 0 0 to 500. The increase in magnetisation (M3) is found to cause large destabilisation, because both magnetic and thermal mechanism favour destabilisation. This can be observed from Tables 1 and 2 in which the increase in M3, decreases No. The addition of the magnetic salt increases magnetic properties of multicomponent fluids, which in turn interacts with temperature and applied magnetic field. The effect of the temperature is very much pronounced on the magnetic field. The addition of salt and increase in temperature tend to oppose each other. Therefore, increase in M5 shows destabilising effect, which is obvious from Tables 1 and 2. The increase in M5 decreases N~. If a magnetic salt of larger magnetic moment is dissolved in the solvent, it increases the magnetisation which in turn promotes instability. As the ratio of mass transport to heat transport (~) is increased, the onset of instability is favoured, as Nc decreases with increase of r (Tables 1 and 2 and in Figs. 1 3). From Tables 1 and 2 it is seen that as salinity Rayleigh number is negatively increased (stably stratified or salted from below), the thermal Rayleigh number N~ ( = RM1)c is found to increase. This means that the system is getting stabilised. This would imply that the solution is denser at the bottom than at the top, which favours stability. From Tables 1 and 2 it is seen that the positive increase in salinity Rayleigh number (Rs) decreases No. The positive increase in salinity Rayleigh number would mean that the system is salted from above (unstable stratification). This makes the upper layer heavier than the lower. This leads to early onset of convective instability. From Fig. 3 it is seen that for all values o f t and that there is no Change on destabilisation, because the system is completely destabilised even for a very small temperature gradient very close to zero. One may conclude that the system can be destabilised purely by salinity gradient itself without the large value of
R s = - 500:
I] M 3 = 5
M 4 = 0.1;
× M 3 = 15
M 5 = 0.1;
o M3 =
25
z£ Nc
1
0.03
|
I
I
0.05
0.07
0.09
Fig. 1. V a r i a t i o n of No versus r for M5 = 0.1.
0.11
G. Vaidyanathan et al. /Journal c f Magnetism and Magnetic Materials 176 (1997) 321-330 13
11
R~ = -500 ;
~ M3= 5
M4=0.1 ;
× M3=15
M 5=0.5;
o M3=25
M 6 = 0.1. 9
5
3
1
o
0.03
0.05
0.09
0.07
Fig. 2. V a r i a t i o n of N~ versus ~ for M5 = 0.5.
N¢ x 10 -3 M 5 =0.1 4.0
3.5
/'~=0.05
I:
~-~ 3.0
M 3 = 25
2.0
0.5
-400
-300
-200
-100
0
100
200
300
Fig. 3. V a r i a t i o n of Nc versus r for R~ for M5 = 0.1.
400
~, R~
329
330
G. Vaidyanathan et al. /Journal o['Magnetism and Magnetic Materials 176 (1997) 321 330
magnetisation and for any value of ratio of salinity to mass diffusivity. This is the rare phenomenon where the system can favour convection by salinity concentration itself.
Acknowledgements The last author is grateful to the University Grants Commission for extending the financial facilities for partly meeting out the expenditure to carry out this work. The authors are grateful to the referees for their valuable suggestions and comments but for this, the paper would not be in the present form.
References [1] R. Moskowitz, ASLE Trans. 18 (2) (1975) 135. [2] R.E. Rosensweig, in: L. Marton (Ed.), Advances in Electronics and Electron Physics, vol. 43, Academic Press, New York, 1979, p. 103. [3] D.B. Hathaway, Sound Eng. Mag. 13 (12) (1979) 42. [4] Y. Morimoto, M. Akimoto, Y. Yotsumoto, Chem. Pharm. Bull. 30 (8) (1982) 3024. [5] R.L. Baily, J. Magn. Magn. Mater. 39 (1,2) (1983) 178. [6] R.E. Rosensweig, Ferrohydrodynamics, Cambridge University Press. Cambridge, 1985. [7] B.A. Finlayson, Int. J. Fluid Mech. 40 (4) (1970) 753. [8] M.S. Das Gupta, A.S. Gupta, Int. J. Eng. Sci. 17 11979) 271. [9] G. Vaidyanathan, R. Sekar, R. Balasubramanian, Int. J. Eng. Sci. 19 (1991) 1259. [10] R. Sekar, G. Vaidyanathan, A. Ramanathan, Int. J. Eng. Sci. 31 (2) (1993) 241. [11] R. Sekar, G. Vaidyanathan, Int. J. Eng. Sci. 31 (8) (1993) 1139. [12] R. Sekar, G. Vaidyanathan, A. Ramanathan, J. Magn. Magn. Mater. 149 (1995) 137. [13] P.G. Baines, A.E. Gill, J. Fluid Mech. 37 (1969) 289. [14] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford University Press, London, 1961.
