Thermal convection in a magnetic fluid in an annular Hele-Shaw cell

Journal of M a g n e t i s m and Magnetic Materials 122 (1993) 319-322 North-Holland

Thermal convection in a magnetic fluid in an annular H e l e - S h a w cell

S. Aniss, J.P. B r a n c h e r a n d M. S o u h a r

L E M T A - E N S E M - U R A 875 CNRS, 54504 Vandoeucre les Nancy Cedex, France

This p a p e r deals with an experimental investigation of thermoconvective instability in a magnetic fluid. T h e onset of horizontal R a y l e i g h - B e n a r d convection in ferrofluids, and its d e p e n d e n c e on magnetic field gradient, are measured.

1. Introduction

T h e natural convection in a magnetic fluid is a new technique that allows the control of heat transfer. This p r o b l e m has led to several recent works [1-5]. T h e fluid motion is driven by mag- netic volume forces I~oMVH, where ~0 is the permeability of the vacuum, M is the magnetiza- tion of the liquid and H is the modulus of the total magnetic field. These forces depend on the thermal state of the fluid because M = M(T, H), where T is the t e m p e r a t u r e . We consider a sym- metrical model in which the magnetization is parallel to the magnetic field.

O n e of the problems most studied in this do- main is the R a y l e i g h - B e n a r d thermoconvective instability. Two situations can occur: an external uniform magnetic field, applied to a layer of magnetic fluid subjected to a t e m p e r a t u r e gradi- ent, produces a magnetic field gradient. T h e r m o - convective instability therefore occurs by the ap- p e a r a n c e of R a y l e i g h - B e n a r d cells, when a cer- tain Rayleigh n u m b e r is reached. This case was examined in ref. [1] and m o r e recently in ref. [4].

A n o t h e r interesting case that has not yet been studied, as far as we know, consists of applying an external magnetic field of constant gradient, par- allel to a t e m p e r a t u r e gradient, in a horizontal H e l e - S h a w cell configuration (fig. la). In this situation the gravitational forces can be ne- glected, and we take into account only the mag-

Correspondence to: D r S. Aniss, L E M T A - E N S E M - U R A 875 CNRS, 2 A v e n u e de la For~t de Haye, BP 160, 54504 Van- doeuvre les Nancy Cedex, France.

netic forces. It is therefore possible to simulate micro- or macrogravity in this situation. This pa- per presents an experimental contribution.

2. Problem formulation

T h e magnetic liquid is contained in an annular horizontal H e l e - S h a w cell (fig. la): R 1 < r < R 2 , 0 < z < e where E<<R~, and r, 0 and z are the cylindrical polar coordinates. The inner side (r = R~) and the outer one (r = R 2) are conductors

and are maintained at t e m p e r a t u r e s T~ and T 2 (T~ > T 2) respectively. The two horizontal faces are rigid and are perfect insulators. The cavity is placed in an applied magnetic field H~, as shown in fig. 1. Since E <<R~, we consider that the radial magnetic field gradient acting on the layer is nearly constant and parallel to the t e m p e r a t u r e gradient. This field gradient creates an induced field, h~ = - M through the liquid layer so that the total field in the magnetic liquid is H = H a - M.

At the onset of convection, the equation sys- tem governing the flow derives from the N a v i e r - Stokes equations, the Boussinesq approximation, and the energy equation where the magne- tocalorific effect can be neglected, since the working t e m p e r a t u r e is well below the Curie tem- perature. Viscous dissipation is also neglected, and we consider the case where M = x H .

Making the approximation AR <<R~, where AR = R 2 - R~, the p r o b l e m is applied to the case of an infinitely thin cavity. The theoretical study [6] of linear stability is carried out by asymptotic

I)304-8853/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

320 S. Aniss et al. / Thermal convection in a magnetic f l u i d

analysis, and gives a critical magnetic Rayleigh n u m b e r defined by:

Ram* ⁼ t%K A T AR31VH,, I e*:,

Poua( 1 + X )

where Ram* = 48"rr 2 in the presence of free boundaries, and Ram* = 825.6 in the presence of rigid b o u n d a r i e s (see [7,8]). T h e cell aspect ratio E * = E / A R , Po the density, p the kinematic vis-

cosity, a the thermal diffusivity, the differential magnetic susceptibility X - (~)M/~)H),r, the pyro- magnetic coefficient K = - ( a M / a T ) u, and AT T 1

T 2 .

3. E x p e r i m e n t a l a p p a r a t u s

T h e H e l e - S h a w cell (fig. lc) comprises two a n n u l a r parallel t r a n s p a r e n t Plexiglass plates (5).

f

Ferrofluid

H a

Ho H1

R]

"x.

