Planar Geometry Ferrofluid Flows in Spatially Uniform Sinusoidally Time-varying Magnetic

Fields

Shahriar Khushrushahi, Alexander Weddemann, Young Sun Kim and Markus Zahn

Massachusetts Institute of Technology, Cambridge, MA, USA

Presented at the 2011 COMSOL Conference in Boston

Ferrofluids

• Ferrofluids – Nanosized particles in

carrier liquid (diameter~10nm)

– Super-paramagnetic, single domain particles

– Coated with a surfactant (~2nm) to prevent agglomeration

• Applications – Hermetic seals (hard

drives) – Magnetic hyperthermia for

cancer treatment

Rp

d

N

S

Md

adsorbeddispersant

permanentlymagnetized core

solvent molecule

2 S. Odenbach, Magnetoviscous Effects in Ferrofluids: Springer, 2002.

Motivation

• Prior ferrofluid problems solved in COMSOL are usually in spherical and cylindrical geometries

• Ferrofluid pumping in planar geometry subjected to perpendicular and tangential magnetic fields

– Well posed problem with analytical solutions

• Traditionally solved using mathematical packages such as Mathematica

– Can COMSOL replicate these results?

3

Planar Geometry Setup

z yv x , ω x z yv i i

4

How to impose Bx field?

(a)

DC Current source gives H=NI/s

(b)

V=Λ0δ(t)->B=Λ0/A

X. He, "Ferrohydrodynamic flows in uniform and non-uniform rotating magnetic fields," Ph.D thesis, Dept. of Electrical Engineering and Computer Science, MIT, Cambridge, MA, 2006. S. Khushrushahi, "Ferrofluid Spin-up Flows in Uniform and Non-uniform Rotating Magnetic Fields," PhD, Dept. of Electrical Engineering and Computer Science, MIT, Cambridge, 2010.

5

• Extended Navier-Stokes Equation • Boundary condition on v,

• Conservation of internal angular momentum • Boundary condition on ω unless η’=0,

ρ [kg/m3] is the ferrofluid mass density, p [N/m2] is the fluid pressure, ζ [Ns/m2] is the vortex viscosity, η [Ns/m2] is the dynamic shear viscosity, λ [Ns/m2] is the bulk viscosity, ω [s−1] is the spin velocity of the ferrofluid, v is the velocity of the ferrofluid, J [kg/m] is the moment of inertia density, η ’ [Ns] is the shear coefficient of spin viscosity and λ’[Ns] is the bulk coefficient of spin viscosity, φ[%] is the magnetic particle volume fraction

Governing Equations

20( • ) ( • ) 2 ( ) ( • ) ( )p g

t

xv v v M H v v i

20 ' '( • ) ( ) 2 ( 2 ) ( ') ( • )J

t

v v

M H

6

Incompressible flow =0

=0

=0

=0

Neglecting Inertia

Neglecting Inertia

32

)( 0wallr R v

( ) 0wallr R ω

Magnetic Field Equations

• Maxwell’s equations for non-conducting fluid

• Assumption

• Magnetic Relaxation Equation

• Langevin Equation

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01• ( ) 0efft

v

MM M M M

0 00

1[coth( ) ], d p

s

H M Va a

a kT

M M

1 1 1

eff B N 0

0

,3 1 a

B N

p

h

K Ve

k f kTV xp

T

Ms [Amps/m] represents the saturation magnetization of the material,Md [Amps/m] is the domain magnetization (446kA/m for magnetite), Vh is the hydrodynamic volume of the particle,Vp is the magnetic

core volume per particle, T is the absolute temperature in Kelvin, k = 1.38 × 10−23 [J/K] is Boltzmann’s constant, f0 [1/s] is the characteristic frequency of the material and Ka is the anisotropy constant of the

magnetic domains

0 0xx

dBB constant

dx B

0 B H M

, , j t j t j tRe e Re e Re e B HHMM B

0 0zz

dHH constant

dx H

eq fluidM H

Substituting in Relaxation Equation

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0

0

00

0 0

2

0

0 0 0

2

0

χτ τ

χτ τ

( )

χ ( 1) /

1 ( 1 χ )

χ 1 χ /

1 ( 1 χ )

xx z xy

zz x zy

xx x x x x

z xy

x

y

z x y

z

y

MM M H

MM M H

BB H M H M

H

j

j

j

j j

j

BM

HM

j

B

j

( )

( )

[ ( ) ]

