The Rayleigh-Taylor Instability with Magnetic Fluids:
Experiment and Theory
G. Pacitto [1,2], C. Flament [1,2], J.-C. Bacri [1,2], M. Widom [3,1]
[1] Université Paris 7 - Denis Diderot
UFR de Physique (case 7008), 2 place Jussieu, 75251 Paris cedex 05, France
[2] Laboratoire des Milieux Désordonnés et Hétérogènes (case 78)
Université Paris 6 - UMR 7603 CNRS, 4 place Jussieu, 75252 Paris cedex 05
[3] Department of Physics, Carnegie Mellon University,
Pittsburgh, PA 15213, USA
Abstract: We present experiments showing the Rayleigh-Taylor instability at
the interface between a dense magnetic liquid and an immiscible less dense
liquid. The liquids are confined in a Hele-Shaw cell and a magnetic field is
applied perpendicular to the cell. We measure the wavelength and the growth
rate at the onset of the instability as a function of the external magnetic field.
The wavelength decreases as the field increases. The amplitude of the interface
deformation grows exponentially with time in the early stage, and the growth
rate is an increasing function of the field. These results are compared to
theoretical predictions given in the framework of linear stability analysis.
PACS number(s): 47.20.Ma, 75.50.Mm
1
1 Introduction
When a dense fluid lies above a less dense fluid, a gravitational instability
causes fingering at the interface between the fluids. Rising fingers of the lighter
fluid penetrate the heavier fluid and, conversely, fingers of the heavier fluid fall
into the lighter one. In three dimensions, the fingers of each fluid take place at
the vertices of a hexagonal lattice on the two dimensional interface. In a HeleShaw cell, modeling a quasi-two dimensional system, the one dimensional
interface is destabilized by the growth of fingers regularly spaced on a line with
a well defined wavelength. This wavelength results from the competition
between the stabilizing capillary force, and the destabilizing gravitational force.
At the threshold of the instability, the wavelength, is proportional to the
capillary length,  0  2 3l c , with l c 