ELSEVIER
Journalof magnetism ~ i ~ and magnetic materials
Journal of Magnetism and Magnetic Materials 176 (1997) 321-330
Ferrothermohaline convection G. Vaidyanathan a'*, R. Sekar b, A.
Ramanathan
c
aDepartment of Physics, Pondicherry Engineering College, Pondicherr), 605014, India b Department of Mathematics, Pondicherry Engineering College, Pondicher~ 605014, India c Department of Mathematics, Mahatma Gandhi Government Arts College, Mahe, U. T ofPondicher 0" 673311, India Received 9 February 1996; received in revised form 14 May 1997
Abstract
The ferroconvective instability of two-component fluid heated from below and salted from above has been analysed. This is a familiar example of ferrofluids in which the ferric components are dissolved into organic solvants acting as carrier. Moreover, the salt is magnetic which modifies the established magnetic field. The effect of salinity has been included in magnetisation and density of the ferrofluid. A linear stability analysis has been carried out to study the onset of ferroconvection both by stationary as well as oscillatory modes. It is found that the salinity of ferrofluid enables the fluid to get destabilised more when it is salted from above. Numerical and graphical results are presented. )2) 1997 Elsevier Science B.V. All rights reserved.
Keywords." Magnetic fluid; Convection; Linear stability
1. Introduction
The importance of ferrofluids was realised soon after the method of formation of ferrofluids during mid-sixties. This is because it had a very large potential application in various fields. However, due to the availability of colloidal magnetic fluids (ferrofluids), many other uses of these fascinating liquids have been identified which are governed with the remote positioning of magnetic field controlling the magnetic fluid. Due to these wide range of application of ferrofluids, its commercial usage includes vacuum feed throughs for semi-conductor manufacturing and related uses [1], pressure seals for compressors and blowers [2]. It is also used in liquid-cooled loudspeakers which involves small bulk quantities of the ferrofluid to conduct heat away from the speaker coils [-3]. This innovation increases the amplifying power of the coil, and hence, it leads to the loudspeaker to produce high-fidility sound. In order to bring the drug to a target site in human body, a magnetic field applied acted on a drop of ferrofluid injected in human body [-4]. The novel zero-leakage rotating-shaft seals are used in computer disk drives [5]. However, there are many more applications in various fields [-6].
* Corresponding author. 0304-8853~/ 97~/ $17.00 ~, 1997 Elsevier Science B.V. All rights reserved PII $ 0 3 0 4 - 8 8 5 3 ( 9 7 ) 0 0 4 6 8 - X
G. Vaiclvanathan el al. / Journal qf Magnetism and Magnetic Materials" 176 (1997) 321 330
322
Finlayson [7] studied the effect of ferroconvection of a single-component fluid. He explained the concept of thermomechanical interaction in ferrofluids. Das Gupta and Gupta [8] have shown the stabilising effect of rotation on setting up of convective instability in ferrofluids. Recently, Vaidyanathan et al. [9] illustrated the inhibiting effect of onset of convection in ferrofluids in the presence of a porous medium. Sekar et al. [10] and Sekar and Vaidyanathan [11] obtained the condition for rotation to determine the nature of instability that may set in ferrofluids. The above authors analysed convective instability in a single-component ferrofluid. Normally, ferrofluids are suspension of magnetic salts in a carrier organic fluids [12] and hence, it is appropriate to study the convective instability in two-component fluids in which the ferric salts are treated as solute and organic carrier as solvents. Hence, the study of convection in the two-component ferrofluids will throw more light on convective instability. This is referred to as a new type of convection known as ferrothermohaline convection studied by Baines and Gill [13]. In the present paper, onset of convection of two-component ferrofluid is studied. The linear stability analysis is carried out to study the onset of ferroconvection both by stationary as well as oscillatory modes.
