Magnetic Benard convection under microgravity

0273—1177/91 $0.00+ .50 Copyright C 1991 COSPAR

Adv. 5oaceRes. VOL 11,No. 7,pp.(7)251—(7)254, 1991 Printed in Great Britain. All rights reserved.

MAGNETIC BENARD CONVECTION UNDER MICROGRAVITY W. V. Hörsten, S. Odenbach and K. Stierstadt Universitat MOnchen, Sektion Physi/c Schellingstrasse 4, D-8000 München 40, Germany

ABSTRACT

We have performed an experiment to investigate the onset and the flow pattern of magnetic Bénard convection under microgravity. Contrary to ground based Bénard convection, gravity and the density gradient are replaced by magnetic forces and a gradient of magnetization respectively. The experiment is carried out with a ferrofluid (a suspension of small magnetite particles) in a gap between two concentric cylinders heated from the axis and cooled on the outside. Periodic boundary conditions not feasible on earth are established in this experiment. The investigations are carried out during the approximatly 360 seconds of microgravity time in a TEXUS sounding rocket flight. Magnetic convection is clearly observed but the flow profile does not agree with what had been expected. INTRODUCTION

The subject of our experiment is the magnetic Bénard convection under microgravity. In a horizontal fluid layer under the influence of gravity, with the bottom of the layer heated and the upper side cooled, one observes normal Bénard convection /1,2/. The temperature difference between the top and the bottom of the fluid gives rise to a density gradient. Due to the interaction between gravity and this density gradient a volume element of the fluid displaced parallel or antiparallel to gravity feels a force in the direction of its displacement. If the temperature difference exceeds a critical value the fluid layer becomes unstable and convection rolls appeare. The resulting flow pattern depends strongly on the boundary conditions and the geometry of the convection cell. Under microgravity it should be possible to establish periodic boundary conditions wich are more suitable for theoretical investigations. We realised periodic conditions by a ferrofluid between two concentric cylinders under the influence of a radialmagnetic force. In this case the interaction of gravity and a density gradient is replaced by a similar interplay between a magnetic force and a gradient in magnetization. A ferrofluid is a suspension of very small magnetite particles (typical diameter 10 nm) in an appropriate carrier fluid. The magnetite particles are covered with a surfactant to prevent agglomeration. Such fluids show liquid behaviour coupled with superparamagnetic qualities /3/. ORIGIN OF THE RADIAL FORCE

We will explain the origin of the radial magnetic force in a ferrofluid between two concentric cylinders (fig.1). The inner cylinder is heated to a temperature T 1 while the outer one is cooled to T2 < T1. This temperature difference gives rise to a radial gradient of magnetization in the ferrofluid. In addition a circular magnetic field is applied by a currentleading wire in the cylinder axis. Such a field has a radial gradient of azimuthal fieldstrength Hp. The gradient of azimuthal magnetization Mp in this field caused by the temperature difference is antiperalle]. to the fieldgradient. If a volume element with large magnetization Mcp is displaced adiabatically in the direction of the fieldgradient it will be surrounded by hotter fluid with lower magnetization Mc - t~Mcp (see fig.1). This gives rise to a force ~r on the volume element AV due to the gradient of the field Hç. The resulting force ~r is given by the difference between the force on IN and the force on its surroundings: -

~s

i. t~M~ (aHc/ar) t~V ~r

(7)251

•

(1)

W. V. HOrsten eta!.

(7)252

The gradient of the magnetic field is antiparallel to the unit vector in radial direction in cylindrical coordinates ~ aHq~/ar< 0. Equation (1) yields a force in the direction of the fieldgradient. Alternatively one may displace the volume element antiparallel to the fieldgradient and get a resulting force also antiparallel to it. The resulting force always acts in the same direction as the displacement and causes magnetic buoyancy.

OUter

Fig.1: The origin of the destabilizing magnetic force. If a volume element with a Certain magnetization Mc (azimuthal arrows) is displaced adiabatically in radial direction it will be surrounded by fluid with a different magnitude of magnetization. Consequently in the applied fieldgradient VHrp a radial force ~r acts on the particle. Due to this destabilizing force a disturbation in the fluid can grow but it is Opposed viscous effects and by thermal diffusivity. For a real destabilization of the fluid temperature difference or the fieldgradient must exceed certain critical values. situation of the system is characterized by the magnetic Rayleigh number R 5 /4,5/. 3 Rm Ii. KG~Td

by the The (2)

Where K is the negative derivative -aMy/aT of the magnetization by the temperature, G the magnetic fieldgradient, j~ the permeability of free space, T the temperature difference. d the thickness of the fluid layer, ~ the dynamic viscosity and K the thermometric conductivity of the ferrofluid. If the dimensionless ratio R 5 exceeds a critcal value depending on the boundary conditions the fluid layer becomes unstable. EXPERIMENTAL SETUP