g
where is
the surface
tension,is the density difference between the two fluids, and g is the
gravitational acceleration. This instability, called the Rayleigh-Taylor instability
(RTI) [1,2], plays an important role in subjects like astrophysics, fusion and
turbulence [3,4,5]. Although the phenomenon has been studied for decades,
much remains to be learned about it.
The development of patterns resulting from the RTI can be divided in
three stages: the early linear stage, where the lengths of the rising and falling
fingers are small compared to the wavelength, the middle, weakly non-linear
stage, and the strongly nonlinear late stage. The linear stage is well described
theoretically and experiments seem to be in good agreement. The nonlinear
stages are not fully understood.
2
Several theoretical studies start from the Navier-Stokes equation and
perform a linear analysis of the instability [6,7]. Particular issues studied address
the compressibility of the fluids [8,9], density gradients [6,10], and viscosity
effects [6,7,11,12,13]. In the non-linear regime, Ott [14], Baker and Freeman [15]
and Crowley [16], describe the motion of the fingers. Studies using miscible
fluids, performed by Petitjeans and Kurowski [17] observe similarities with
immiscible fluids in the development of the instability, even though the
wavelength and the growth rate differ greatly. They believe the similarity arises
because the density gradient between the two fluids acts like an equivalent
surface tension at the onset of the instability. Authelin, Brochard and de Gennes
[18] describe the interface melting of two miscible fluids by the RTI. This
instability generates micron-sized drops which then dissipate by diffusion.
One of us [19], used a mode-coupling analysis of Darcy’s law to describe
the weakly nonlinear evolution of the viscous fingering patterns obtained in a
Hele-Shaw cell. This study, describing the RTI, is also applicable for the
Saffman-Taylor instability (STI) [20], with a low viscosity fluid pushing a more
viscous one in a Hele-Shaw cell.
Recent works [21] determine the length scale of the fingers, show a
difference between the width of the rising fingers and the width of the falling
fingers, and explain their amalgamation in terms of spatial modulations. For
miscible fluids, the turbulent mixing zone is numerically studied by Youngs in
2D [22] and 3D [23], and experimentally by Read [24]. Ratafia [25] studied the
non-linear regime, and described the destabilization of fingers by the presence
of Kelvin-Helmoltz instability (KHI) [26], resulting from the jump of the
tangential velocity between the two fluids at the edges of the fingers. This
3
interpretation can explain the fractal structure obtained after nonlinear
evolution, which is the result of a KHI cascade.
Recent Rayleigh-Taylor experiments using a mixture of water and sand
[27], modeled as a Newtonian fluid, determine the viscosity of the suspension,
and find results in agreement with other experimental measurements. To our
knowledge, this Rayleigh-Taylor experiment is the only experiment that uses
complex fluids.
Magnetic fluids (MF, also called ferrofluids) are colloidal suspensions of
magnetic nanoparticles. An applied magnetic field provides a new external
parameter that can stabilize or destabilize the fluid interface, causing interesting
hydrodynamic instabilities. One can distinguish two kinds of instabilities in
ferrofluids: static instabilities caused by the magnetic field, which are not
present in ordinary fluids; dynamic instabilities that occur even in the absence of
magnetic fields, but are modified by applied fields..
The first static instability observed in MF is the peak instability [28]. A
static magnetic external field, Hext, applied tangent to a free surface, generally
stabilizes the surface. However, Hext applied normal to a horizontal surface
causes the peak instability above a critical value of Hext A line (in 2D) or a lattice
(in 3D) of spikes arises from the competition between the destabilizing magnetic
forces and the stabilizing capillary and gravitational forces.
Now, let the MF be confined in a two-dimensional Hele-Shaw cell.
Another instability can appear if the external field is applied in the direction
perpendicular to the cell. This phenomenon, called the labyrinthine instability
[29,30], occurs above a critical value of the applied field and with a critical
wavelength. The threshold value of Hext results from a balance between the
4
destabilizing magnetic dipole-dipole repulsion and the stabilizing surface
tension (and possibly gravity in a vertical cell).
MF can be used as a dynamic system if a time-dependent magnetic field
is applied. Different surface phenomena are observed like surface waves [31,32],
the Faraday instability [33], or a period doubling in the case of the peak
instability [34].
Hydrodynamic instabilities may occur when the MF flows. For example,
Saffman-Taylor
fingering
has
been
studied,
both
experimentally
and
theoretically, with a MF. In this configuration, the external magnetic field can be
applied normal to, or within, the plane of the cell. The situation is stabilizing if
Hext is tangent to the interface within the plane of the cell [35]. Experiments
performed with a field applied in a direction perpendicular to a circular HeleShaw cell show a destabilizing behavior [36].
The aim of this article is to study the influence of a homogeneous
magnetic field applied perpendicular to a vertical Hele-Shaw cell filled with a
dense ferrofluid above a lighter oil. In a recent paper [37], one of us describes
theoretically
the
general
viscous
fingering
pattern
obtained
in
this
configuration. In this dynamic situation, the magnetic force is added to the
gravitational force to destabilize the interface, while the capillary effects
stabilize it.
2 Linear stability analysis
Consider a vertical Hele-Shaw cell of gap h filled with oil of density and viscosity - at the bottom and an immiscible MF of density +and viscosity
+ on top. We use a coordinate system in which the Hele-Shaw cell lies parallel
5
to the xy plane, the y axis is vertically upwards, and the z axis is perpendicular
to the Hele-Shaw cell. Gravity acts downwards parallel to the y axis and a
uniform external magnetic field, H ext  Hext zˆ , is parallel to the z axis (see Figure
1). We present equations of motion and boundary conditions then we perform a
linear stability analysis of these equations. We show that both gravitational
(provided +>-) and magnetic instabilities deform an initially flat interface.
To begin, we derive Darcy's law for the flow of magnetic fluids. The
analysis begins with the basic equation governing the 3-dimensional fluid flow
v(x,y,z) , the Navier-Stokes equation

Dv
 p  g  v  fm
Dt
(1)
From this equation we derive Darcy's law assuming sufficiently high viscosity
that the flow velocity is small so the inertial term on the left-hand side may be
neglected. The idea is to average equation (1) over the gap, resulting in a 2-D
flow equation for the gap-averaged velocity u . The gap average of the
three-dimensional pressure gradient yields a two-dimensional gradient of the
gap-averaged pressure, which we continue to represent as p . As is usual in
derivations of Darcy's law, the gap average of the viscous drag force, subject to
no-slip boundary conditions imposed at z=±h/2, is -(12/h2) u .


