Presented at the 2011 COMSOL Conference in Boston
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

permanently magnetized core •
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 d
N
M d
R p
S solvent molecule adsorbed dispersant
S. Odenbach, Magnetoviscous Effects in Ferrofluids: Springer, 2002.
2

•
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

v    i     i y
4

x
(a)
(b)
DC Current source gives H=NI/s
V=Λ
0
δ(t)->B=Λ
0
/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.
5
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.

•
Extended Navier-Stokes Equation
=0



 v
 t
 v v

 p

0
=0
( • ) H 2   (       )
Incompressible flow v  g i x
Neglecting Inertia
•
Boundary condition on v, v ( r

R wall
)  0
•
Conservation of internal angular momentum
J


 
 t
=0
 v 


 
0
(
=0
M

H
 v    '   ' 2   
3
2

Neglecting Inertia
•
Boundary condition on ω unless η’=0, ω ( r

R wall
ρ [kg/m
[Ns/m 2
3 ] is the ferrofluid mass density, p [N/m
] is the bulk viscosity, ω [s
2 ] is the fluid pressure, ζ [Ns/m 2 ] is the vortex viscosity, η [Ns/m 2 ] is the dynamic shear viscosity, λ
−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
’
6

•
Maxwell’s equations for non-conducting fluid
M 
Re

M e

, B 
Re

B e

, H 
Re

H e

B 0 dB dx x
0
B x
 constant
H dH z
0
H z
 constant dx
B  
0

•
Assumption
M eq
  H fluid

•
Magnetic Relaxation
Equation

M
 t
 v •
M
 
M


1 eff
(
M

M
0
•
Langevin Equation
M
0

M s a

1 a
], a


0
H M V d p kT

1 eff


1
B


1
N

B
 3
V h
 k
0
T
, 
N
 f
1
0 e xp


K V a kT p


M s
[Amps/m] represents the saturation magnetization of the material, M d
[Amps/m] is the domain magnetization (446kA/m for magnetite), V h
is the hydrodynamic volume of the particle,V p
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, f
0
[1/s] is the characteristic frequency of the material and K magnetic domains a
is the anisotropy constant of the
7

j

M x
  y
M j

M z
  y
M x z

M x 
χ
τ τ
0

M z 
χ
τ τ
0
H x
H z
B x
 
0
(
H x

M x
) 
H x

B
 x
0

M x

0
M x
M z


χ
0
H z j



  2  j
  j

χ
0
H z
 j
 1 χ
0
 
B x


  2  j
  j

B y
0


B
 [ x i x
 z
( ) ] e
(
H
 [ x
( ) i x

H z i z
] e
(
)
)
8

μ
2
0
μ
2
0
Re
Re


 
H
* 

 F x
 


M H
* 

 T y

1
2
Re d

μ dx 4
0


M B x
*
M
 x
2
μ
0


M x
*

M z

H z
 

0  

 p z
'

  



2  d v dx z d
 y dx

    
2  y



  ' d d 2 dx v
2 z
2  y dx 2 p
'  
μ
4
0
M x
2
 ρgx
9

  
T y


T y
, H

0
H
0
2

,  
, M 
H
0
2 

0
H
0
2 
H
0
,  
, B



0
H
0
 '

0
H
0
2  d
2
, x
, 

 x d
, v z

2 

0
H
0
2  v z
 d
,
,
 p

 z
 y
   y

, d
 p


0
H
0
2  z
M x
M z


 y
2 
 y
H z


 j
  1 j
  1 j
1

B x

0

0
 y
2

 j
 j
  1


0
 j
H z

B
 x y
1 
0



T y

M z
B
* x

M x
*

H z

M z

0  
 p
'
 z

T y
  
 

 d
 y dx dv z d x



1
2
 
 2  y
   d
2  y dx
2 d
2 v dx
2 z
 0
10

•
Analytical Torque
Density
•
Small spin limit Torque
Density
 lim y
 1

T y
T
0


T
0
  y
 

0
R e


2
0


B x

0
2

1 

1


0
 j
H z

  2
2
2
2
1 
0
 
0
2  2

1  
0
  2
2
 2

  H B z x
*
 1  
0
 2

11

•
Linear Momentum
Equation
–
2D Incompressible
Navier Stokes Module
0  
 p
'
 z
 

 d
 y dx



1
2
  d
2 dx v
2 z
Inlet BC – Pressure, No viscous Stress p o
  p
',

 p z
'
 0
COMSOL
Subdomain quantities
Value
No Slip BC v
 z
0
Outlet BC, Normal Stress f
0
=0
ρ
η
Fx,Fy
0

1
2
 

 d
 y dx

 12

•
Angular Momentum
Equation
–
General PDE Equation

T y
   dv z dx
 2  y
   d
2  y dx
2
 0
Boundary
Conditions
  
  
COMSOL Quantities condition
 boundary condition
G=0
COMSOL
Subdomain quantities
Г
F e a
,d a
Value
0,0

T y
   dv z dx
 2  y
   d
2  dx
2 y
0,0
13

•
Magnetic Relaxation
Equation
–
2D Perpendicular
Induction Currents,
Vector Potential
COMSOL Subdomain quantities Value
Boundary
Conditions
All walls
COMSOL Quantities
H
0
= z
, x

B x

M x
M

 y
2

0

 2 y

 j
H y z
 

1

 j
  j
1

1
B x

0

 j

 j
  1

0

 j
H z

B x
 y
1 
0


,
14

Parameters used –

0
 1,   1,   1,
 p
'
 z
 0.00001,   
15
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.

Parameters used –

0
 1,   1,   1,
 p
'
 z
 0.00001,   
16
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.

Parameters used –

0
 1,   1,   1,
 p
'
 z
 0.00001,   
17
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.

Small Spin Velocity limit does not hold
Parameters used –

0
 1,   1,   1,
 p
'
 z
 0.00001,   
18
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.

Parameters used –

0
 1,   0.0592,
 p
 z
'
1,  
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.
19
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.

•
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
20