Biomagnetic Fluid Flow in a Driven Cavity

Biomagnetic Fluid Flow in a Driven Cavity

E.E. Tzirtzilakis

and M.A. Xenos

1

Department 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

2

Department of Mathematics, University of Ioannina Ioannina, 45110, Greece

e-mail: [email protected] ;

web page: http://www.math.upatras.gr/~maik/

Introduction

* Galinos (129-201 A.D)

* Franz Anton Mesmer (1734-1815)

Magnet as purgative

Influence of biological magnetism (Mesmerism)

* Durval (end 19th century ) Magnetic bracelet

* Moscow 1976 Hypertension - Headache

Magnetic field – hemoglobin of red blood cells

* Pauling, Coryell 1936

* Neuringer, Rosensweig 1964 FerroHydroDynamics

* 1940- Synthesis of magnetic fluids

* Russians (Zaitsev, Shliomis, Cvetkov)

* Rosensweig 1980 Book: “Ferrohydrodynamics”

* 1983 - Magnetic field → hemoglobin of red

blood cells

* Y. Haik, C.J. Chen, V. Pai 1996 Biomagnetic Fluid Dynamics

Introduction

APPLICATIONS

Therapy -Hyperthermia (cancer cells, eye injuries

without medication)

-Increment of contrast, clearer imaging, addition of magnetic particles (hollow organs)

-MRI (Magnetic Resonance imaging) -X-Rays

Diagnosis

-Addition of magnetic particles in the arteries

Reduction of bleeding Isolation of organs

-Blood pumps

-Cell separation (red blood cells or ill natured)

-Technical muscles Medical devices

-cancer cells -clotted blood

-Magnetaphaeresis Drug targeting

(nano-particles + drug)

Introduction

APPLICATIONS

• E.E. Tzirtzilakis, “A mathematical model for blood flow in magnetic field”, Physics of Fluids, Vol. 17, 077103, 2005.

Mathematical Model

• Haik, Y., Chen J.C. and Pai, V.M., 1996. Development of bio- magnetic 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.

Biomagnetic Fluid Dynamics (BFD)

Mathematical Model

Mathematical Model (E. Tzirtzilakis, FHD, MHD)

V 0

    

2

o

DV p F V J B M H

 Dt        

        Continuity

Momentum

MHD

  FHD

H J V B

     ^{    }

 

B H M 0

   ^{   }  ^  Magnetic Field

2

p o

DT M DH J J

C T k T

Dt T Dt

 

     

 

 

2 2 2 2

2 2 2

u v w v u w v u w 2 u v w

2 x y z x y y z z x 3 x y z

                           

                                                          

Magnetization: M(ρ,Η,Τ) Energy

M   H M K T  

 T 

c 1

1

T T

M M T

  

  

 

o

o

mH T

M mN coth

T mH

     

           





M  K H T   T

Mathematical Formulation

  

 

u v x y = 0

 

                              

2 2

u u p H 2 1 u u

u v = Mn H N uH vH H

F y x y 2 2

x y x x Re x y

 

                     

         

2 2

v v p H 2 1 v v

u v Mn H N vH uH H

F x x y 2 2

x y y y Re x y

  

   

   

   

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

Boundary conditions : Dimensionless numbers :

r

2 2 2

o o

r 2

o o

F 2

r

= L u (Reynolds number), Re

H L Ha

N = = (Stuart number, MHD),

u Re

Mn = H (FHD Magnetic number).

u





 



 



2 2

( , ) = | | .

(  )  (  ) H x y b

x a y b

Stream Function-vorticity formulation?

 

                              

2 2

u u p H 2 1 u u

u v = Mn H N uH vH H

F y x y 2 2

x y x x Re x y

 

                     

         

2 2

v v p H 2 1 v v

u v Mn H N vH uH H

F x x y 2 2

x y y y Re x y

          

       

H H H

H

Mn H Mn Mn H

F F F

y x y x y x

 

        

       

H H H

H

Mn H Mn Mn H

F F F

x y x y x y

(1) (2)

(1) (2)

(3) (4)

(4)-(3) = 0 ???

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, 3

ed, 2002.

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

Re=400

MAGNETIC PARAMETERS

B

and corresponding values of Mn

B and corresponding values of N

Results

Results

Results