2. Mathematical formulation A horizontal layer of an incompressible ferromagnetic fluid of thickness 'd' in the presence of transverse applied field, heated from below and salted from above is considered. The temperature and salinity at the bottom and top surfaces z = -T-½dare To +_AT/2 and So-'-AS/2, respectively. Both the boundaries are taken to be free and perfect conductors of heat and solute. The mathematical equations governing the above investigation are as follows. The continuity equation for an incompressible fluid is V.q=0.
(1)
The momentum equation as given by Finlayson [7] is
Dq Po Dt -
V p + PO +
V. (HB) + flVZq.
(2)
The temperature equation for an incompressible ferrofluid is
[ p o C , , , ~ - / ~ o H \(~M) oT..
]dT/Jat
+/toT
(~M) ~,,u--dt dH =
KxV2T + q)"
(3)
The mass flux equation is given by
~tt + q" V
S = g s V2S,
(4)
where, Po, O, q, t, p, I~,H, B, C~.u, T, M, K1, S, K, and q~ are the density, acceleration due to gravity, velocity, time, pressure, dynamic viscosity (constant), magnetic field, magnetic induction, heat capacity at constant volume and magnetic field, temperature, magnetisation, thermal conductivity, salinity, mass diffusivity and viscous dissipation factor containing second-order terms in velocity, respectively.
323
G. Vaidyanathan et al. / Journal of Magnetism and Magnetic Materials 176 (1997) 321-330
Using Maxwell's equation for non-conducting fluids [12] one can assume that the magnetisation is aligned with the magnetic field and depends on the magnitude of the magnetic field, temperature and salinity, so that H
(5)
M = ~ M(H, T, S).
The magnetic equation of state is linearised about the magnetic field Ho, the average temperature To and the average salinity So to become M = Mo + z ( H - Ho) - K ( T -
(6)
To) + K2(S - So).
The density equation of state for a Boussinesq two-component fluid is (7)
p = po[-1 -- ~t(T -- To) + ~s(S -- So)],
where $~t is the thermal expansion coefficient and ~ is the solute analog of at. Basic state is assumed to be quiescent state and the basic state quantities are obtained by substituting, the velocity of quiescent state in the governing Eqs. (1)44) and the solutions of Eqs. (17(7) are obtained following the techniques of [7, 10-12]. Further, for the stability of the above equations one studies using linear theory. Following the analysis of [7, 11] one can use normal mode techniques for all dynamical variables. This can be written as
(s)
f (x, y, z, t) = f (z, t) exp i(kxx + kyy),
where the wave number k is given by, k 2 = k~ + k 2
(9)
The vertical component of the momentum equation can be written as
po~ Uz2 - k : w - -(1- +Z) + P°g°~skZS
~z-KO
]k2
, oK2 sI
(1 + ) / ) ~ z + K 2 S
-pogoqk20+~
poKK2 (82 ) p ° K K 2 flsk20 + - fisk2S + !~ -- k 2
(1 + Z)
(1 + Z)
~
w.
]
k2
(10)
The modified Fourier heat conduction equation is
poN-.oKToat
az =/¢' ~ z : - k2 0+
pc/~,
p°K2T°fi' +
(l+z)
w,
(I+z)
J
(11)
where pC = poC~,,n + poKHo. The salinity equation is
8~-+/~sw=K~ Uz:-k 2 S.