I I I

L - -

4 .I ^R2

(a) (b)

CL

,i

R~=105 m m

R2=125 m m

, l U l l m m'$~t 1'2

AR=20mm

- 2 ( Q

Lt~ E

eq

R:

tl

T T T T

l DetaiI.A

R]

T

(c)

Fig. 1. The H e l e - S h a w cell configuration. (a) Ferrofluid crown in an external magnetic field with constant gradienl. (b) Electro Aimant. (c) Magnet Experimental set-up. 1-1', circulation of heated and cooled water; 2, ferrofluid: 3 - 3 ' , two amlular copper

plates; 4, joint; 5, Plexiglass plates. Detail A: five azimuthal thermocouples at the centre of the cell, distance (ram).

S. A n i s s et al. / T h e r m a l concection in a m a g n e t i c f l u i d 321

A slot ( w i d t h A R = 20 mm, t h i c k n e s s e / 2 = 2 m m ) was m a c h i n e d o n e a c h Plexiglass p l a t e to h o l d t h e a n n u l a r cavity, with r e c t a n g u l a r s e c t i o n a n d t h i c k n e s s e = 4 mm, e n c l o s i n g t h e f e r r o f l u i d . A n n u l a r p i p e s (1, 1'), with r e c t a n g u l a r section, allow t h e c i r c u l a t i o n o f h e a t e d a n d c o o l e d w a t e r to m a i n t a i n t h e a n n u l a r c o p p e r p l a t e s (3, 3 ' ) at c o n s t a n t t e m p e r a t u r e . T h e two a n n u l a r c o p p e r p l a t e s a r e i n s e r t e d i n t o g r o o v e s m a c h i n e d on e a c h Plexiglass p l a t e . T h e w a t e r t i g h t n e s s is as- s u r e d by toric seals (4) which a r e p l a c e d in grooves. E a c h c o p p e r p l a t e is 4 m m thick a n d 7 m m high. T o m a i n t a i n t h e i d e a l c o n d i t i o n o f a d i a b a t i c h o r i z o n t a l p l a t e s , t h e t h i c k n e s s o f Plexi- glass p l a t e s is c h o s e n a b o u t 22.5 mm. T h e fer- r o f l u i d (2) is t h e r e f o r e c o o l e d f r o m t h e i n n e r c o p p e r s u r f a c e a n d h e a t e d f r o m t h e e x t e r n a l one.

W a t e r t e m p e r a t u r e is c o n t r o l l e d with e i g h t C r - A I t h e r m o c o u p l e s m o u n t e d flush with t h e e x t e r n a l c o p p e r surface.

W e use t h e E l e c t r o A i m a n t t y p e m a g n e t E A F - 16-NC (fig. l b ) . N e a r t h e e d g e o f t h e g a p b e - t w e e n R i = 105 m m a n d R e = 150 mm, t h e m a g - netic field g r a d i e n t is c o n s t a n t (fig. 2) a n d v a r i e s l i n e a r l y with t h e coil c u r r e n t I (fig. 3). T h i s c a l i b r a t i o n was c a r r i e d o u t by a g a u s s m e t e r t y p e M G - 5 D , w h i c h o p e r a t e s on t h e H a l l effect p r i n c i - ple. F o r t h e c o n f i g u r a t i o n o f fig. l(a), t h e fer- r o f l u i d c h a n n e l is p l a c e d at t h e e d g e o f t h e gap.

T h e m a g n e t i c fluid u s e d was an E M G 905-type light h y d r o c a r b o n - b a s e d f e r r o f l u i d ( t h e s a m e as t h a t u s e d by Stiles [4]), w h e r e t h e m a g n e t i c sus- c e p t i b i l i t y X = 1, s a t u r a t i o n m a g n e t i z a t i o n M S = 3 . 1 8 × 1 0 4 A / m , d e n s i t y p = 1 . 2 6 × 1 0 3 k g / m 3, d y n a m i c a l viscosity # = 8.4 × 10 3 kg m - ~ s - ~ ,

200

100

Ha/~u~

(Gauss)

----x.

r(mm)

0 i i

0 100 200

Fig. 2. Leakage field for coil current 1 = 3 A.

5000 4000 3000 2000 1000

IGrad Hal/uo (Gauss~m)

I (A/m)

• I I I " I " | ' l " I

0 1 2 3 4 5 6 7

Fig. 3. Calibration curve in the zone R I < r < R~.

t h e r m a l diffusivity a = 8.6 × 10 s m 2 / s , pyro- m a g n e t i c c o e f f i c i e n t K ~ 27.3 A m - 1 K - 1 [4].