The final term in equation (1), fm   0 M   H , represents the magnetic
body force on a fluid element, neglecting compressibility and self-induction of
the fluid [38]. In this approximation, M is constant and parallel to z, and the gap
average of fm reduces to (0M/h)(H(x,y,h/2)-H(x,y,-h/2)). Because the applied
field H ext is spatially uniform, it drops out of this difference and the magnetic
force arises entirely from the demagnetizing field Hd  H  H ext caused by the
6
surface magnetic poles. Express the demagnetizing field as the gradient of a
magnetic scalar potential, Hd   (x,y,z) , and take  as an odd function of
z. The gap average of magnetic force is
fm  (20M / h) (x, y, h / 2)
where now the gradient acts only on x and y coordinates [39].
Collect all averaged terms and isolate the velocity u on the left-hand side,
12
2M
ˆ
 (x,y) .
2 u  p gy  0
h
h
(2)
Here all vectors lie in the xy plane, and the scalar potential (x,y)(x,y,h/2) is
evaluated at the top plate. Further simplification of Darcy's law eq. (2) occurs if
we exploit the irrotational flow to introduce the velocity potential u   so
that
12
2M
(x,y)
2   p  gy  0
h
h
(3)
Now we apply Darcy's law (3) within each fluid evaluated at the interface
between the two fluids, y=(x). Subtract equation (3) for the oil (fluid -) from
the same equation for MF (fluid +) and find
h2
2M  h 2 (+ - - )g
          

A 


(p

p
)


 


0
2   2  12(     )  
h  (12(+ + - ))
(4)
The viscosity contrast A =(+--)/(++-),. The pressure jump across the
interface, p+- p-, is the surface tension  times the mean interface curvature .
For a thin gap h, we need only consider the curvature of (x), which we may
approximate 2/x2 for the purpose of linear stability analysis [37].
Represent the net perturbation (x, t) in the form of a Fourier mode
7
(x,t)   k(t)cos(kx) .
(5)
The velocity potentials ±must obey Laplace's equation 2±=0, because the
fluids are incompressible. The boundary conditions at y±, so that ±=0. We
give ± the appropriate wavevector and phase to be consistent with the
perturbation . The general velocity potentials obeying these requirements are
  ( x, y, t )   k exp(  ky) cos(kx) .
(6)
In order to substitute expansions (6) into the equation of motion (4), we need to
evaluate them at the perturbed interface. To first order in the perturbation it
suffices to simply set y=0 in eq. (6).
To close equation (4) we need additional relations expressing the velocity
potentials in terms of the perturbation amplitudes. To find these, consider the
kinematic boundary condition that the interface moves according to the local
fluid velocities. To first order in  we simply note /t = -i/y [
Substituting
equation (6)
for
±
and
Fourier
transforming
].
yields
Ýk  k k   k k . Then equation (4) reads

1  k
h2
2M  h 2 (   )g
 2

k



 

k
0
k t 12(   ) 
h k  12(   ) k
(7)
To obtain k, the Fourier transform of the magnetic scalar potential, we
write the magnetic scalar potential
 ( x, y) 
M
4




1
1

dy 

 ( x )
 ( x  x ) 2  ( y  y ) 2
( x  x ) 2  ( y  y ) 2  h 2 
(8)
dx 


The expansion of (x,(x)) to first order in  is
8
 ( x)   0 
M
4

1
1

d
x
  ( x  x) 2  ( x  x) 2  h 2



 ( x)   ( x)


(9)
and its Fourier transform k=2 M J(kh)k where
J(x)  ln(x / 2) K0 (x)  CEuler
(10)
with K0 a Bessel function and CEuler=0.5772… is the Euler constant [40].
Inserting k into equation (7) for the growth of the cosine mode, the
differential equation of the interface is
Ýk  (k) k

