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Proceedings of ICHIT- 06

26 February - 5 March 2006, Moscow, Russia




ON WAVE REGIMES IN FERROFLUID CONVECTION



A. Bozhko1, G. Putin1, T. TynjДlД2

1 Perm State University, Perm, Russia, 614990
2 Lappeenranta University of Technology, Lappeenranta, Finland, 53851
bozhko@psu.ru


Abstract
The investigation of Rayleigh convection in a thin cylindrical layer has
been conducted for a ferrofluid containing magnetite single domain
particles suspended in kerosene carrier liquid. Near the onset of
convection the wave oscillatory convection was observed in experiments and
numerical simulations using a two-phase mixture model. The influence of a
homogeneous longitudinal magnetic field on the convective instability and
structure of flows has been studied for horizontal and inclined
orientations of the layer. The most fascinating effect in real ferrofluid
convection is spontaneous formation of localized states, those where the
convection chaotically focuses in confined regions and is absent in
remainder of cavity.

introduction
By tradition, under consideration of ferrofluid convection are taken into
account only temperature induced driving mechanisms such as buoyancy,
magnetic and thermodiffusion [1-3]. The experiments [4, 5] in part shown
that at terrestrial conditions the heat-mass transfer in magnetic colloids
is essentially complicated for the most part because of uncontrollable
gravitational sedimentation of magnetic particles and their aggregates. The
competitive action of density gradients of thermal and sedimentation nature
results in oscillatory and traveling wave, mostly spatiotemporally chaotic,
convection close to threshould. Previously similar irregular behaviour near
the convection onset so-called spatio-temporal chaos was revealed in gase
and binary mixtures, nematic liquid crystal et cetera [6]. The disordered
patterns in spatio-temporal chaos have a characteristic wave number and
appreciably differ from fully developed turbulence.

wave convection in ferrofluid

Near-threshold chaos in a thermal convection

Experiments were performed with a kerosene-based magnetic fluid having the
following parameters: mean particles size 10 nm, magnetic phase
concentration 10 %, density 1.25[pic]103 kg/m3, magnetic saturation
MS = 48 kA/m, dynamical viscosity in zero magnetic field 0,006 kg/m(s,
Prandtl number 6[pic]102.
The cylindrical fluid layer with a thickness 3.50(0.03 mm and diameter
75 mm is used for study of heat transfer and convection patterns. It was
confined between copper and transparent heat exchangers from below and
above. The circular sidewall of the layer was made of plexiglass. The
patterns were visualized by the liquid crystal sheet. It undergoes its
entire color change from brown to blue at temperature interval 24 to 27 0C.
The temperature difference (Т measured in the center of ferrofluid layer
with the help of thermocouples.
In the model the ferrofluid is treated as a two-phase mixture of magnetic
particles in a carrier phase. For the mixture phase are solved the
conservation equations for mass, momentum and energy. In addition, a mass
conservation equation for the suspended particles and an algebraic
expression for the relative velocity between the fluid and particles are
solved [7].
Numerical simulations were conducted using a finite volume simulation
method, where the governing equations are integrated about each control
volume, resulting discrete equations that conserve each quantity on a
control-volume basis. Second-order upwind scheme was used for continuity,
momentum and energy equations, whereas the first order scheme was used for
the calculation of magnetic potential.
In contrast to the single component fluid, the convection in a
horizontal ferrofluid layer appears "hard" and with hysteresis [8]. When
temperature difference is increased quasistatically, the convection starts
at (( ( ((С and (( changes within wide limits in the dependence of
experiment prehistory. The reproducible critical temperature (TC = 4,5 K
turns out at decreasing ((. In the entire investigated range of temperature
differences (( ( 4((С only oscillatory convection was observed.
The sample of typical irregular temperature oscillations and spatio-
temporal convection patterns are shown in figs.1 and 2. The temperature
sygnal (fig.1) consists of a superposition of low and high frequency
oscillations. The wavelet-analysis revealed that along with periods 8- 15
min there are periods from 1 to 6 hours. The existence of large and small
periods is typical for other values of (T as well.