(12)
Using the analysis similar to Finlayson [-7] one gets ,82q~ (1 + Z)~z2
k2(
l+~
Mo) 80 8S ° q5 - K~zz + K2~zz = 0 ,
(13)
G. Vaidyanathan et al. /Journal of Magnetism and Magnetic" Materials 176 (1997) 321 330
324
w h e r e the n o n - d i m e n s i o n a l t e r m s used are
vt
w*
t* = ~ ,
wd = ~-'
a a=kd,
D-
~z*'
T * - K1 aR1/2 p C fltvd O,
(1 + z ) K l a R 1/2 KpCfitvd2 05,
05,
z z*
d'
K aRJ/2 S * - - - S . pCfl~vd
(14)
T h e n o n - d i m e n s i o n a l f o r m of g o v e r n i n g e q u a t i o n s c a n be w r i t t e n as
~t*
(D 2 - a2)w * = MlaR1/2D05 * - (1 + M1)aR1/2T * + M1MsaR1/2D05 * + (1 + M4)aR~/2S *
ST* P,~- - M2P, ~
-- M 1 M s a R 1 / 2 T * + M 4 M 5 laR~/2S* + (D 2 - a2)2w *,
(15)
(DO*) = (D 2 - a2)T * + [1 - M2 -- M 2 M s ] aR1/2w *,
(16)
~S* Pr ~ - = z( D2 - a2) S* - M 5 M 6 laR~/2w*
(17)
D205 * - M3a205 * - D T * + - - ~c
(18)
and
DS* = O,
where the d i m e n s i o n l e s s p a r a m e t e r s used are
M1
poK2i, (1 + Z)Po~tg
M3 -
(1 + (Mo/Ho)) (1 -]- Z) ' K 2fl s
Ms -
Kilt '
lxoK2itTo (1 + z ) p C '
M2
t~oK2 il~ (1 + Z ) P o ~ g
M4
K, M6 = ~ ,
R
pC P" - K I '
pCoqiltgd 4 vK~
Rs -
poCoc~fisgd 4 Ksv
Ks z = --K1 pC.
(19)
atz=-T-½.
(20)
2.1. Exact solution f o r f r e e boundaries T h e b o u n d a r y c o n d i t i o n s are
w* = D2w * = T* = D05* = S* = O T h e exact s o l u t i o n s satisfying w* = A e ~t* c o s ~z, D05" = O e ~* cos r~z,
Eq. (20) are
T* = B e ~t* cos rcz, 05* = D_ e~t, sin r~z, 7~
S* = C e at* cos gz, (21)
w h e r e A, B, C a n d D are c o n s t a n t s . I n the a b o v e s o l u t i o n the lowest m o d e of sin(nrcz), for n = 1 is a s s u m e d as s o l u t i o n . T h e s o l u t i o n c a n be o d d or even m o d e s c o m p a t i b l e with b o u n d a r y c o n d i t i o n s . In the p r e s e n t case of c h o o s i n g reference at the c e n t r e e n a b l e s to c h o o s e lowest e v e n m o d e , n a m e l y cos(~z) for all d y n a m i c a l
G. Vaidvanathan et al. / Journal of Magnetism and Magnetic Materials 176 (1997) 321 330
325
variables. Substitution of Eq. (21) in Eqs. (15)-(18) leads to A(/r2 + a2)[-(/r2 + a2) + 0-1 -- B[(1 + m l ) + m l m s ] a R 1/2
+ C[(1 + M4) + M 4 M ~ 1]aR~/2 + DMI(1 + M s ) a R 1/2 = 0, AaR1/2(1 - M2 - M 2 M s )
--
B[(1r 2 + a 2) +
(22)
Pral + D M z P r o = 0,
Am6aRls/2 + C[z(~ z + a 2) + Pra] = 0, -- BTrZRls/2 + D( g2 + a 2 M 3J~RX/2 s
+
(23) (24)
CMsM61R1/2,j~2 =
(25)
0.
For the existence of non-trivial eigenfunctions the determinant of the coefficients of A, B, C and D in Eqs. (22)-(25) must vanish. Following the techniques and analysis of Refs. [7-91 on Eqs. (22)-(25) leads to (26)
Ua 3 + Va 2 + W a + X =O.
