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An Accurate Mode-Selection Mechanism for Magnetic Fluids in a Hele-Shaw
Cell
David P. JacksonDickinson College, Carlisle, PA USA
José A. MirandaUniversidade Federal de Pernambuco, Recife,
Brazil
Slide 2
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The Birthday Girl!
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What is a Ferrofluid?
• Colloidal suspension of tiny magnets (10 nm) coated with a molecular surfactant
• Thermal motion keeps the dipoles uniformly distributed and randomly oriented unless there is a magnetic field present
• The dipoles align in a magnetic field
For details, see Ferrohydrodynamics, Ronald E. Rosensweig (Cambridge University Press, 1985),
(Dover, 1997)
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Basic Physical Situation
Ferrofluid is confined between two closely spaced glass plates and placed in a
perpendicular magnetic field
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Experimental Setup
Hele-Shaw cell
Light
Video Camera
Helmholtz Coils
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Qualitative Description
No magnetic fieldUniform magnetic field
Parallel-plate capacitorCurrent RibbonUniform magnetization collinear with field
Outward magnetic pressure competes with surface tension that results in a fingering instability
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Sample Evolution
Single drop experimental example
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Controlling the Instability• How can we control the fingering instability?
• Add an azimuthal field that falls off with distance
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Essential Physics
• Outward force caused by a magnetic pressure due to dipole alignment from normal field
• Inward force caused by surface tension that tends to minimize surface area
• Inward force caused by the radial gradient of the azimuthal field
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Governing Equations
Navier-Stokes:
Hele-Shaw Approximations:
Laplace’s Equation:
Interfacial BC:
€
ρ Dr v
Dt= −
r ∇P + η∇ 2r
v + Mr
∇H
€
rv x,y( ) = −
b2
12η
r ∇Π
€
ˆ n ⋅∂
r r α , t( )
∂t= ˆ n ⋅
r ∇Π r
r α ,t( )€
∇2Π = 0€
Π=1
hP x,y,z( )dz
0
h
∫ +2Mn
bψ n x, y,h( ) −
1
2μ0χ
I
2πr
⎛
⎝ ⎜
⎞
⎠ ⎟2
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Conformal Mapping
• Solve Laplace’s Eq. on unit disk (Poisson integral formula)• Map exists from complex (simply connected) domain to unit
disk• Interfacial BC gives evolution equation for domain boundary
• Equation looks like:
€
z = f t ω( )
plane
€
z plane
€
ω
€
∂f
∂t ω= e iα
= ω∂ω f Aℜ ω∂ω A Π α( ){ }[ ]
ω∂ω f2
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪ω= e iα
Bensimon et. al., Rev. Mod. Phys. 58, 977 (1986)
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Numerical Evolution
Destabilizing (normal) field only!
QuickTime™ and aPhoto - JPEG decompressor
are needed to see this picture.
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Linear Stability Analysis
Specifying and linearizing the equation of motion leads to growth rates
where
€
˜ λ n =12ηR0
3
σb2
⎛
⎝ ⎜
⎞
⎠ ⎟λ n = n NB
⊥Dn p( )− NB − (n2 −1)[ ]
€
p =2R0
b
€
NB⊥ =
μ0M 2b
2πσ
€
NB =μ0χI2
4π 2σR0
€
r θ, t( ) = R0 + ς n cos nθ( ) eλnt
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Growth Rates I
UnstableStable
Destabilizing (normal) field only!
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Growth Rates II
UnstableStable
Single unstable mode!
Possible mode-selection mechanism!
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Stability Phase Portrait
• Solid lines are neutral stability curves
• Gray areas denote regions where a particular mode is the fastest growing
• Diamonds denote specific values used for simulations
n=3
n=2
n=4
n=5
Single Unstable Modes
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Precisely Selected Modes
Simulations run with identical initial conditions - Bond numbers chosen so that there is only a single unstable mode
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Simulations with NB=1.5
• Simulations with NB
=1.5
• Initial condition is left-right n=2 mode
• As NB increases, more modes become stable
• When only a single mode is unstable, the initial condition is drown out
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Simulations with NB=2.5
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Summary
• An azimuthal magnetic field can be used to control the normal field fingering instability of a magnetic fluid in a Hele-Shaw cell
• By tuning the azimuthal and normal fields, one can produce a situation in which a single unstable mode exists
• Numerical simulations demonstrate that mode growth can be accurately selected
• Large enough azimuthal fields completely stabilize the interface
An Accurate Mode-Selection Mechanism for Magnetic Fluids in a Hele-Shaw
Cell
David P. JacksonDickinson College, Carlisle, PA USA
José A. MirandaUniversidade Federal de Pernambuco, Recife,
Brazil
Slide 2
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