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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: etzirtzilakis@teimes.gr ;

web page: www.tzirtzilakis.myp.teimes.gr

2Department of Mathematics, University of IoanninaIoannina, 45110, Greece

e-mail: mxenos@cc.uoi.gr ; web page: http://www.math.upatras.gr/~maik/

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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

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IntroductionAPPLICATIONS

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 bleedingIsolation 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)

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Introduction

APPLICATIONS

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• E.E. Tzirtzilakis, “A mathematical model for blood flowin 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 IXInternational Symposium on Transport Properties in ThermalFluids Engineering, Singapore, Pacific Center of Thermal FluidEngineering, 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 FluidDynamics, In: Fluid Dynamics at Interfaces, W. Shyy and R.Narayanan (Eds.), Cambridge University Press, 439-452.

Biomagnetic Fluid Dynamics (BFD)

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Mathematical Model

Mathematical Model (E. Tzirtzilakis, FHD, MHD)

V 0

2o

DV p F V J B M HDt

Continuity

Momentum

MHD

FHD H J V B

B H M 0 Magnetic Field

2p o

DT M DH J JC T k TDt T Dt

2 2 2 22 2 2u v w v u w v u w 2 u v w2x y z x y y z z x 3 x y z

Magnetization: M(ρ,Η,Τ)

Energy

M H cM K T T

c1

1

T TM MT

o

o

mH TM mN cothT mH

cM K H T T

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Mathematical Formulation

u v = 0x y

2 2u u p H 1 u u2u v = Mn H N uH vH HF y x y 2 2x y x x Re x y

2 2v v p H 1 v v2u v Mn H N vH uH HF x x y 2 2x 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 2o o

r2

o oF 2

r

L u= (Reynolds number),Re

H L HaN = = (Stuart number, MHD),u Re

HMn = (FHD Magnetic number).u

2 2

| |( , ) = .( ) ( )

bH x yx a y b

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Stream Function-vorticity formulation?

2 2u u p H 1 u u2u v = Mn H N uH vH HF y x y 2 2x y x x Re x y

2 2v v p H 1 v v2u v Mn H N vH uH HF x x y 2 2x y y y Re x y

2H H H HMn H Mn Mn HF F Fy x y x y x

2H H H HMn H Mn Mn HF F Fx y x y x y

(1)

(2)

(1)

(2)

(3)

(4)

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

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Primitive variables approachSimple – 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 temperaturedependence, 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 412271-2293.

ρ=1050kgr/m3, μ=3.210-3 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

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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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