Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 6.1.3 - Observations on the Faraday Instability
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 Material Information
Title: 6.1.3 - Observations on the Faraday Instability Instabilities
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Batson, W.
Zoueshtiagh, F.
Narayanan, R.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: interfacial instability
Faraday waves
 Notes
Abstract: Faraday instability is a result of inertial interactions between the bulk phase and the interface. The instability is excited by external oscillatory forcing acing upon density differences between phases. We examine the effect of fluid height on the instability and see that even in a viscous medium the predictions are remarkably close to what one expects from an ideal fluid. We also examine the effect of multiple frequencies and show the existence of new islands of instability.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Bibliographic ID: UF00102023
Volume ID: VID00145
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: 613-Batson-ICMF2010.pdf

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7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010



Observations on the Faraday Instability


William Batson*, Farzam Zoueshtiagh** and Ranga Narayanan*
*University of Florida, Department of Chemical Engineering
P.O. Box 116005, Gainesville, FL 32611, USA
ranga @ufl.edu

and

**University of Lille, IEMN, Villeneuve d'Ascq, France


Keywords: Interfacial Instability, Faraday waves


Abstract

Faraday instability is a result of inertial interactions between the bulk phase and the interface. The instability is excited by
external oscillatory forcing acing upon density differences between phases. We examine the effect of fluid height on the
instability and see that even in a viscous medium the predictions are remarkably close to what one expects from an ideal
fluid. We also examine the effect of multiple frequencies and show the existence of new islands of instability.


Introduction


The Faraday instability has many applications from
microfluidic reactor mixing to dislodging of drops and
bubbles in capillary loops during space enabling
operations. This instability is visualized as standing waves
at a fluid interface when a fluid compartment is subject to
external vibrations. The original work on the instability is
due to Michael Faraday (1831) who first observed and
described the phenomenon when he was investigating the
pattern formation of particulate grains on plates when he
held a vibrating violin bow to the plate. In subsequent
work he discovered what he termed "beautiful crispations"
on liquid surfaces by holding the violin bow to plates
containing fluid layers. Among the important descriptions
he made included the effects of liquid "fluidity" (the
reciprocal of viscosity), liquid height, and the imposed
frequencies. A figure of the problem helps to understand
the physics of the instability.


upper light fluid


unstable, defecing Interface


vertical external
osclllaton


~--- - -- "

lower heavy fluid
flat interface


Figure 1: Depiction of the instability


Figure 1 depicts a liquid bilayer system that is top light and
that is subject to vertical vibrations. For low accelerations
the interface is flat but for high enough accelerations the
flatness gives way to standing waves of a defined
wavelength. The instability can be traced to a trade-off
between two effects. The first is the force due to the
imposed periodic acceleration which causes the instability.
The second effect is stabilizing and is made of pressure
differences between crest and troughs which in turn are due
to gravity and interfacial tension, the latter acting on short
wavelength disturbances. The two effects can be in perfect
resonance when frictional damping due to viscosity is
ignored. Otherwise they can only come into "near
resonance". It is this resonance that is at the heart of the
physics of Faraday instability and we elaborate on this later.
The mathematical characterization of the problem was first
given for ideal fluids by Benjamin and Ursell (1954), and
later studied for viscous fluids by Kumar and Tuckerman
(1994).

Due to the nonlinear nature of the governing equations,
linear stability theory is used to gain a perspective on the
governing physical aspects of the problem. In other words
the equations are linearized around a known base state
which here is a quiescent base state of no flow from the
perspective of an observer in motion on the erstwhile flat
surface. Using this approach, Benjamin and Ursell were the
first to reduce the problem to a series of Mathieu equations,
which is also the model for understanding a pendulum with
an oscillating base plate. This is an important analogy
because both Faraday and Rayleigh had already made the
observation that the resulting frequency of the standing
waves was half that of the excitation frequency, also
characteristic of a pendulum. Now for a given fluid system,
there exist any number of modes of periodic deformations
with a wave number k characterizing the onset of instability,









and a proper stability analysis of the system requires each
mode of deformation to be analyzed. The power of linear
stability theory is its ability to predict the onset of
instability, and the physics governing the onset. Nonlinear
analysis is needed to describe the dynamics and magnitude
of the instability.