[pic]

[pic]
Figure 1
Temperature oscillations measured by thermocouples and corresponding them
wavelet-transform at ((/((С ~ 2

As to the time evolution of patterns there are slow movement of roll
systems as a whole and high-speed reconstruction of the convection rolls
because of the cross-roll instability [6]. The breaking-up of the spiral
roll pairs and their subsequent recombination proceed through a cellular
structure (fig.2). Temperature drop from cool (black) to warm (white)
liquid is approximately 3 K. Each white (black) strip in photographs
corresponds to the same handedness of two neighboring rolls. Figure 3
presents the spatio-temporal structures arisen at the applied concentration
gradient in numerical calculation. When (T is increased suddenly from below
subcritical to supercritical values, the one- or two-armed giant spirals
can appear [9].
In order to demonstrate that the oscillatory motions conditioned by
behavior of magnetic fluid itself and not the features of heating et
cetera, in fig. 4 the stationary patterns for the case of single fluid both
in experiment and theory are shown.

|[pic] |[pic] |
|(a) |(b) |

Figure 2
Liquid crystal visualization of spatio-temporal patterns in ferrofluid at
((/((С ~ 1.5. The time intervals between snapshots is 40 min

|[pic] |[pic] |
|(a) |(b) |

Figure 3
Numerical simulations of patterns at ((/((С ~ 2. The time interval between
snapshots is 15min

|[pic] |[pic] |
|(a) |(b) |

Figure 4
Convection patterns in single fluid at ((/((С ~ 2: (a) liquid crystal
visualization for the transformer oil with Prandtl number 3[pic]102; (b)
numerical simulation
Localized states under interaction of thermal, hydrodynamic, magnetic and
concentration fields

A mean shear flow and a longitudinal magnetic field may exert identical
orientation influence upon gravitational magnetic fluid convection drawing
up convection rolls along the background flux [5,12] or force lines
[8,10,11], respectively (fig.5). In this paragraph is discussed the
situation when longitudinal magnetic field is superimposed to convection
flow in an inclined layer so the direction of force lines is perpendicular
to axes of Rayleigh convection rolls aligned with the upslope direction.
Therefore, the interaction or the "competition" of longitudinal (fig.5(a))
and horizontal (fig.5 (a)) convection rolls are observed.



|[pic] |[pic] |
|(a) |(b) |

Figure 5
Schematics of roll motions: (a) in an inclined layer,
(b) in a horizontal layer in the presence of uniform longitudinal magnetic
field


Figure 6 shows the stability boundaries of convection regimes in an
inclined ferrofluid layer subjected to longitudinal magnetic field in the
parameters (T/(TC, ( and M/MS. Here (((((C - dimensionless value of
temperature difference (((С is the threshold of Rayleigh convection at
( = 00 and H = 0 kA/m), ( - inclination angle from the horizontal, M/MS -
dimensionless value of magnetization. Region "a" - outside of a shaded "top-
boot" space - corresponds to mechanical equilibrium at ( = 00 and thermally
driven shear flow at ( > 00 (fig. 7). When ( > 00 within the shaded volume
the secondary flows are superimposed onto the basic unicellular motion -
regions "b" and "c". As it is visible from the plot, the size of the region
of secondary convection motions decreases with the increasing of
inclination angle and magnetic field strength. Therefore, in the case of
tilted layer the longitudinal magnetic field extinguishes the convection
perturbations along the field direction and stabilizes Rayleigh flows. This
is in contrast to the situation of horizontal layer where longitudinal
magnetic field doesn't influence on convective instability and only renders
oriented effect [8,10].



[pic]

Figure 6
Stability boundaries of thermally driven shear flow in an inclined
ferrofluid layer in the presence of a longitudinal magnetic field: a -
shear flow; b - convection rolls aligned with the shear flow; c -
convection rolls aligned with the magnetic field

[pic] [pic]

Figure 7
Shear flow: schematic and photograph from the direction of lateral wide
side at ( = 900 and (T = 20 K (region "a" in figure 6)

