Thermomagneticconvectionandopenquestionsinrheology ......Paramagnetic melts or solutions (MnCl2) χ...

Thermomagnetic convection and open questions in rheology

of ferrocolloids

Sergey A. Suslov

Swinburne University of Technology, Australia

Alexandra A. Bozhko, Alexandre Sidorov and Gennady F. Putin

Perm State National Research University, Russia

October 24, 2012

1

Outline:

• What is ferrofluid?

• What is magneto-convection?

• Potential applications.

• Problem definition and approach.

• Stability and energy analyses.

• Physical mechanisms discovered.

• Open questions of rheology.

2

Natural magneto-polarisable fluids:Fluid Magnetic susceptibility

Diamagnetic protein solutions (Lysozime) χ ∼ −10−5

Paramagnetic melts or solutions (MnCl2) χ ∼ 10−4 − 10−3

Ferro-magnetic fluid (ferro-colloid):

Base: kerosene or mineral oil;

Solid magnetic phase: single domain mag-

netite, iron or cobalt particles (size ∼10

nm, concentration ∼ 10%);

Surfactant: mono-layer of oleic acid.

χ ∼ 5

3

Nature of magnetic ponderomotive (Kelvin)

force

Electrostatic analogy: Magnetic force:

~F = M∇H ,

M—magnetisation,

H—magnetic field

Stronger magnetised particles

are forced to regions with a

stronger magnetic field.

4

Two types of convection

Gravitational convection:

• Local heating leads to fluid expansion

∆ρ = −β∆T ;

• Less dense fluid rises due to buoyancy.

Hot Cold

∆ρ+ ρ

g

∆ρ− ρ

Magnetoconvection:

• Local heating leads to partial de-

magnetisation ∆M = −K∆T ;

• Stronger magnetised fluid is driven to

regions with stronger magnetic field.

H∆

Hot

∆M+ MCold

∆M− M

5

Applications: Geometry sketch:

• Targeted drug delivery in cancer

treatment

• Magnetic sealing of joints and

gaps

• Cooling of high-power loud-

speakers

• Heat exchangers in low gravity

conditions (spacecrafts)

• Convection control in protein

crystal growth

T*+Θ

Hg

2d

−ΘT*

y

x

z

0

6

Non-dimensional governing equations:

∂~v

∂t+ ~v · ∇~v = −∇P +∇2~v −Grθ~eg −Grmθ∇H ,

∂θ

∂t+ ~v · ∇θ =

1

Pr∇2θ , ∇ · ~v = 0 , ∇× ~H = ~0 ,

(1 + χ∗)∇ · ~H + (χ− χ∗)∇H · ~e∗ − (1 + χ)∇θ · ~e∗ = 0 ,

~M = [(χ− χ∗)(H −N)− (1 + χ)θ]~e∗ + χ∗~H ,

~eg = (0,−1, 0) , ~n = ~e∗ = (1, 0, 0)

Boundary conditions:[

~He − [(χ− χ∗)(H −N)∓ (1 + χ)]~e∗ − (1 + χ∗) ~H]

· ~n = 0 ,

~v = ~0 , θ = ±1 , at x = ∓1

Dimensionless parameters:

Gr =ρ2∗β∗Θgd3

η2∗

, Grm =ρ∗µ0K

2Θ2d2

η2∗(1 + χ)

, P r =η∗

ρ∗κ∗

, N =H∗(1 + χ)

KΘ

7

Basic flow solutions:

−1 0 1−0.5

0

0.5

x

v0

−1 0 1−1

0

1

2

3

4

xθ 0,

H0,

M0

−1 0 1−4

−3

−2

−1

0

x

P0

M0

H0

θ0

Typical parameter ranges:

Gr ∼ 0− 8 , Grm ∼ 0− 15 , P r ∼ 130 , χ ∼ 5 , χ∗ ∼ 5

• Uniform external magnetic field results in a nonuniform field inside the layer;

• Stronger magnetised fluid is in a weak magnetic field region.

8

Experimental equipment:

9

Experimental convection chamber:

1. Ferrofluid layer

2. Copper heat exchanger

3. Plexiglas heat exchanger

4. Plexiglas frame

5. Thermo-sensitive liquid crystal sheet

6. Thermocouples

10

Experimentally observed instability in uniform

transverse magnetic field:

11

Schematic fluid motion:

Basic buoyancy-induced flow: Flow in magnetic field:

��

��

��

��

��

��

��

��

H

12

Linearised perturbation equations (normal mode form):

σu +(α2 + iαv0 −D2

)u+DP +GrmDH0 θ +GrmΘ0D

2φ = 0 ,

σv + Dv0 u+(α2 + iαv0 −D2

)v + iαP −Grθ + iαGrmΘ0Dφ = 0 ,

σθ + DΘ0u+

(α2 −D2

Pr+ iαv0

)

θ = 0 ,

Du+ iαv = 0 ,

(

D2 −1 + χ∗

1 + χα2

)

φ−Dθ = 0 ,

σ = σR + iσI— complex amplification rate ,

α — disturbance wavenumber

Boundary conditions:

u1 = v1 = w1 = θ1 = 0 , (1 + χ)Dφ1 ±√

α2 + β2φ1 = 0 at x = ±1 ,

where D ≡ d/dx and ~H = (Dφ, iαφ).