(11)
where
 h2 (   )g


(12)

2 BM J (kh)  (kh) 2  ,
 12(     ) 12(     )

is the linear growth rate multiplying the first order term in 
 2M2h
with BM   0 
the magnetic Bond number An asymptotic expression for
4  
kh << 1 which is more convenient for the data analysis can be obtained by
expanding J(kh) for small kh [41]:

k
B

(13)
 (k ) 
BG   M (1  C Euler  ln( kh / 2))( kh) 2  (kh) 2 

12(     ) 
 2


 (k )  k 

(     )gh 2
with BG 
the gravitational Bond number

3 Experimental set-up
The experimental geometry is sketched in Figure 1. The cell is located in a
gap between two coils in the Helmholtz configuration to achieve good axial
homogeneity for the magnetic field, H ext . The radial homogeneity of the field is
9
better than 3%. The amplitude of the external field is nearly constant. The cell is
mounted so it can be rotated around the x axis, which passes through the
middle of the gap.
We use an ionic magnetic fluid made of cobalt ferrite particles (Co Fe2 O4)
dispersed in a mixture of water and glycerol. This MF is synthesized by S.
Neveu [42] following the Massart’s method [43]. The magnetization of the MF as
a function of Hext is obtained by the use of a calibrated fluxmeter.
The rectangular cell consists of two parallel plates made of altuglass
(plexiglass) with a spacing between the two plates of h = 500 m. The cell is
initially filled with an oil (White Spirit or WS) of low density (compared to the
MF) which wets the altuglass walls. A thin film (of micron width) of WS
separates the MF from the walls, avoiding pinning of the MF-WS contact line on
the walls.
The mass density of MF is + = (1686 ± 85) kg m-3, compared to - = 800 kg
m-3 for WS. The dynamic viscosity of the MF is + = 0.14 kg m-1s-1 at room
temperature. The viscosity of the WS is two orders of magnitude lower, so we
may take   0 .
Image processing is used to measure the wavelength and the growth
increment. The images are recorded by a CCD camera (charge couple device)
and digitized by an acquisition card in a computer. We use the public domain
software NIH Image [44] to analyze the images.
4 Results
4.1 Wavelength measurements
The experiments are conceptually simple: the cell is placed vertically with
the heavier liquid (MF) below. We rotate the cell by 180 degrees around the x
10
axis and apply the magnetic field. The selected wavelength depends on how and
when the magnetic field is applied. If Hext is applied during the rotation, we get
a different measurement than if Hext is switched on at the end of the rotation.
The response of the MF-WS interface to the magnetic field is usually much faster
than its response to a gravitational field. That is, a longer time is needed to
observe the classical RTI without external magnetic field than to observe the
labyrinthine instability [29,30]. If Hext is applied during rotation when the cell is
momentarily horizontal, the normal field instability [28] appears before the RTI.
To avoid these difficulties we apply the field only after the rotation is complete,
but before the RTI appears. The duration of rotation is about 1 second, and the
time constant for ramping up the magnetic field is about 1 second.
We collected data for thirteen different values of Hext, with two
independent runs for each value of Hext. The wavelength, 0=2/k0, is measured
at the onset of the instability until the amplitude, , of the interface deformation
remains small: k0 < 0.1. We compared two different methods: an FFT of the
interface gives the fundamental mode k0; a direct measurement of the average
peak to peak distance gives 0. In the second approach, we reject the peaks
located close to the edges of the cell, and we omit certain peaks which are
dominated by others (« finger competition »). Both methods give similar results
within the errors bars.
Figure 2 displays a sequence of pictures of the destabilizing MF-WS
interface for Hext = 7.9 kA/m. A comparison of interfaces for different values of
the external field is shown in Figure 3. Both the wavelength and the width of the
MF fingers decrease as Hext increases. The experimental values of 0 as a
 2M2 h
function of the magnetic Bond number, BM  0
, are reported in Figure
4 
4.
11
To determine the value of BM for a given value of Hext, we need to know
the MF-WS surface tension, . The value of  can be deduced from the
wavelength at the onset of the RTI with Hext = 0 using formula (12). We find  =
(11.7 ± 3.4) mN m-1 (REPLACE WITH 8.6+-???) by this method. We can also
determine the value of  from the wavelength 0  ???? at the threshold of the
normal field instability, which is linked with the capillary length [45]. This
method yields  = (12.0 ± 1.3) mN m-1, which is consistent with the previous
value. The value of the magnetization M(Hext) is directly deduced from the
magnetization curve of the MF. Taking the latter value of , we deduce a
capilary length lc  0.12 cm, and the gravitational Bond number B g  0.18 .
To compare experimentally observed wavelengths with the linear
stability analysis, consider the growth rate, k. Maximizing expression (13) for
k versus the wavevector, gives the fastest growing mode k0:
 
