Tzirtzilakis2013-07

Biomagnetic Fluid Flow in a
Driven Cavity
E.E. Tzirtzilakis1 and M.A. Xenos2
1Department
of Mechanical and Water Resources Engineering,
Technological Educational Institute of Messolonghi,
Messolonghi, 30200, Greece
e-mail: [email protected] ;
web page: www.tzirtzilakis.myp.teimes.gr
2Department
of Mathematics, University of Ioannina
Ioannina, 45110, Greece
e-mail: [email protected] ;
web page: http://www.math.upatras.gr/~maik/
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Introduction
* Galinos (129-201 A.D)
Magnet as purgative
* Franz Anton Mesmer (1734-1815)
Influence of biological magnetism (Mesmerism)
* Durval (end 19th century )
Magnetic bracelet
* Moscow 1976
Hypertension - Headache
* Pauling, Coryell 1936
* 1940-
Magnetic field – hemoglobin of red blood
cells
Synthesis of magnetic fluids
* Neuringer, Rosensweig 1964
FerroHydroDynamics
* Russians (Zaitsev, Shliomis, Cvetkov)
* Rosensweig 1980
Book: “Ferrohydrodynamics”
* 1983 -
Magnetic field → hemoglobin of red
blood cells
* Y. Haik, C.J. Chen, V. Pai 1996
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Biomagnetic Fluid Dynamics
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Introduction
APPLICATIONS
Drug targeting
(nano-particles + drug)
Medical devices
Reduction of bleeding
Isolation of organs
Diagnosis
Therapy
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-cancer cells
-clotted blood
-Magnetaphaeresis
-Blood pumps
-Cell separation (red blood cells or ill
natured)
-Technical muscles
-Addition of magnetic particles in the
arteries
-Increment of contrast, clearer imaging, addition
of magnetic particles (hollow organs)
-MRI (Magnetic Resonance imaging)
-X-Rays
-Hyperthermia (cancer cells, eye injuries
without medication)
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Introduction
APPLICATIONS
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Mathematical Model
Biomagnetic Fluid Dynamics (BFD)
• Haik, Y., Chen J.C. and Pai, V.M., 1996. Development of biomagnetic fluid dynamics, In Proceedings of the IX
International Symposium on Transport Properties in Thermal
Fluids Engineering, Singapore, Pacific Center of Thermal Fluid
Engineering, S.H. Winoto, Y.T. Chew, N.E. Wijeysundera,
(Eds.), Hawaii, U.S.A., June 25-28, 121--126.
• Haik, Y., Pai, V. and Chen, C.J., 1999. Biomagnetic Fluid
Dynamics, In: Fluid Dynamics at Interfaces, W. Shyy and R.
Narayanan (Eds.), Cambridge University Press, 439-452.
• E.E. Tzirtzilakis, “A mathematical model for blood flow
in magnetic field”, Physics of Fluids, Vol. 17, 077103,
2005.
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Mathematical Model
Mathematical Model (E. Tzirtzilakis, FHD, MHD)
Continuity
Momentum
Magnetic Field
 
MHD
V  0



2  

DV

 p  F   V  J  B  oMH
Dt
  
 
FHD
 H  J   V  B
    
 B   H  M  0




 
M DH J  J
DT
Cp
 oT

 k2T  
Dt
T Dt

Energy
 u 2  v 2  w 2   v u 2  w v 2  u w 2 2  u v w 2
  2       
   

    
   


x

y

z

x

y

y

z

z

x
3

x

y
z 
  

    
 
 

Magnetization: M(ρ,Η,Τ)
M  K  Tc  T
M  H

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 Tc  T 
M  M1 

 T1 
M  KH  Tc  T 

T 
  mH 

M  mN coth  o




T
mH


o


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Mathematical Formulation
u v
 =0
x y
1   2u  2 u 
u
u
p
H


2


u  v =   Mn H
 N  uH  vH H  

F
y
x
y
2
2

x
y
x
x

 Re x
y 

v
v
p
H
1   2v  2v 


2


 N  vH  uH H  

u  v    Mn H
F
x
x
y
2
2
x
y
y
y

 Re  x
y 

Dimensionless numbers :
Boundary conditions :
UpperWall ( y = 1,0  x  1) : u = 1, v = 0. 
LowerWall ( y = 0,0  x  1) : u = 0, v = 0. 

LeftWall ( x = 0,0  y  1) : u = 0, v = 0. 
RightWall ( x = 1,0  y  1) : u = 0, v = 0 
H ( x, y ) =
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Re =
L  ur

(Reynolds number),
o2 H o2 L Ha 2
N=
=
(Stuart number, MHD),
Re
 ur
| b | o  H o2
. (FHD Magnetic number).
Mn2F =
22
u

( x  a )  ( y  br)
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Stream Function-vorticity
formulation?
1   2u  2 u 
u
u
p
H


2


u  v =   Mn H
 N  uH  vH H  

F
y
x
y
2
2
x
y
x
x

 Re  x
y 

v
v
p
H
1   2v  2v 


2


 N  vH  uH H  

u  v    Mn H
F y
x
x
y
2
2

x
y
y

 Re x
y 

(1)   Mn H H
y  F x
(1)
(2)
H H
2H

 Mn H
  MnF
F yx
y x


H 
H H
2H
 Mn
 Mn H
(2)  MnF H
F x y
F xy
x 
y 
(3)
(4)
(4)-(3) = 0 ???
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Primitive variables approach
Simple – staggered grid – upwind scheme – “differed correction” approach
Difficulties with source term of FHD
J.H. Ferziger, M. Peric, “Computational Methods for Fluid Dynamcs”, Springer
Verlang, Berlin, 3rd ed, 2002.
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MAGNETIC PARAMETERS
Saturation Magnetization M0=40Am-1.
Haik Y., PAi V Chen CJ, 1999. Biomagnetic fluid dymanics. In: Fluid Dynamics at Interfaces,
Shyy W. and Narayanan R. (eds), Cambridge University Press, pp. 439-452.
σ = 0.8sm-1
Jaspard F. and Nadi, M., 2002. Dielectric properties of blood: an investigation of temperature
dependence, Physiological Measurement 23 547-554.
Gabriel, S., Lau R.W. and Gabriel, C., 1996. The dielectric properties of biological tissues: III.
Parametric models for the dielectric spectrum of tissues, Physics in Medicine and Biology 41
2271-2293.
ρ=1050kgr/m3, μ=3.210-3 kgm-1s-1
Pedley, T. J., 1980. The fluid mechanics of large blood vessels, Cambridge University Press.
L=5x10-2m b=2.5 10-3m
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Re=400
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MAGNETIC PARAMETERS
Bo and corresponding values of MnF
Bo and corresponding values of N
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Results
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Results
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Results
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