13

Computational stability results:

0 5 10 150

10

20

30

40

Grm

Gr

0 5 10 151

1.5

2

2.5

3

3.5

4

Grm

α c

1 2

3

4 5-78

1 23

4

5

6

7

8

Thermo-gravitational+Thermo-magnetic waves

Thermo-magnetic stationary rolls

Thermo-magnetic waves

2D Stability

• Full 3D map is obtained using inverse Squire’s transformation

(Suslov, Phys. Fluids, 2008).

How does this compare with experiment?

14

Experimental stability map:

5 10 15 20 25

8

10

12

14

16

18

20

22

24

∆ T, K

H, k

A/m

• Suslov et al, Phys. Rev. E, 2012.

Need to convert H and ∆T to Gr =ρ2

∗β∗Θgd3

η2∗

and Grm = ρ∗µ0K2Θ2d2

η2∗(1+χ) .

15

Two experimental scenarios

Grm

Gr

H1

H2

(a)

H2< H

1

Grm

Gr

∆ T1

∆ T2

(b)

∆ T2>∆ T

1

• Viscosity is not constant and is not known.

• Accurate determination of Gr and Grm is currently impossible.

Practical design problems!

16

Disturbance energy balance equation:

σRΣk = Σuv +Σm1 + Σm2 +ΣGr +Σvis ,

Σk =

∫ 1

−1

(|u|2 + |v|2

)dx > 0 , Σuv = −

∫ 1

−1

Dv0ℜ(uv) dx ,

Σm1 =

∫ 1

−1

−GrmDH0ℜ(θu)︸︷︷︸

Em1

dx , Σm2 =

∫ 1

−1

GrmDθ0ℜ(Dφ u)︸︷︷︸

Em2

dx ,

ΣGr =

∫ 1

−1

Grℜ(θv)︸︷︷︸

EGr

dx , Σvis = −α2Ek −

∫ 1

−1

(|Du|2 + |Dv|2

)dx = −1 .

17

Disturbance energy integrals for selected

marginal stability points:

0 5 10 150

10

20

30

40

Grm

Gr

1 2

3

4

5−78

Σuv Σm1 Σm2 ΣGr

1 -0.006 0 0 1.006

2 -0.007 0.004 -0.002 1.005

3 -0.011 0.302 -0.086 0.795

4 -0.003 1.141 -0.318 0.180

5 0.001 1.795 -0.759 -0.036

6 0.002 1.812 -0.595 -0.219

7 -0.003 1.516 -0.350 -0.163

8 0 1.584 -0.584 0

18

Disturbance energy integrands:

−1 0 1

0

0.5

1

1.5

2

2.5

3

3.5

4

−1 0 1

0

0.5

1

1.5

2

2.5

−1 0 1−0.5

0

0.5

1

1.5

−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

−1 0 1

−1

0

1

2

3

x−1 0 1

−3

−2

−1

0

1

2

3

4

x−1 0 1

−1

−0.5

0

0.5

1

1.5

2

2.5

3

x−1 0 1

−0.5

0

0.5

1

1.5

2

x

1 2 3 4

5 6 7 8

Em 1

Em 2

EGr

Em 1

Em 2

EGr

Ponderomotive forcing

Magnetic fielddistortion

Buoyancy

0 5 10 150

10

20

30

40

Grm

Gr

1 2

3

4

5−78

19

Thermo-gravitational waves (point 1):

−1 0 1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

t= 0

y

−1 0 1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

t=T/ 4

y

−1 0 1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

t=T/ 2

y

−1 0 1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t= 3 T/ 4

x y

20

Thermo-magnetic waves (point 4):

−1 0 1

0

0.5

1

1.5

2

2.5

3

3.5

x

t= 0

y

−1 0 1

0

0.5

1

1.5

2

2.5

3

3.5

x

t=T/ 4

y

−1 0 1

0

0.5

1

1.5

2

2.5

3

3.5

x

t=T/ 2

y

−1 0 1

0

0.5

1

1.5

2

2.5

3

3.5

t= 3 T/ 4

x y

21

Summary:

• The instability in a vertical ferro-fluid layer is caused by two physical

mechanisms: thermo-gravitational (buoyancy) and thermo-magnetic.

• Three instability patterns are found: counter-propagating thermal waves

(large Gr, small Grm), (new) counter-propagating thermo-magnetic waves

(large Grm, intermediate Gr) and stationary magneto-convection rolls

(intermediate to large Grm, small Gr).

• The propagating thermal or thermo-magnetic instability waves form

horizontal or inclined convection rolls; stationary magneto-convection rolls

remain vertical.

• Knowledge of rheological properties of ferrofluids is crucial for practical

design of applications.

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Surprise, surprise:

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Thermomagneticconvectionandopenquestionsinrheology ......Paramagnetic melts or solutions (MnCl2) χ...

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