Discussion of the Physics:


To visualize the physics governing the existence of Faraday
waves, one should first imagine a layer of fluid bounded by
a rigid surface at the bottom and a surface free to deflect at
the top. When there is no external forcing the free surface is
flat and stable because there is no destabilizing force to
cause it to deflect. An example of a destabilizing force is if
the gravity vector were suddenly switched reversed in
direction. The system would then become gravitationally
unstable with the only stabilization coming from surface
tension inherent to the free surface that wants to maintain a
flat surface. This is the classic Rayleigh-Taylor "heavy over
light" instability. In the case of Faraday waves, the external
vibrations can be understood as adding an oscillatory part to
the gravity vector, which is visualized as the fluid layer
being shaken up and down. When the system is accelerated
upwards with respect to gravity, the system feels a larger
effective gravity and is "more" stable. But when the system
is accelerated downwards, and the oscillatory part of the
acceleration overcomes the constant acceleration due to
gravity, snapshots in time show the system resembles the
Rayleigh-Taylor instability, and the destabilizing effect
responsible for Faraday waves is understood.

The slight fallacy in this description lies in the fact that
Faraday waves can appear in systems where the magnitude
of the oscillatory acceleration is less than the acceleration
due to gravity. This arises due to a resonance of the natural
frequencies inherent to the fluid and the frequency of the
oscillatory acceleration. For a perturbed free surface, there
exists a natural time scale at which the surface will want to
"accelerate" back towards its flat state, which is a function
of the fluid properties and the mode of the perturbation. If
the time scale of this restoring acceleration matches up with
the excitation frequency, then resonance can occur and the
surface will overshoot the unexcited equilibrium state.
Therefore one can expect that certain modes of instability
can be excited with smaller accelerations of excitation than
others due to this resonance. Benjamin and Ursell's result
that the inviscid problem reduces to a series of Mathieu
equations validate these notions. The linear stability
analysis of the problem produces a stability diagram where
the acceleration of excitation is plotted versus the wave
number of the deflection for a given frequency.

Characteristic of Mathieu equation solutions is that there
are thin regions of instability near infinitesimal excitation
accelerations and as the excitation acceleration increases
these regions of instability broaden, implying that
resonance can cause a system to become unstable for
certain wave numbers and that any system can be forced
unstable for a large enough acceleration. Also worth
noting is the existence of regions of stability above the
point where the excitation acceleration is greater than the


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

acceleration due to gravity. The explanation here is simply
that an "anti-resonance" occurs and the time scale of the
restoring acceleration is out of phase with the excitation
frequency. The standard form of the Mathieu equation is

2 +[ + ecos(t})]u = 0
at2
where the free parameters 8 and s determine the stability of
the variable u for a given initial condition. Stable solutions
oscillate and increase without bound in time and stable
solutions oscillate but neither grow nor decay. In the case
above equation
k tanh(kH) j2 +2


S=k-tanh(kH)A

where k is the scaled wave number H is the scaled fluid
depth, co is the input unsealed vibrational frequency, p and
y.are the actual interfacial tension and A is the amplitude of
the imposed motion. Resonance occurs when 6 takes on
values of n2 where n is an integer. At these points of
resonance it takes only an infinitesmal amplitude to trigger
an instability..

The above equations illustrate the relation between the
system and fluid properties and the forcing acceleration on
the stability of the interface. Note that a full description of
the stability of a fluid system involves analyzing the
stability of each wave number to which the system has
access. By specifying the form of the deformation, one
restricts the wave numbers that are accessible in finite
system. An infinite system theoretically has access to an
infinite set of wave numbers, while a finite system, i.e. a
container with walls, only has access to integer multiples
of wave numbers, ensuring the boundary conditions at the
walls are met.

Discussion of results:

Calculations on the instability were done numerically for
both ideal and viscous fluids using a spectral method. Figure
2 shows a comparison of ideal to viscous fluids for the
critical amplitude versus imposed wave number. The axes
are deliberately plotted unsealed to give a rough idea of
physical magnitudes required to trigger the instability. What
stands out from the graph are the similarity in the
characteristics of the ideal fluid and viscous fluid cases.
Observe first that the wave numbers of resonance occur at
zero amplitude forcing when an ideal fluid is considered.
The dotted lines at resonance project to the abscissa but are
not shown at these extreme limits.