[ ( ) ]

j tx z

j tx z

B Re B B x e

H Re H x H e

x z

x z

i i

i i

Force and Torque Densities

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2*0 0

x z

* * *0

y 0

μ μdRe F ,F 02 dx 4μ 1Re T Re μ2 2

x

z x x z z

M

M B M M H

F M

T

H

M H

2

2

2

y 2

2' 0

d'0 2dxdd0 T 2 2 '

dx dxμ ρgx4

y z

yzy

x

d vp

z dx

v

p p M

Linear and Angular Momentum Eqns

Normalization and Substitution

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0 0 0 0

2 2 2 2 2 20 0 0 0 0 0 0 0 0 0

ˆ ˆ ˆ, , , , , , ,

2 ' 2, , , ,

zz y y

y

y

v

T

xx v

H H H d d

p d pT

H H H d H z H z

H M BH M B

0

20

0 0

20

* *

1

1 1

1

1 1

1 Re2

y z x

x

y

z x y

y

y z z

z

x x z

H BM

j j

j H BM

j j

T MM MH

j

B

2

2

2

2

' 12

0

0

2

y z

yzy y

d vp

z dx dx

ddv

d

dT

x dx

Torque Density

• Small spin limit Torque Density

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• Analytical Torque Density

01

2 2 *0 0 0

0 22 2 20 0

2 2 22 20

022 2 2

0 0

1

1

lim

R

1

1

12

e

yy y

z x

x z

T

HT

B H

T

j B

COMSOL Setup

• Linear Momentum Equation

– 2D Incompressible Navier Stokes Module

COMSOL Subdomain quantities

Value

ρ 0

η

Fx,Fy 12

2

2

' 102

zy dd vp

z dx dx

12

yd

dx

Inlet BC – Pressure, No viscous Stress

Outlet BC, Normal Stress f0=0

'', 0o

pp p

z

No Slip BC

0zv

COMSOL Setup

• Angular Momentum Equation

– General PDE Equation

COMSOL Subdomain quantities

Value

Г 0,0

F

ea,da 0,0 13

2

22 0yzy y

ddv

dx dxT

2

22 yzy yT

ddv

dx dx

Boundary Conditions

COMSOL Quantities

All walls (if ) Dirichlet boundary condition R= - , G=0

All walls (if ) Neumann boundary condition G=0

y

0

0

COMSOL Setup

• Magnetic Relaxation Equation

– 2D Perpendicular Induction Currents, Vector Potential

COMSOL Subdomain

quantities

Value

M

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Boundary Conditions

COMSOL Quantities

All walls H0 =

0

20

0 0

20

1,

1 1

1

1 1

y z x

y

z x y

y

H B

j j

j H B

j

j

j

,z x x xH H B M

η’≠0 Results, Weak Rotating Fields

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Parameters used – 0'1, 1, 1, 0.00001, 1, ' 0.01p

z

S. Khushrushahi, "Ferrofluid Spin-up Flows in Uniform and Non-uniform Rotating Magnetic Fields," PhD, Dept. of Electrical Engineering and Computer Science, MT, Cambridge, 2010.

η’≠0 Results, Weak Rotating Fields

16

Parameters used – 0'1, 1, 1, 0.00001, 1, ' 0.01p

z

S. Khushrushahi, "Ferrofluid Spin-up Flows in Uniform and Non-uniform Rotating Magnetic Fields," PhD, Dept. of Electrical Engineering and Computer Science, MT, Cambridge, 2010.

η’=0 Results, Strong Rotating Fields

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Parameters used – 0'1, 1, 1, 0.00001, 1, ' 0p

z

S. Khushrushahi, "Ferrofluid Spin-up Flows in Uniform and Non-uniform Rotating Magnetic Fields," PhD, Dept. of Electrical Engineering and Computer Science, MT, Cambridge, 2010.

η’=0 Results, Strong Rotating Fields

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Parameters used – 0'1, 1, 1, 0.00001, 1, ' 0p

z

Small Spin Velocity limit does not hold

S. Khushrushahi, "Ferrofluid Spin-up Flows in Uniform and Non-uniform Rotating Magnetic Fields," PhD, Dept. of Electrical Engineering and Computer Science, MT, Cambridge, 2010.

“Kinks” for special parameters

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Parameters used – 0'1, 0.0592, 1, 5, ' 0p

z

L. L. V. Pioch, "Ferrofluid flow & spin profiles for positive and negative effective viscosities," Masters of Engineering, Dept. of Electrical Engineering and Computer Science, MIT, 1997. M. Zahn and L. Pioch, "Ferrofluid flows in AC and traveling wave magnetic fields with effective positive, zero or negative dynamic viscosity," J. Magn. Magn. Mater., vol. 201, p. 144, 1999. M. Zahn and L. L. Pioch, "Magnetizable fluid behaviour with effective positive, zero or negative dynamic viscosity," Indian Journal of Engineering & Materials Sciences, vol. 5, pp. 400-410, 1998.

Conclusions

• Ferrohydrodynamic flows are difficult to model – Coupling of five vector equations

• Linear and angular momentum equations • Gauss’s law for magnetic flux density • Ampere’s law with no free current • Ferrofluid magnetic relaxation equation

• Solving the basic planar geometry ferrofluid pumping problem is valuable before moving to cylindrical and spherical geometries

• COMSOL gives identical results to prior software of choice - Mathematica

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