2
 0  0 
.
k k k
k
0
0
We get the following non-algebraic equation:
BG
4


2
3
2 x0  BM a  2 ln(x 0 )  3  0 ,
(14)
with, a  1  32 C Euler  32 ln( )  0.79 , and x0 = 0/h. The roots of the equation
(14) for different values of BM are reported in Figure 4 for comparison with the
experimental data. Both are qualitatively coherent. We get a good agreement for
low values and high values of BM The discrepancy for the intermediate values
should result from the omission of the demagnetizing effect in the magnetic
forces (MAYBE NCLUDE THIS EFFECT AND SEE WHAT HAPPENS).
12
THE FOLLOWING DISCUSSION MIGHT BE REMOVED IF THE PLOT OF
WAVELENGTH
USING
THE
DEMAGNETIZING
EFFECT
LOOKS
PROMISING. IF WE DO NOT REMOVE THE DISCUSSION, I WILL REVISE IT.
A better way to evaluate magnetic forces is to performed experiments
with the same device ( H always remain perpendicular to the cell) but with
different orientations. In the first one, the labyrinthine experiment, we eliminate
the gravitational term in (14) putting the cell in a horizontal position. Measuring
the obtained wavelength l, we obtain a deduced value for the magnetic Bond
number, and thus a better estimation of the magnetic forces :
BM 
3
3
a  ln(x l )
2
(15)
with xl=l/h
In the second one, the labyrinthine-gravitational experiment, we put the cell
vertically, with the ferrofluid below. This situation (changing the sign of gravity,
so the sign of BG in (14)) leads to

BG
4


2
3
2 xlg  BM a  2 ln( xlg )  3  0
(16)
with xlg=lg/h where lg is the measured wavelength. Combining (15) and (16),
we obtain :


182 ln x lg / xl 
BG  2
xlg a  3 ln xl 
 2



(17)
Including (15) and (17) in (14), we obtain a theoretical value of x0 :
x 20  x2lg
ln( x0 / x l )
ln( x l / xlg )
(18)
In the inset of Figure (4), we show a comparison between the
experimental results obtained in the Rayleigh-Taylor experiments and
13
wavelengths deduced from (18). A good agreement between both is observed. It
is important to notice that, if the situation is always unstable in the RayleighTaylor experiment, a threshold value of the magnetic field exists in the
labyrinthine [30] and the labyrinthine-gravitational [38] experiments, below
which no wavelength are measured. This results from the competition between
destabilizing magnetic effects and stabilizing effects (surface tension and gravity
in the second experiment).
4.2 Growth rates
Now, let us study the growth rate, , of the instability. We measure the
length  of the falling fingers, and divided by (t=t0); t0 corresponds to the
first time where the interface deformation is detectable and is actually given by
the resolution of the video recorder. Plotting this relative depth to which the
instability penetrates the lighter fluid as a function of time, we can clearly
separate two distinct stages (Figure 5)
Just after the onset of the instability, when the amplitude of the growth
(the length of the spikes) is small compared to the wavelength, we see an
exponential growth over time. This occurs for all values of the external applied
field. We observe an augmentation in the growth rate values as the field
increases, as is predicted by the linear analysis given by the formula (13). In this
equation, the growth rate is a function of Hext, and also a function of x0. As we
have seen in the previous part, x0 is an implicit function of Hext, and we can not
find an explicit expression of (Hext ). Nevertheless, the equation (13) can be
written
  *
 BM
˜

