# Biomagnetic fluid flow in a rectangular duct

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4

th

GRACM Congress on Computational Mechanics

GRACM 2002

Patras 27-29 June, 2002

BIOMAGNETIC FLUID FLOW IN A RECTANGULAR DUCT

E. E. Tzirtzilakis

, N. G. Kafoussias

and P. M. Hatzikonstantinou

Department of Mathematics, Section of Applied Analysis, University of Patras, 26500 Patras, Greece

*

Department  of Engineering Science, University of Patras, 26500 Patras, Greece

e-mails:

stratis@math.upatras.gr

and

nikaf@math.upatras.gr

Keywords: Incompressible, Biomagnetic fluid, rectangular duct.

Abstract. In this work the fundamental problem of the biomagnetic fluid flow taking place in a rectangular

duct under the influence of an applied magnetic field is studied. It is assumed that the magnetization M of the

fluid is varying linearly with temperature T and magnetic field intensity H. For the numerical solution of the

problem, which is described by a coupled and non linear system of PDEs, with their appropriate boundary

conditions, the stream function-vorticity formulation is adopted and the solution of the problem is obtained

developing an efficient numerical technique based on the upwind finite differences joint with a line by line

implicit method. Results concerning the velocity and temperature field, skin friction and rate of heat transfer

indicate that the presence of magnetic field appreciable influence the flow field.

1 INTRODUCTION

Biomagnetic Fluid Dynamics (BFD) is a relatively new area in fluid mechanics investigating the fluid

dynamics of biological fluids in the presence of magnetic field. The applications in bioengineering and medicine

seem to be numerous and the research work is rapidly growing

[1-3]

A biomagnetic fluid is a fluid that exists in a living creature and its flow is influenced by the presence of a

magnetic field. Biofluids are poor conductors thus, BFD deals with no electric current and the flow is affected

only by the magnetization of the fluid in the magnetic field. Magnetization is the measure of how much the

magnetic field is affecting the magnetic fluid and generally is a function of the magnetic field intensity H and the

temperature T.

Mathematical models have been developed for the (BFD) in order to examine the flow of a biomagnetic fluid

under the action of an applied magnetic field. The implementation of these models is based on the modified

Stokes principles and on the assumption that besides the three thermodynamic variables P, ρ and T the

biomagnetic fluid behavior is also a function of magnetization M



. The derived governing equations for

incompressible fluid flow are similar to those derived for FerroHydroDynamics (FHD)



Unlike MagnetoHydroDynamics (MHD), which deals with conducting fluids, the mathematical model of

BFD ignores the effect of polarization and magnetization and the induced current is negligibly small. Thus,

unlike MHD, Lorentz force is much smaller in comparison to the magnetization force.

The most characteristic biomagnetic fluid is the blood, which can be considered as a magnetic fluid because

the red blood cells contain the hemoglobin molecule, a form of iron oxides, which is present at a uniquely high

concentration in the mature red blood cells.  It is found that the erythrocytes orient with their disk plane parallel

to the magnetic field



and also that the blood possesses the property of diamagnetic material when oxygenated

and paramagnetic when deoxygenated



. So, blood possesses the property of a magnetic material, and under

some circumstances, can be considered as diamagnetic, paramagnetic or ferromagnetic fluid



.

In the present study a simplification of this mathematical model of BFD is used to obtain numerical solution

of the differential equations describing the fluid flow (blood) in a rectangular duct under the action of a magnetic

field. We assume that the magnetization is described by a linear equation involving the magnetic intensity H and

temperature T

In order to proceed to the numerical solution the stream function-vorticity formulation is adopted

and the solution of the problem is obtained numerically developing an efficient numerical technique based on the

upwind finite differences joint with a line by line implicit method.

The results concerning the velocity and temperature field, skin friction and rate of heat transfer presented,

showed that the flow is appreciably influenced by the magnetic field. A vortex is arising and the temperature is

increasing near the area where the dipole is located. These results indicate that application of a magnetic field, in

the flow of a biomagnetic fluid, could be useful for medical and engineering applications.

E.Tzirtzilakis, N. Kafoussias, P. Hatzikonstantinou

2 MATHEMATICAL

FORMULATION

We consider the viscous, steady, two-dimensional, incompressible, laminar biomagnetic fluid (blood) flow

taking place between two parallel flat plates (duct). The length of the plates is L and the distance between them

is h, such that L/h=7. We assume that the flow at the entrance is fully developed and that the upper plate is kept

at constant temperature T

u

, while the lower at T

w

, such that T

w

u

. The origin of the Cartesian coordinate system

is located at the leading edge of the lower plate and the flow is subjected to a magnetic dipole, which is placed

very close to the lower plate and below it (see Figure 2). We assume blood to be Newtonian fluid and we discard

the rotational forces acting on the erythrocytes when entering the magnetic field. The equations describing the

flow field in this duct, under the action of the applied magnetic field, as in FHD, are the continuity, the

momentum and the energy equations.

