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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

Ferrofluids

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

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

v    i     i y

4

How to impose B

x

field?

(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.

Governing Equations

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

Magnetic Field Equations

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

Substituting in Relaxation Equation

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

Force and Torque Densities

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

 

Linear and Angular Momentum Eqns

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

Normalization and Substitution

  

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

Torque Density

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

COMSOL Setup

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

COMSOL Setup

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

COMSOL Setup

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

,

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η’≠0 Results, Weak Rotating Fields

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.

η’≠0 Results, Weak Rotating Fields

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.

η’=0 Results, Strong Rotating Fields

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.

η’=0 Results, Strong Rotating Fields

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.

“Kinks” for special parameters

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.

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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