An Accurate Mode-Selection Mechanism for Magnetic Fluids in a Hele-Shaw Cell

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

Aug. 15, 2007 Mar del Plata, Argentina

D.P. Jackson 2/20

The Birthday Girl!


D.P. Jackson 3/20

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)


D.P. Jackson 4/20

Basic Physical Situation

Ferrofluid is confined between two closely spaced glass plates and placed in a

perpendicular magnetic field


D.P. Jackson 5/20

Experimental Setup

Hele-Shaw cell

Light

Video Camera

Helmholtz Coils


D.P. Jackson 6/20

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


D.P. Jackson 7/20

Sample Evolution

Single drop experimental example


D.P. Jackson 8/20

Controlling the Instability• How can we control the fingering instability?

• Add an azimuthal field that falls off with distance


D.P. Jackson 9/20

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


D.P. Jackson 10/20

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


D.P. Jackson 11/20

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)


D.P. Jackson 12/20

Numerical Evolution

Destabilizing (normal) field only!

QuickTime™ and aPhoto - JPEG decompressor

are needed to see this picture.


D.P. Jackson 13/20

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


D.P. Jackson 14/20

Growth Rates I

UnstableStable

Destabilizing (normal) field only!


D.P. Jackson 15/20

Growth Rates II

UnstableStable

Single unstable mode!

Possible mode-selection mechanism!


D.P. Jackson 16/20

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


D.P. Jackson 17/20

Precisely Selected Modes

Simulations run with identical initial conditions - Bond numbers chosen so that there is only a single unstable mode


D.P. Jackson 18/20

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


D.P. Jackson 19/20

Simulations with NB=2.5


D.P. Jackson 20/20

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

Documents

An Accurate Mode-Selection Mechanism for Magnetic Fluids in a Hele-Shaw Cell