thermal convection in a ferrofluid supported by thermodiffusion _odenbach(2005)

8/19/2019 Thermal Convection in a Ferrofluid Supported by Thermodiffusion _Odenbach(2005)

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Journal of Magnetism and Magnetic Materials 289 (2005) 122–125

Thermal convection in a ferrofluid supported

by thermodiffusion

S. Odenbach, Th. Vo ¨ lker

ZARM, University of Bremen, Am Fallturm, D-28359 Bremen, Germany

Available online 26 November 2004

Abstract

Convective motion has been investigated in a ferrofluid layer heated from below. In contrast to normal experiments

on thermal convection in fluids, the focus of this investigation was devoted to the destabilizing effect of the density

redistribution in the fluid forced by thermal diffusion of the magnetic particles relative to the carrier liquid. It could be

shown that the fluid layer is destabilized at a temperature difference well below the critical limit for a one-component

fluid. It was found that the convective flow is stable in time and the measured convection amplitude could be fitted with

a theory given by Hollinger et al. (Phys. Rev. E 57 (1997) 4).

r 2004 Elsevier B.V. All rights reserved.

PACS: 75.50.Mm; 47.27.i; 44.90.+c

Keywords: Soret effect; Thermal convection

1. Introduction

The phenomenon of thermal convection appearing in

a flat fluid layer heated from below and cooled at the

upper surface is well known from the literature since

more than 100 years [2]. The driving force for the

flow can easily be understood if one remembers that

the temperature gradient causes a density gradient in the

fluid (see Fig. 1). If it is now assumed that a volume

element of the fluid is displaced adiabatically in the

direction of the density gradient, it will experience aresulting body force in the direction of the displacement

due to the buoyancy in the gravitational field. Equally a

displacement in the opposite direction will also cause a

resulting force in the direction of the displacement, and

it is thus clear that the buoyancy has destabilizing

character for the fluid layer since it can amplify

stochastic disturbance of the stratification. A convective

flow will set in as soon as the destabilizing buoyant force

will overcome the stabilizing effects of viscous friction

and thermal conductivity in the fluid. The actual

situation of the system is usually described by a

dimensionless parameter, the Rayleigh number Ra:

Ra ¼ bTrgDTd 3

kZ

; (1)

where bT denotes the thermal expansion coefficient, r

the density, Z the dynamic viscosity, x the thermometric

conductivity of the fluid, DT the temperature difference

between the plates, d their spatial distance, and g the

gravitational acceleration. Convection appears if the

actual Rayleigh number exceeds a certain critical value

Ra which depends on the boundary conditions.

Since, in an experiment with a given geometry and a

fluid with fixed properties, the driving force for the

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doi:10.1016/j.jmmm.2004.11.036

Corresponding author. Tel.: +49 421 2184785;

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E-mail address: [email protected]

(S. Odenbach).

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convective flow is only determined by the temperature

difference applied between the bounding plates, thecritical Rayleigh number gives directly a critical

temperature difference necessary to drive the convec-

tion. This critical temperature difference DT is equiva-

lent to a critical density difference Dr:In a binary mixture the temperature gradient will not

only lead to a density gradient due to thermal expansion

of the fluid, but also to a separation of the components

due to thermodiffusion. This separation will create an

additional density difference in the fluid which has to be

combined with the one driven by thermal expansion. If

the Soret coefficient S T —describing the thermodiffusive

demixing of the fluid—is positive, the material transport

is directed towards the cold plate of the arrangement.

Thus, assuming that the component transported by the

Soret effect has a higher density than the surrounding

medium—as it is obviously true for a ferrofluid where

the magnetic particles are moving with respect to the

carrier liquid—the thermodiffusive transport will en-

hance the density gradient established by the tempera-

ture difference in the fluid. This can lead to a situation in

which the temperature difference is smaller than the

critical one for the one-component fluid in a certain

arrangement DT 0 but the density gradient is large

enough to drive convection. The relative influence of

thermal and concentrational density changes is de-scribed by the separation ratio c ¼ S Tbc=bT; where bc

denotes the concentrational expansion coefficient. The

phenomenon of convective flow driven by thermodiffu-

sion is well known from molecular mixtures where the

diffusion coefficient is large—leading to a fast demixing

of the system—while the Soret coefficient is small,

producing only small concentration differences and

therefore also small density differences in the fluid.

For such liquids the separation ratio gains values in the

order of c 0:1: In contrast, the Soret coefficient has

been found to be very large in ferrofluids [3] where S T ¼

0:15 K1

has been measured. Together with the great

density difference between the particles and the carrier

liquid, this leads to high separation ratios in the range

order of 100–1000. For such high values of c;remarkably large convection amplitudes have been

predicted in Refs. [1] and [4].

2. Experimental setup

To investigate Soret-driven convection in ferrofluids,

we have set up an experiment consisting of a flat layer of

magnetic fluid with a diameter of 150 mm and a fluid

layer thickness of 5 mm. The layer is cooled at the top

and heated from the bottom by means of water loops

yielding a temperature stability of the bounding plates of

0.01K. The detection of the thermal flow in the fluid is

carried out by a set of temperature probes located at the

cold side of the fluid layer. The temperature probes—so-

called microthermistors—have a mean diameter of 0.5 mm and allow a resolution of the temperature

measurement of approximately 1 mK. Thirteen of these

probes have been arranged in cross form in the center of

the upper bounding plate.