14
(19)
with the growth rate for the mode of wavelength =h x0 in the absence of magnetic
field
* 
b BG
( 2 - x -2
0 )
x 0 4
~ 
b
(a   ln x0 ) .
2 x 03
and a characteristic time
2 3
We define b 
and a’=1-CEuler-ln  = -0.72 .
3 h
If we insert the theoretical values of the wavelength, x0th, in the expressions of
*
˜ , we can compare the theoretical linear analysis to the experimental
 and 
measurements of exp.
These experimental results of the growth rate exp measured near the
onset of the instability for different values of the external applied field are
 exp  *
shown in Figure 6, where we plot
versus the magnetic Bond number.
˜

The theoretical continuous line results from the linear analysis and is the first
bisectrix of the Figure 6. We observe a linear behavior in agreement with this
theory. The experimental results are below the theoretical curve, but are
included in the error bars. These experimental uncertainties are large due to the
difficulty of measuring the amplitude .
The systematic discrepancy between the two curves can be explain by the
fact that the calculation does not take into account the demagnetizing field
effect. The demagnetizing factor of an infinite plane with an external field
applied perpendiculary to the plane is equal to D = 1. Consequently, the local
field is: H  Hext  DM Hext  M. The magnetization, M, is linked to the local
15
field by the relation: M  H , it leads to M   /1  Hext . A local susceptibility
can be determined by the use of the magnetization curve: (H ext )  M(Hext )/ H ext
[46] , and subsequently a magnetic Bond number including the demagnetizing
 2 M 2 h
 exp  *
effect is BM   0
. The plot of
versus BM’ is also reported in
˜
4 