0. 4




Sideal luid
III "
0 50 100 150 zoo
wavenuber k, m1
Figure 2: The effect of viscosity on the instability










As noted earlier these extreme modes are due to the absence
of frictional damping, thus rendering the system to act like a
simple pendulum with an oscillating base plate. When
viscosity is taken into account, the shift from ideality takes
place principally at higher wave numbers. The reason is that
only gravity is dominant at low wave numbers. And when
gravity is dominant the system acts like the linear
pendulum. At low wave numbers, interfacial tension and
viscosity play a small role as both are diffusive in character
and diffusion is enhanced only at large wave numbers.
Observe also that low input vibrational frequencies lead to
low wave numbers at resonance where we already know that
gravity is dominant.


0 50o 100O 150 200zoo
wavenumber Rk m-1
Figure 3: The effect of fluid height on the instability


Other observations can be made if the height of the fluid is
varied. Figure 3 presents two curves showing the boundaries
of neutral stability which separate the regions of stability
and instability for a viscous fluid layer of two different
heights. The main feature is the presence of fins of
instability, where the increase of the fluid depth shifts the
fins at low wave numbers but saturates at higher wave
numbers. This qualitative behavior follows from the ideal
fluid laws and from the definition of 6 above. Notice that the
height of the fluid H appears only as tanh(kH) and this
function saturates to nearly unity for kH as low as 2. This
means that if the wave number is low enough then the
height has a chance of influencing the points of resonance.
In fact a low k approximation would have us believe that k2
goes as 1/H. neutral stability. Remarkably this scaling law
seems to continue to hold even for viscous fluids,
principally because it is valid at low k where viscosity does
not matter much anyway. The upshot of this is that declaring
the depth to be infinite at the outset of a calculation runs the
risk that for certain container geometries this may not be
true. The geometry sets the allowable wavelengths when the
side walls are treated as periodic and if the most unstable
mode of instability appears near the first point of resonance
then the fluid depth matters a lot.

We now move on to the last point of our discussion. It
pertains to the role of multiple frequencies of the same
amplitude on the instability.

Consider induced motion where two non commensurate
frequencies are imposed on the system. To be concrete take,
as an example, these to be 2 and 3 Hz. The response to the
individual frequencies is plotted on the same graph and one
sees a shift to the right of the resonant wavenumbers as the
imposed frequencies increase. This behavior could have


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

been predicted from the ideal fluid formulas where for low
wavenumbers the increase is linear and for higher wave
numbers the increase is proportional to the square root of
the frequency. Such behavior is nearly carried over for the
viscous fluids case which is what one might observe from
Figure 4. When both frequencies act in concert with each
other there is as expected a change in the instability with
signatures of each frequency being seen and with the
additional surprise of islands of instability in regions where
stability might have been expected. The precise reason for
the occurrence of these islands is currently under
investigation. What is clear is that the behavior of combined
frequency response is markedly different than what is
observed in a similar situation where the liquid is subject to
a heating arrangement and buoyancy flow is staved off.
There it turns out that regions of greater stability can
actually be created as flow reversals can be introduced.



E 4







2 Hz

90 100 150 200 250
wavelength k, m-1

Figure 4: Response to two individual frequencies




6








0 10 20 30 40 50 60
wavenumber k, m-
Figure 5: The instability with combined multiple frequencies


Summary.:

The ideal fluid is an excellent predictor of the qualitative
behavior of Faraday instability for viscous fluids and in
some cases is even a reasonable predictor of the quantitative
behavior. Assuming infinite depths is not always safe and
depends upon the allowable wavenumbers that, in turn,
depend upon the container geometry. Their response to
multiple-frequency forcing predictably contains the
signature of the individual frequencies but surprisingly
islands of increased instability appear. This is in contrast
with frequency forcing on the classical B6nard problem.
(Shukla and Narayanan, 2002)


f ---h=0.05 cm
---h=0.02 em
II


11 i


250






7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


References

[1] Faraday, M. 1831 On the Forms and States assumed by
Fluids in contact with fibrating elastic surfaces. Phil. Trans.
R. Soc. Lond. 52, 319-340.

[2] Benjamin, T. B. and Ursell, F. 1954 The stability of a
plane free surface of a liquid in vertical periodic motion.
Proc. R. Soc. Lond. A 225, 505-515.

[3] Kumar, K. and Tuckerman, L. S. 1994 Parametric
instability of the interface between two fluids. J. Fluid.
Mech. 279, 49-68.

[4] Shukla, P. K. and Narayanan, R. 2002 The effect of
time-dependent gravity with multiple frequencies on the
thermal convective stability of a fluid layer. Intl. J. Heat
Mass Transfer 45 (19), 4011-4020.




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