Measuring the temperature distribution along the

lines spanned by the microthermistors, one can obtain

the convection amplitude a from the sinoidal tempera-

ture signal generated by the fluid flow (see Fig. 2).

The ferrofluid used in the experiments is a commercial

fluid APG516A from Ferrotec containing 2 vol% of

magnetite in oil. The Soret coefficient of this ferrofluid

has been determined to be S T ¼ 0:16 K

1

: In theexperiments, the fluid has been mixed carefully before

the start of the experiment. Afterwards, both plates have

been set to a common mean temperature of 25C: This

temperature has been kept constant for 2 h to allow

equilibration of the system. Afterwards the temperature

difference has been established by a stepwise, symmetric

temperature change on both bounding plates. The

temperature profile measured by means of the micro-

thermistors has been monitored until the equilibrium

situation has been reached. From this final profile the

amplitude of the convective flow can be estimated and

thus, measuring temperature profiles for various tem-

perature differences between the plates, one can

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Fig. 1. Principle sketch of the experimental situation for an

explanation of the convective driving force.

Fig. 2. Sketch for the explanation of the technique used to

measure the convection amplitude.

S. Odenbach, Th. Vo lker / Journal of Magnetism and Magnetic Materials 289 (2005) 122–125 123



determine the dependence of the convection amplitude

on the Rayleigh number.

3. Experimental results

Fig. 3 shows an example of temperature distribution

along one of the lines of thermistors measured for a

Rayleigh number larger than Ra0 : Here Ra

0 denotes the

critical Rayleigh number for the experimental setup and

a one-component fluid. It should be noted that the

wavelength of the temperature distribution does not

equal two times the thickness of the fluid layer. This is a

result of the fact that the convection rolls have a finite

angle with the thermistor line. Taking the results of both

lines together, one finds a wavelength of the convective

flow of ð10:1 0:2Þ mm which fits well to the fluid layer

thickness of 5 mm.

Measuring these temperature distributions for varioustemperature differences and plotting the square of the

amplitude of the distribution against DT (see Fig. 4)

provides the possibility to determine the critical tem-

perature difference DT 0 and thus the critical Rayleigh

number Ra0: This is possible since the influence of the

Soret effect on the convective flow is small well above

Ra0: Thus the normal linear extrapolation of a2ðDT Þ to

a2 ¼ 0 can be used to find DT 0 ¼ ð5:0 0:2Þ K for the

system investigated here.

It is obvious from Fig. 4 that a significant increase of

the convection amplitude appears for temperature

difference below or close to D

T

0 —a fact clearlyindicating the presence of a contribution of the Soret

effect to the destabilizing forces driving the convective

flow.

To compare the measured effects with the theory in

Ref. [1], Fig. 5 shows the amplitude of convection—

measured over a wide range of temperature differ-

ences—as a function of the reduced Rayleigh number ;which is defined by

¼ Ra Ra

0

Ra0

: (2)

In the figure the dotted line represents the typical

square root increase of the amplitude as is expected for a

one-component fluid. Again it is clearly observed that a

strong convective flow appears below DT 0 : The solid

line is a fit of the theoretical dependence of the

amplitude on as is given in Ref. [1]. The free fit

parameter is the separation ratio c which is determined

from the fit to be c ¼ 1850 100: It should be

noted that long-term measurements have shown that

the convective flow is stable over time and that

no oscillations of the convection amplitude can be

observed.

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Fig. 3. Temperature distribution measured along one of the

lines of thermistors for a temperature difference of 9 K between

the upper and lower bounding plate of the fluid layer.

Fig. 4. Amplitude of the temperature distribution as a function

of the temperature difference. The solid line is a fit to the

experimental data for values of DT above 7.5K.

Fig. 5. Amplitude of convection as a function of with a fit of

the theory in Ref. [1].

S. Odenbach, Th. Vo lker / Journal of Magnetism and Magnetic Materials 289 (2005) 122–125124



4. Conclusion and outlook

The influence of the Soret effect on thermal convec-

tion in magnetic fluids has been investigated. From a

comparison between the experimental data and the

theory in Ref. [1], it has been found that the separation

ratio in the fluid investigated here takes a high value of

c ¼ 1850: This opens for the future the possibility to

perform an experimental proof of the square roll flow

pattern which has been predicted in Ref. [1]. Further-

more, the influence of magnetic field on the Soret effect

as well as on the convection itself opens a wide field for

new investigations.

Acknowledgment

The authors are thankful to Prof. M. Lu ¨ cke for

inspiring discussions.

References

[1] S. Hollinger, M. Lu ¨ cke, H. Mu ¨ ller, Phys. Rev. E 57 (1997) 4.

[2] H. Be ´ nard, Revue gen. des Science, 1900.

[3] Th. Vo ¨ lker, E. Blums, S. Odenbach, Magnetohydro-

dynamics 37 (2001) 3.

[4] A. Ryskin, H. Mu ¨ ller, H. Pleiner, Magnetohydrodynamics

39 (2003) 51.

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S. Odenbach, Th. Vo lker / Journal of Magnetism and Magnetic Materials 289 (2005) 122–125 125

thermal convection in a ferrofluid supported by thermodiffusion _odenbach(2005)

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