The aim of our experiment is to study the onset of convection in the absence of gravity by varying the azimuthal magnetic fieldstrength. In addition we wanted to investigate the resulting flow pattern by measuring the temperature distribution on the inner surface of the outer cylinder. The convection cells have the form of two hollow—cylindrical containers with different diameters (see fig.2 a). The containers are filled with ferrofluid and they are heated from the center and cooled at the outer surface of the outer cylinder. It is possible to apply a magnetic field in the direction of the axis and another one in circular direction. The axial field is used to align the expected convection rolls along the cylinder axis (see /4/). The circular field yields the radial fieldgradient and the azimuthal component of magnetization. In this geometry we expected to find a flow profile like that shown in fig.2 b. The investigation of the flow profile is realised by measuring the temperature distribution on the inner surface of the outer cylinder by means of 224 microthermistors. The thermistors are embedded in small glass tubes of about 0.5 millimeter diameter. They are arranged along two circular lines and one axial line on the outer wall of each container. The dimensions of the two containers are given in table 1. TABLE 1

Dimensions of the Containers length of radius of Container 1 : radius of gap width Container 2 radius of gap width

each container inner cylinder outer

cylinder

outer cylinder

200.0 mm 10.0 mm 25.5

mm

15.5 nun 22.5 nun 12.5 nun

Magnetic Bénard Convection

(a)

ø45n1Tl

(7)253

(b)

4~111~~ ‘t..-020 I mm 051mm

Fig.2: a) Convection cells with inner (1) and outer (2) cylinder; axial (3) and azimuthal (4) field coils and the ferrofluid (5) between the cylinders. b) Cross section with the expected convective pattern.

RESULTS OF THE ig TEST

The whole setup was controlled in a lg test. During this test the module was positioned with the cylinder axis horizontal, i.e. perpendicular to gravity. In this case the expected flow profile is shriwn in fig.3 a. Hot fluid rises from the inner to the outer cylinder at the top while cold fluid streams towards the inner cylinder at the bottom. The measured temperature profile is shown in fig.3 b. The temperature resolution is better than 0.01 K.

(a) thermi tors 20

(b)

‘~ 1

0

0

i’o ~o 310 4~0 number of thermistor

Fig.3: a) Expected flow profile and b) measured temperature distribution during the ig test. PRELIMINARY RESULTS FROM THE TEXUS-25 jig EXPERIMENT was flown in the TEXUS-25 campaign in spring 1990. During the 351 seconds of pg time the azimuthal magnetic field was increased from 0 A/m to 1.5 kA/m in 15 steps. Data acquisition for all microthermistors was taken one times at each step. After changing the fieldstrength a delay time of about 4 seconds to the next data acquisition was observed to allow the flow pattern to stabilize.

The experiment

++.

.~

13.7

~136 13.5

+ +++ +4*4~4-

.~

+14+14.

~~+++~ +

r~imberof thermistor

1S7 155

~

+

15.3

6i22b~e36 rumber of thermistor

Fig.4: Temperature distribution along a circular line (a) and the axial line (b) of microthermistors under microgravity. Insets: possible flow patterns corresponding to the measured temperature profiles.

w.V. Horsten eta!.

(7)254

An example of the first preliminary results is shown in fig.4, the temperature distributions of the axial line and one of the two circular lines of thermistors of the larger container (fig.2). The axial fieldstrength is about 6 kA/m while the circular field was 0.6 kA/m in 2. The temperature the

middle

between

the

cylinders

with

a

gradient

of

about

34

khi/m

difference is 20 K. Only two longitudinal convection rolls are established instead of the eight expected. In addition one pair of azimuthal rolls is observed at the end of the cylinder. The amplitudes of the rolls increase proportional to the azimuthal magnetic fieldstrength. The described results will be subject to further investigation. ACKNOWLEDGEMENTS The TEXUS microgravity experiment was supported by BMFT and DLR. The experiment built by the MBB ERNO TEXtJS team. Dr. Weilner from Bayer AG Leverkusen supplied

module was a special

varnish to protect the inner surface of the containers from the ferrofluid. Dr. L. Schwab of the University of Munich has proposed this experiment and has made many valuable contributions to its realization. We thank all institutions and their representatives for their valuable support wich made this research possible. REFERENCES

1. M. Velarde, C. Normand, Convection, Sc

nUi~A.~. ~ji~jj~

243, 78 (1980)

2. F.M. Busse, Transition to Turbulence in Rayleigh-Bénard

Convection,

ln.~ta~LVtLe.~ and the. Txan~t~onto Tu.’thule.nce., 3.P. Gollub,

3. R.E. Rosensweig,

Topics

in applied

Fe.nAohydkodynanuc4,

4. L. Schwab, KOnvJzt~oR .~tFe

physics 45, Springer

Cambridge

University

o~u.4de.n,Dissertation

Press,

in

Hydiwdynanuc

ed. H .L. Swinney and 1981 Cambridge

1985

LMU MGnchen, MOnchen 1989

5. B.A. Finlayson, Cony. Instab. of Ferromagnetic Fluids, 3. FZ.wLd Me.ch. 40, 753 (1970)