4. Experimental results and discussion

In o r d e r to m e a s u r e t h e critical R a y l e i g h n u m - b e r at t h e o n s e t o f c o n v e c t i o n , five t h e r m o c o u - pies w e r e e m b e d d e d in t h e c e n t e r o f t h e c h a n n e l to o b t a i n t h e a z i m u t h a l t e m p e r a t u r e , as shown in fig. l(c), d e t a i l A . In t h e b a s e s t a t e , t h e t e m p e r a - t u r e d i s t r i b u t i o n c o r r e s p o n d s to a c o n d u c t i v e r e g i m e . W h e n c o n v e c t i o n occurs, t h e t e m p e r a - t u r e field o b e y s a p e r i o d i c law [8]. T h e c o n v e c t i o n is also v i s u a l i z e d t h r o u g h the t r a n s p a r e n t hori- z o n t a l p l a t e s using liquid crystals t h a t exhibit b r i l l i a n t c o l o r c h a n g e s a c c o r d i n g to c h a n g e s in t e m p e r a t u r e . T h e o n s e t o f c o n v e c t i o n was de- f i n e d w h e n t h e i s o t h e r m s o f the s t a t e o f p u r e h e a t c o n d u c t i o n w e r e slightly d e f o r m e d to a w a w s h a p e , i n d i c a t i n g t h e a p p e a r a n c e o f r a d i a l flow c o m p o n e n t s a n d t h e d e v e l o p m e n t o f a r e g u l a r p a t t e r n o f cells (fig. 4).

T2

m

T| I

R1 [

..~ -AR -- i

R2

(a) Ib~

Fig. 4. Visualization of isotherms using liquid crystals by taking a color photo. (a) Conduction regime VII,, = 0; (b) convective regime response to the onset of instability, VH,, _>

VH~, critical.

322 S, Aniss et al. / Thermal convection in a magnetic fluid

Table 1

Experimental results

A T ( K ) I t"H,, [ I ~'H,, I'~,T Ram*

( × 1 0 4 A / m 2) ( × 1 0 a A / m 2 ) . k )

9.4 6.57 61.7 4936

7.5 7.95 59.6 4768

6.2 1.03 63.8 5014

5.0 1.60 64.1) 5146

4.7 1.45 68.1 5448

The onset of convection in the fluid layer within the gap was determined by increasing or decreasing the t e m p e r a t u r e at the inner or exter- nal copper plate, in small steps, and by control- ling the field gradient, in order to reach marginal state. It was approached repeatedly by increasing the t e m p e r a t u r e difference AT between the two interface f e r r o f l u i d - c o p p e r plates, from the state of heat conduction, and by decreasing AT start- ing from a state of slow convective motion. The experimental results are summarized in table 1.

Note that in these runs the values of H~ were smaller than 201.) G, corresponding to the case M = x H , and that the values of VH~,. AT remains almost constant and exhibit a relative dispersion of less than _+6%. Thus it is possible to deter- mine precisely Ram* when the physical proper- ties are well known. This is very important since this experiment represents a great improvement on the classical method for m e a s u r e m e n t s of Ra*, where the dispersion can reach +_30% [8]. We have also calculated Ram* using the physical properties given in ref. [4], but the result was disappointing in that the experimental values of Ra* were 5 - 6 times the theoretical value. These discrepancies were most likely due to:

1) the imprecision of the values of K, u and a which were not measured, and

2) the cell curvature ratio AR/R~ was ne-

glected in our Ram* theoretical approach. These points will be reconsidered in the future.

The values of the Rayleigh numbers depend upon e / A R = 0.2 and the ratio of the thermal conductivities of the material and the fluid [7]. In our H e l e - S h a w cell, which is bounded by high- conductivity Plexiglass plates with AR2/a>>

AR2/ap, where ap is the thermal diffusivity of the Plexiglass material, the conductive heat trans- fer is much faster in the Plexiglass plates than in the fluid. Thus heat is normally transferred through the Plexiglass plates into the fluid, and the t e m p e r a t u r e profile in the plates is main- tained almost linear at any given time during slight transient heating processes. Since the con- dition (0.5E)2/a << ( A R)/a p holds, a quasi-steady heating rate can easily be achieved in these cells.

5. Conclusion

The onset of horizontal R a y l e i g h - B e n a r d in- stabilities has been detected in this experimental study. Therefore, Ram* can be determined with good accuracy, provided that the values of the physical quantities that a p p e a r in the definition of Ram* are known.

References

[I] B.A. Finlayson, J. Fluid Mech. 41) (1970) 753.

[2] V.G Bashtovoy, B.M. Berkovsky and A.N. Vislovich, ln- troductkm to the Thermomechnics of Magnetic Fluids (Hemisphere, New York, 1988).

[3] B.M. Berkovsky, V.E. Fertman, V.E. Polevikov and S.V.

Isaev, lnt. J. Heat Mass Transfer 19 11976) 981.

[4] P.J. Stiles and M. Kagan, J. Magn. Magn. Mater. 85 (1991)) 196.

[5] k. Shwab, J. Magn. Magn. Mater. 85 (19911) 199.

[6] S. Aniss, J.P. Brancher and M. Souhar, C.R. Acad. Sci.

(Paris) (submitted).

17] J.N. KosteL Int. J. tteat Mass Transfer, 25 11982) 426.

[8] M. Souhar, J.P. Brancher and S. Aniss, lnt. J. Ileal Mass

Transfer 35 11992) 749.