u

v

0

x

y

+

=

(1)

2

2

0

2

2

u

u

p

H

u

u

u

v

M

x

y

x

x

x

y

ρ

+

= −

+ µ

+ µ

+

,

(2)

2

2

0

2

2

v

v

p

H

v

v

u

v

M

x

y

y

y

x

y

ρ

+

= −

+ µ

+ µ

+

,

(3)

p

0

2

2

2

2

2

2

T

T

M

H

H

c u

v

T

u

v

x

y

T

x

y

T

T

u

v

v

u

k

2

2

.

x

y

x

y

x

y

ρ

+

+ µ

+

2

=

+

+ µ

+

+

+

(4)

The boundary conditions of the problem are:

Inflow conditions

(

x 0

=

,

0 y h

≤ ≤

) :

u u(y)

=

v 0

=

T / x 0

∂ ∂ =

Outflow conditions

(

x L

=

,

0 y h

≤ ≤

) :

(R) / x 0

∂ =

(5)

Upper plate

(

y h

=

,

0 x L

≤ ≤

) :

u 0

=

v 0

=

,

u

T T

=

Lower plate

(

y 0

=

,

0 x L

≤ ≤

) :

u 0

=

v 0

=

,

w

T T

=

In the above equations

u(y

is a parabolic velocity profile corresponding to fully developed flow,

)

u

and

v

are

the velocity components of the fluid in x and y direction, respectively (

(

)

, v

=

q

u

G

), R stands for T, u or v,

p

is

the pressure,

ρ

is the biomagnetic fluid density,

µ

is the dynamic viscosity,

o

µ

is the magnetic permeability,

p

c

the specific heat at constant pressure,

k

the thermal conductivity,  T  the temperature and the bar above these

quantities denotes that they are dimensional.

The terms

o

M H x

µ

∂ ∂  and

o

M H y

µ

∂ ∂  in (2) and (3), respectively, represent the components of the magnetic

force, per unit volume, and depend on the existence of the magnetic gradient.

For the variation of magnetization M ,

with the magnetic field intensity  H  and temperature T , we consider the

relation derived experimentally



.

c

M KH(T

T)

=

(6)

where  K  is a constant and

c

T

is the Curie Temperature.

For the magnetic field intensity  H  we use the expression

2

1

H(x, y)

2 (x a)

(y b)

γ

=

π

+

+

+

2

(7)

where γ is the magnetic field strength at the pole and (a,b) is the position where the dipole is located.

3 TRANSFORMATION

OF

EQUATIONS

In order to proceed to the numerical solution we introduce the non dimensional variables

E.Tzirtzilakis, N. Kafoussias, P. Hatzikonstantinou

x

x

h

=

,

y

y

h

=

,

r

u

u

u

=

,

r

v

v

u

=

2

r

p

p

u

=

ρ

o

H

H

H

=

u

u

w

T

T

T

T

T

=

(8)

where

r

u

is the maximum velocity at the entrance and

o

H  is the magnetic field strength at the point (3,0),

whereas the dipole is placed at the point (3,-0.02). The contours of the dimensionless magnetic field strength H

are shown in Figure 2.

For the numerical solution we adopt the stream function-vorticity formulation, by introducing the

dimensionless vorticity function J=J(x,y) and the dimensionless stream function Ψ=Ψ(x,y) defined by the

expressions

(

)

v

J x, y

x

y

u

=

, (9)

u

y

∂Ψ

=

,

v

x

∂Ψ

= −

,

(10)

Thus, equation (1) is automatically satisfied and equations (2), (3) and (4) produce, by eliminating the pressure p

from the first two and substituting (10) in (4) and (9), the following system of equations

2

J

∇ Ψ = −

(11)

2

J

J

H T

H T

J Re

Mn  Re  H

x y

y x

x y

y x

∂ ∂Ψ ∂ ∂Ψ

∂ ∂

∂ ∂

∇ =

+

∂ ∂

∂ ∂

∂ ∂

∂ ∂



(12)

2

2

2

2

2

2

2

T

T

H T

H T

T Pr   Re

Mn Pr Re Ec H ( -T)

x y

y x

x y

y x

+ Pr Ec

4

x y

y

x

∂ ∂Ψ ∂ ∂Ψ

∂ ∂

∂ ∂

=

+

ε

∂ ∂

∂ ∂

∂ ∂

∂ ∂

∂ Ψ ∂ Ψ

∂ Ψ

+

∂ ∂

2

(13)

where

is the two dimensional Laplacian operator (

2

(

)

2

2

2

2

/ x

/ y

∇ ⋅∇ = ∂ ∂ + ∂ ∂

2

∇ =

G G

).