Eq. (26) helps one to obtain eigenvalue R for which solution exists. If oscillatory instability exists, a is taken to be equal to i0-1 if and only if al = 0. By using the technique similar to [7 9] the Rayleigh number for stationary mode has been obtained using (~2 q_ a2)3 _ a2Rs(1 + M4 + M 4 M ~ 1) R = aZ(1 + M1 + M 1 M s ) - TrZMla2(1 + Ms)(1 + (Ms/r))/(~z 2 + aZM3)"
(27)
When the salinity Rayleigh number is taken as zero, this tends to the critical Rayleigh number obtained in Ref. [7] for single component ferrofluid. When M1 = 0 the classical Rayleigh problem for bouyancy-induced convection is obtained [a 2 = ~2/2 and Re = 277r41 [14]. When all the magnetic parameters M t - M 6 vanish, this reduces to double-diffusive convection [13]. For M1 very large one gets the results for the magnetic mechanism, and the critical thermal Rayleigh number for stationary mode is obtained using N = RM1 =
((7~2 + a2) 3 -- aZRs(1 + m ~ + m 4 m ~ 1))(I~[2 q- aZM3) a2(1 + Ms)[a2M3 _ TcZM3.c - 11
(28)
Following the analysis and techniques of Refs. [7-9] the critical Rayleigh number for oscillatory mode has been calculated using (29)
R = ZZ4~4 + ( X Z 2 + Z Z 3 ) a 2 + X Z ~ . Dr
when M1 is very large one gets the thermal Rayleigh number as N = RM1
2122(74 -- ( X l l Z 4 q- 2 1 1 2 3 ) 0 -2 q- X l l Z 1
Dr,
'
(30)
where Dr = X 2 + a2Z 2,
2 2 Dr, = X21 + 0-1Zll,
X = (~2 + a2)a2(1 + M I
+
0-21
X2Y1
+ X1Y3
= X 1 Y 2 + X 3 Y 1'
M1Ms)(/1; 2 + a 2 M 3 ) ' c -- aeTrZMl(1 +
X l 1 = (/-c2 -}- aZ)a2(1 + Ms)[(~ 2 + a2M3)z + 7r2z + ~2M5], Z = a2(1 + M , + M1Ms)Pr(~ 2 + aZM3) -- Tt2(1 + M s ) z P r M l a 2,
Ms)(z + Ms),
G. Vaidyanathan et al. / Journal of Magnetism and Magnetic Materials" 176 (1997) 321 - 330
326 Z1 =
_ (~2 _1_ a2)2(,/.~2 _}_ a 2 M 3 ) [ . r ( g 2 q_ a2)2 q_ a2(1 + M 4 +
M4Ms 1 ) R s M 6 ] ,
Z 2 = (~z2 + a2)(~ 2 + a2M3)P~[(rc 2 + a 2) + (1 + r ) ] , Z 3 = a2(1 q- M1 + M , M s ) P r ( g 2 + a 2 M 3 )
-
-
(,1~2 __ a e ) 3 [ ( g 2 + aeM3){Pr(1 + "r) + z } ] ,
Z 4 = ( r d + a2)(rc 2 + a 2 M 3 ) P 2, Zll
= a2(1 + M s ) P r [ - - ( r c
2 + a 2 M 3 ) + ~2(1 + M s ) ] .
(31)
Table l Stationary mode of instability of two-components ferrofluids (R)~
M3
M5 = 0.1, M 4 = 0.1, Ratio of mass transport to heat transport z = 0.05
r = 0.09
~ = 0.1
a~
N~
a~
Nc
ac
Nc
ac
N~
l 5 t0 15 20 25
6.73 4.33 3.71 3.43 3.24 3.08
11404.95 4266.68 3481.12 3201.48 3053.18 2959.93
5.98 3.91 3.39 3.12 3.00 2.87
7968.60 3200.94 2643.79 2444.20 2338.61 2272.24
5.50 3.67 3.20 2.96 2.83 2.74
6250.61 2629.60 2190.22 2032.45 1949.16 1897.11
5.32 3.57 3.12 2.92 2.78 2.69
5682.42 2433.55 2033.73 1890.18 1814.46 1767.23
l 5 10 15 20 25
6.25 3.71 3.16 2.92 2.78 2.69
7882.70 2012.28 1500.47 1333.22 1248.77 1197.50
5.52 3.43 2.96 2.78 2.65 2.60
5505.94 1622.59 1252.90 1130.19 1068.03 1030.22
5.05 3.24 2.83 2.69 2.60 2.55
4361.36 1419.15 1121.65 1021.75 971.02 940.25
4.90 3.16 2.78 2.65 2.55 2.50
3990.88 1350.45 1076.90 984.59 937.73 909.17
1 5 10 15 20 25
6.06 3.46 2.96 2.74 2.65 2.55
6880.21 1377.37 965.27 839.31 778.23 741.92
5.34 3.20 2.78 2.65 2.55 2.50
4808.78 1190.31 885.66 789.15 741.56 713.12
4.90 3.08 2.74 2.60 2.50 2.45
3830.61 1093.79 843.25 762.15 721.73 697.43
4.74 3.04 2.69 2.55 2.50 2.45
3517.69 1061.32 828.77 752.90 714.85 691.97
100
1 5 10 15 20 25
5.85 3.00 2.60 2.45 2.40 2.35
5780.99 647.14 519.42 490.53 490.53 374.40
5.15 2.9 2.6 2.5 2.4 2.4
4048.20 621.56 502.06 439.05 409.31 351.87
4.72 2.87 2.60 2.50 2.40 2.40
3256.33 550.29 457.17 398.29 369.73 322.74