Figure 6. In contrast to the previous case, the data are above the first bisectrix.
Since we have crudely included the demagnetizing fields through a
demagnetizing factor D=1, the demagnetizing effects are naturally overestimated. An exact a calculation would have to deal with the nonuniform
fringe fields at the edge of a paramagnetic slab. Experimentally, we could
approach the limit of uniform demagnetizing fields with D=1 by using a thinner
cell gap h.
After the initial exponential growth of the disturbances, we enter a new
growth regime shown in the inset of Figure 5. A linear growth is observed for
each value of the applied magnetic field. This behavior is observed for long
times, up to the secondary instabilities, where the finger tips split and start to
compete with each other. As magnetic field increases, we observe an increase in
the linear coefficient. For example, for Hext = 0, we get:  = 0{1.9+2.0(t-to)}, and
for Hext = 39.3 kA/m, we get:  = 0{10.8+20.1(t-to)}. Saturation of the
exponential growth is predicted by weakly nonlinear analysis [37]. Crossover
from exponential to linear has been found in numerical simulations for
nonmagnetic fluids [50].
4.3 Far from the threshold
16
This study emphasized the downwards propagating MF fingers, but
upward fingers made of WS also exist. (SENTENCES REMOVE DUE TO
CONCERNS DESCRIBED IN 6/2/00 E-MAIL. ENTIRE PARAGRAPH COULD
GO IN PREVIOUS SECTION). As a matter of fact the heavier liquid, i.e. the MF,
falls down due to the buoyancy forces and consequently, the lighter liquid
which is the less viscous fluid has to penetrate into the viscous one. A finger of
WS grows between each spikes of MF. These WS fingers rise like the ST finger
propagating in a narrow channel [20]. In fact, both instabilities (RTI or STI) can
be described by the same set of equations [37]. The width of the WS fingers is
greater than the MF fingers, but a common feature is that the width is a
decreasing function of the external magnetic field. Such a symmetry-breaking of
the interface is related to the viscosity contrast between MF and WS [37].
The tip of the WS fingers splits into two fingers (the so-called tip-splitting
phenomenon) and the angle between these new fingers is roughly equal to 90
degrees. The evolution of the system exhibits a cascade of tip-splitting: each new
finger divides itself into two fingers which destabilize themselves while they
remain upward. Let us notice that the finger changes slightly its direction after
each tip-splitting: it seems to undulate like a narrow finger confined in a channel
in the oscillating tip regime [47]. No other secondary instability like the sidebranching phenomenon is observed. The tip-splitting cascades acting on a finger
give a pattern which looks like a tree as it is illustrated in Figure 7. This pattern
is somewhat similar to the radial viscous fingering obtained with the STI [36]
with the difference that the system is anisotropic due to the gravity field.
The MF fingers always remain stable because the viscosity contrast is
opposite (a viscous fluid penetrating a less viscous one is stable situation). When
the MF fingers are sufficiently far from each other and for high values of the
17
magnetic field, a bending instability occurs [48]. When the distance between the
fingers is comparable to the finger width, long range magnetic interactions
between fingers are visible: they undulate together for high magnetic field in the
same manner than the MF parallel stripes in the bending instability [49]. Finally,
the pattern is very non-symmetric (Figure 7b) because of all these features. For
high amplitudes of the external field and at long times the top of the cell
becomes a labyrinthine and the bottom is rather well organized as the MF
smectic [49].
At very long times, the MF accumulates in the bottom of the cell,
displacing the WS to the top (figure 8). Presumably the limiting pattern will be a
conventional MF labyrinth [38] with a reservoir MF at the bottom of the cell.
However, long bifurcated fingers of WS initially are trapped in this region and
the dynamics of the pattern evolution becomes dramatically slow. A hierarchical
dynamical behavior [51] emerges because of the tree-like structure of highly
bifurcated fingers of WS. Retraction of a bifurcated finger of WS cannot occur
because its point of bifurcation represents a point of force balance and is
therefore immobile. To undo a bifurcation requires retraction of at least one of
the branches. However, each branch may itself be bifurcated, further slowing
down the pattern evolution. Only at finger tips are forces unbalanced and
dynamics unfrozen. Hierarchically constrained dynamics leads to glassy
behavior [51]. A Kohlrausch stretched exponential law should govern the
evolution in this regime.
5 Conclusion
18
Acknowledgments
We are greatly indebted to Sophie Neveu for providing us with the
magnetic fluid and Patrick Lepert for building the experimental device. We
acknowledge useful discussions with Jose Miranda and support by the National
Science Foundation grant DMR-9732567.
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21
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Figure captions
Figure 1: the experimental set-up consists in a cell located vertically between
two coils. The external magnetic field obtained is horizontal and perpendicular
to cell. The cell which contains the both liquids can be rotated around a
horizontal axis.
Figure 2: several pictures of the destabilizing MF-WS interface for different
times (a: t = 0; b: t = 5s; c: t = 8s; d: t = 11s) for Hext = 7.9 kA/m. The grey bar
equals 1 cm.
Figure 3: several destabilizing interfaces for different values of the applied field
(a: Hext = 4.1 kA/m; b: Hext = 11.9 kA/m; c: Hext = 17.8 kA/m; Hext = 27.7 kA/m).
BETTER TO QUOTE THE ACTUAL TIMES FOR EACH FIELD. The grey bar
equals 1 cm.
Figure 4: Wavelength as a function of the magnetic Bond number
NB 
 0 2M2 h
. Experimental data are measured by the FFT method, at the
4 
onset of the instability.
Figure 5: growth rates for Hext = 23.6 kA/m in the exponential regime and in the
linear regime (shown in the inset)
22
Figure 6: magnetic field dependence of the growth rate; black circles represent
the experimental data (see text) without demagnetising effects; white squares
maximize the demagnetising effects in the expression of the magnetic Bond
number
Figure 7: pictures of the RTI far from the threshold (see text) and for high value
of the applied field: Hext = 40 kA/m. The black bar equals 1 cm.
Figure 8: Late stage evolution of the RTI at high fields Hext = 40 kA/m. (a)
Arrow points to branched finger about to disappear. (b) ??? seconds later, one
branch has retracted. (c) ??? seconds later, the entire finger disappears.
23