When the magnetic field is small enough (stratum "b" of the shaded volume)
the hydrodynamic orientation mechanism predominates over the demagnetising
one, and the axis of convection rolls are lined up along the shear flow,
i.e. perpendicular to the imposed magnetic field (schematic of flow see in
fig.5(a), photograph - fig.8). At strong magnetic fields and not large
inclination angles (column "c" in fig.6), the demagnetising effect
increases which results in a horizontal orientation of the rolls. The
schematic and the visualization of such roll structure are shown in fig.
5(b) and figs.9-11.
Among the various wave regimes which take place in the ferrofluid
convection one should note apart the chaotic localized states. The shape of
these states depends on values of control parameters ((, ( and H.
At the beginning to consider the plane of zero magnetic fields in the
stability map (fig.6). At ( < 500 and near-threshold (( the strong
amplitude modulation of convection rolls can lead to attenuation of roll
motion in the entire cell. In fig.8(a) Rayleigh convection focuses to form
a localized regions of incomplete rolls on the lower part of snapshot. Then
this confined state dies away, returning the cell to the base flow
(fig.8(b)). After some time roll convection begins again in a qualitatively
similar manner as before. During all other runs at the same control
parameters repeated transients from confined Rayleigh convection to the
basic unicellular motion were registered irregularly over the periods of 30
- 40 minutes. The disappearance and the post-forming of roll convection
last some dozens cycles. Previously, similar repeated transients from
convection to conductivity state were registered in binary mixtures [13].

|[pic] |[pic] |
|(a) |(b) |

Figure 8
Localized pattern during repeated transients in an inclined layer at H = 0
for ( = 150, ((/((C = 1.8. The time interval between snapshots is 10 min

The tendency of the pattern to generate the (confined state( in a
horizontal layer subjected to a longitudinal magnetic field is exhibited in
fig.9 (the plane of ( = 00 in fig.6). The convection rolls arising in this
situation align themselves so that their axes tend to be parallel to the
imposed field. At any moment a central portion of the container is nearly
free from convection and the heat transport is mainly confined to edge
regions (fig.9(a)). Then, convection is excited in the central part of
container and partly dies away in the top part of the snapshot (fig.9(b)).

|[pic] |[pic] |
|(a) |(b) |

Figure 9
Confined states in a longitudinal magnetic field at ( = 00 for
(T/(TC = 1.3, H = 17 kA/m. The time interval between snapshots is 15 min.
Magnetic field is directed horizontally in the plane of photos

A sharp bend on the stability surface in fig.6 corresponds to a transition
between hydrodynamic and magnetic mechanisms of convection rolls
orientation. A comparable contribution of both mechanisms in the area of
intersection of zones "b" and "c" leads to formation of different types of
chaotic localized states (or pulses) (figs.10,11).
Figure 10 demonstrates the pattern evolution with the increasing of
magnetic field at fixed values of ( and (T. At weak magnetic field the tick-
like structure may form (fig.10(a)). When the applied field is larger, the
localized traveling pulses occur (fig.10(b)). These pulses appear and die
at irregular locations and times, have unique forms, and vary irregularly
in dimension.

|[pic] |[pic] |
|(a) |(b) |

Figure 10
Evolution of localized states at ( = 25(, (T/(TC = 2 with the increasing of
H: a) 1 kA/m; b) 5 kA/m

The time evolution of the localized pulses directed along the imposed
magnetic field is shown in fig.11. In fig.11(a) only lower left quarter of
the layer is occupied of half-rolls. Then, the convection rolls grow out of
base unicellular motion and increase to a finite amplitude on the right
quarter in fig.11(b). The procedure of appearance and disappearance of
pulses is repeated in an unpredictable fashion. The dark top parts of the
cavity in the photos correspond to the shear flow since the region of
Rayleigh convection shifts to the bottom of the layer with the magnetic
field increasing. Under boundary of the branches "c" and "a" in fig.6 only
one-roll pulse twinkles near the low edge of cavity. The similar localized
states also have been observed in the vicinity of stability boundary in
electroconvection [14].

|[pic] |[pic] |
|(a) |(b) |

Figure 11
Localized pulses at ( = 20, ((/((С = 2, H = 2.5 kA/m. The time interval
between snapshots is 1 min


CONCLUSION

The experimental and numerical results have shown that the concentration
gradients of solid phase due to the settling of magnetic particles and
their aggregates in gravity field can have substantial effect on the
character and stability of flows in magnetic colloids.
Besides, the form and the stability of secondary flows in the inclined
ferrofluid layer may be controled with the help of a longitudinal magnetic
field. The interaction of thermo-hydrodynamic, concentration and magnetic
fields in such situation gives birth to a wealth of localized states.
As the wavelet-analysis revealed the temperature signals consist of a
superposition of low and high frequency oscillations, which correspond to
slow movement of roll system as a whole and high-speed reconstruction of
the convection rolls due to cross-roll instability.
The research described in this publication was made possible in part by
Russian Foundation for Basic Research under grant 04-01-00586, Finnish
Academy grant 110852 and Award No. PE-009-0 CRDF.
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