4.56 2.87 2.60 2.50 2.40 2.40
3007.98 458.61 415.04 317.98 290.02 233.37
500
1 5 10 15 20 25
5.48 5.48 5.48 5.48 5.48 5.48
48.84 15.40 13.87 13.41 13.19 13.06
4.74 4.74 4.74 4.74 4.74 4.74
29.87 9.63 8.62 8.31 8.17 8.08
4.18 4.18 4.18 4.18 4.18 4.18
22.59 6.85 6.07 5.83 5.71 5.65
3.94 3.94 3.94 3.94 3.94 3.94
18.17 4.67 4.57 4.38 4.29 4.24
-- 500
--
~ = 0.07
100
G. Vaidyanathan et al. / Journal (?.['Magnetism and Magnetic Materials' 176 (1997) 321-330
327
There are many types of ferrofluids formed by changing ferric oxides and carrier organic fluids. In the present analysis, the range of values pertaining to ferric oxide, kerosene and other organic carriers are chosen. With the same ferric oxide, the different carriers like alcohol, hydrocarbon, ester, halocarbon, silicone could be chosen. Depending on this, the parametric values of ferrofluid are found to vary within these limits. These values are used for analysis in the paper. Suggestion from Finlayson 1-7] and Das Gupta and Gupta 1-8] have also been taken for variation of these parametric values. In the above analysis M1 takes the value 1000 and M takes values from 0.1 to 0.5, Pr = 0.01. M is allowed to vary from 0.1 to 0.5 and r varies from 0.01 to 0.1.
3. Conclusions The thermohaline convection of ferrofluid has been analysed. The Prandtl number is assumed to be 0.01. The magnetisation parameter Mt is taken to be 1000; for a very large value of M 1 , the effect of magnetic
Table 2 Stationary mode of instability of two-components ferrofluids (R)~
M3
Ms = 0.5, M4 = 0.1, Ratio of mass transport to heat transport z = 0.05
c = 0.07
c - 0.09
"c = 0.1
a¢
N¢
at
N~
ac
N~
a~
N~
- 500
5 10 15 20 25
6.36 5.05 4.50 4.15 3.94
6083.67 3432.10 2703.56 2361.09 2160.30
5.27 4.47 4.00 3.71 3.50
4076.70 2445.12 1977.66 1753.72 1621.13
5.05 4.09 3.67 3.43 3.28
3100.47 1941.76 1600,33 1434.88 1336,36
4.87 3.97 3.57 3.32 3.16
2783.02 1773.87 1473.21 1326.95 1239.63
- 100
5 10 15 20 25
6.00 4.64 4.06 3.71 3.50
4584.66 2188.80 1583.23 1314.05 1162.51
5.27 4.12 3.64 3.35 3.20
3059. t9 1601.21 1214.42 1038.19 937.49
4.80 3.81 3.39 3.16 3.00
2339.50 1310.73 1027.44 896.48 820.93
4.61 3.67 3.28 3.08 2.96
2109.39 1214.33 965.27 849.00 781.78
5 10 15 20 25
5.92 4.47 3.87 3.54 3.32
4175.59 1843.29 1273.32 1027.03 891.55
5.17 4.00 3.50 3.24 3.08
2783.28 1371.30 1008.58 847.13 756.60
4.69 3.71 3.28 3.08 2.92
2134.68 1140.73 875.64 755.29 686.78
4.53 3.57 3.20 3.00 2.87
1928.85 1065.39 831.52 724.42 663.23
100
5 10 15 20 25
5.81 4.30 3.67 3.32 3.12
3744.75 1469.45 936.91 717.14 601.26
5.05 3.84 3.35 3.08 2.92
2493.89 1127.59 791.40 647.25 568.68
4.61 3.57 3.16 2.96 2.83
1921.13 963.73 718.17 609.71 549.34
4.42 3.46 3.12 2.92 2.78
1741.12 910.58 693.66 596.75 542.36
500
5 10 15 20 25
4.42 4.42 4.42 4.42 4.42
68.99 40.29 27.21 17.19 15.06
3.64 3.64 3.64 3.64 3.64
19.86 17.75 15.48 14.76 14.41
2.92 2.87 2.87 2.87 2.87
16.85 15.07 11.83 10.71 8.65
3.81 2.45 2.29 2.24 2.24
14.67 12.57 8.91 7.45 6.23
G. Vaidyanathan et al. / Journal of Magnetism and Magnetic Materials 176 (1997) 321 330
328
mechanism is very large, when compared to bouyancy effect [7]. For such fluids, M 2 is assumed to have negligible value and hence taken to be zero [14]. M3 is varied from 1 to 25 because M3 cannot take a value less than one [9]. m 6 is taken to be 0.1. M 4 is the effect on magnetisation due to salinity. This is allowed to vary from 0.1 to 0.5 taking values less than the magnetisation parameter M3. M5 represents the ratio of the salinity effect on magnetic field and pyromagnetic coefficient.This is varied between 0.1 and 0.5. r is the ratio of mass diffusivity to thermal diffusivity which is varied between 0 and 0.1 [10, 14]. The salinity Rayleigh number Rs is varied from - 5 0 0 to 500. The increase in magnetisation (M3) is found to cause large destabilisation, because both magnetic and thermal mechanism favour destabilisation. This can be observed from Tables 1 and 2 in which the increase in M3, decreases No. The addition of the magnetic salt increases magnetic properties of multicomponent fluids, which in turn interacts with temperature and applied magnetic field. The effect of the temperature is very much pronounced on the magnetic field. The addition of salt and increase in temperature tend to oppose each other. Therefore, increase in M5 shows destabilising effect, which is obvious from Tables 1 and 2. The increase in M5 decreases N~. If a magnetic salt of larger magnetic moment is dissolved in the solvent, it increases the magnetisation which in turn promotes instability. As the ratio of mass transport to heat transport (~) is increased, the onset of instability is favoured, as Nc decreases with increase of r (Tables 1 and 2 and in Figs. 1 3). From Tables 1 and 2 it is seen that as salinity Rayleigh number is negatively increased (stably stratified or salted from below), the thermal Rayleigh number N~ ( = RM1)c is found to increase. This means that the system is getting stabilised. This would imply that the solution is denser at the bottom than at the top, which favours stability. From Tables 1 and 2 it is seen that the positive increase in salinity Rayleigh number (Rs) decreases No. The positive increase in salinity Rayleigh number would mean that the system is salted from above (unstable stratification). This makes the upper layer heavier than the lower. This leads to early onset of convective instability. From Fig. 3 it is seen that for all values o f t and that there is no Change on destabilisation, because the system is completely destabilised even for a very small temperature gradient very close to zero. One may conclude that the system can be destabilised purely by salinity gradient itself without the large value of
R s = - 500:
I] M 3 = 5
M 4 = 0.1;
× M 3 = 15
M 5 = 0.1;
o M3 =
25
z£ Nc
1
0.03
|
I
I
0.05
0.07
0.09
Fig. 1. V a r i a t i o n of No versus r for M5 = 0.1.
0.11
G. Vaidyanathan et al. /Journal c f Magnetism and Magnetic Materials 176 (1997) 321-330 13
11
R~ = -500 ;
~ M3= 5
M4=0.1 ;
× M3=15
M 5=0.5;
o M3=25
M 6 = 0.1. 9
5
3
1
o
0.03
0.05
0.09
0.07
Fig. 2. V a r i a t i o n of N~ versus ~ for M5 = 0.5.
N¢ x 10 -3 M 5 =0.1 4.0
3.5
/'~=0.05
I:
~-~ 3.0
M 3 = 25
2.0
0.5
-400
-300
-200
-100
0
100
200
300
Fig. 3. V a r i a t i o n of Nc versus r for R~ for M5 = 0.1.
400
~, R~
329
330
G. Vaidyanathan et al. /Journal o['Magnetism and Magnetic Materials 176 (1997) 321 330
magnetisation and for any value of ratio of salinity to mass diffusivity. This is the rare phenomenon where the system can favour convection by salinity concentration itself.
Acknowledgements The last author is grateful to the University Grants Commission for extending the financial facilities for partly meeting out the expenditure to carry out this work. The authors are grateful to the referees for their valuable suggestions and comments but for this, the paper would not be in the present form.
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