The non-dimensional parameters entering now into the problem under consideration are

r

h u

Re

ρ

=

µ

(Reynolds number),

2

o

o

u

w

2

r

H K(T

T )

Mn

u

µ

=

ρ

(Magnetic number),

u

u

w

T

T

T

ε =

(Temperature number),

2

r

p

u

w

u

Ec

c (T

T )

=

(Eckert number),

p

c

Pr

k

µ

=

(Prandtl number).

It is worth mentioning here that when the magnetic number Mn=0 we have the common problem of flow in a

duct with heat transfer. Also, for a specific Reynolds number (

r

u

const

=

) increasing Mn is equivalent either

increasing the magnetic field strength

H

at the pole, or the temperature difference between the two plates.

o

3.1 Boundary Conditions

The system of equations (11)~(13) is of elliptic type and boundary conditions are required. All unknown

quantities at the exit of the duct are assumed to be independent on x (

(R) / x 0

∂ =

). Thus, the outflow

conditions for all quantities are determined from the interior grid points of the computational domain. The

boundary conditions for Ψ are also easy implemented from (10), since the

velocity components are known (fully developed flow at the entrance and no

slip conditions on the plates). The dimensionless temperature T at the plates is

also easy calculated from (8) and we additionally assume that in the entrance

of the duct is independent on the x direction.

However, in order to solve the vorticity transport equation (12) it is also

necessary to determine boundary conditions of the vorticity J and this is not an

easy task. We have already mentioned that at the exit no dependence on the x

direction is taken into account. Also, at the entrance the use of (10) enable us

to derive easily the corresponding boundary condition for J. On the contrary,

the derivation of boundary conditions of J on the solid surfaces (plates) is more

complicated. Quartapelle and Valz-Gris



demonstrated that there is no strictly

Figure 1. Grid points for

the calculation of J

i,m

E.Tzirtzilakis, N. Kafoussias, P. Hatzikonstantinou

equivalent local boundary condition available for J. The vorticity is to be computed from the velocity field, but it

cannot be specified at the boundaries before the problem is solved.

So, the solution of the problem under consideration is not as simple as it first appears to be, and special

techniques are needed to utilize the numerical methods. To overcome this difficulty, at each iteration step,

numerical boundary conditions are constructed for the vorticity J, using the stream function Ψ, as follows.

We consider an arbitrary boundary grid point (i,m) as shown in Figure 1. The vorticity at the boundary point

(i,m), provided that the borders are still (u=v=0) is calculated from the formula



( )

(

)

( )

(

i,m

i 1,m 1

i,m 1

i 1,m 1

i,m 2

i,m 1

2

2

1

2

J

2

x

3 y

+

= −

Ψ

− Ψ

+ Ψ

Ψ

− Ψ

)

(14)

The vorticity boundary conditions just derived enable us to solve  (12) provided that the right hand side is

already known from the previous iteration. However, the determination of

Ψ

from (11) depends on the

distribution of vorticity within the bounded domain. Thus Ψ and J are coupled and an iterative scheme is

employed.

4 NUMERICAL

METHOD

For the numerical solution of the system of equations (11)~(13) we developed an efficient technique, based

on a combination of existing numerical methods, as follows. Equation (11) is a Poisson equation and can be

solved fast and efficiently by the use of a simple S.O.R. method. Our main effort will focus on solving the other

two equations of the system, namely (12) and (13). In order to demonstrate the numerical technique used, we

will implement it for the first of the two equations, e.g equation (12).

The unknown function of this equation is the vorticiy J and we use central differencing for the discretization

of the second order derivatives as well as for the first order derivatives except for

J / x

∂ ∂ , where we use an

upwind scheme



. Thus, with respect to (i,j) point the discretization is:

( )

2

i 1, j

i, j

i 1, j

2

2

i, j

J

2J

J

J

x

x

+

+

=

( )

2

i, j 1

i, j

i, j 1

2

2

i, j

J

2J

J

J

y

y

+

+

=

i 1, j

i 1, j

i, j

g

g

g

x

2 x

+

=

i, j 1

i, j 1

i, j

g

g

g

y

2 y

+

=

i, j 1

i, j 1

i, j

J

J

J

y

2 y

+

=

(15)

i 1, j

i 1, j

i 2, j

i 1, j

i, j

i 1, j

i, j

J

J

J

3J

3J

J

J

q

x

2 x

3

x

+

+

+

=

+

where g stands for T, H or Ψ in (12) and q is a constant set to 0.5 rendering the discretization of

of

accuracy O(∆x

J / x

3

), while all the other quantities are descretized with accuracy O(∆x

2

) or O(∆y

2

).

By substituting (15) into (12) an equation of the form

(16)

i, j 1

i, j

i, j 1

AJ

BJ

CJ

Rhs

+

+

+

=

can be obtained, where

nb

A A( x, y,

, Re)

∆ ∆ Ψ

nb

B B( x, y,

, Re)

∆ ∆ Ψ

nb

C C( x, y,

, Re)

∆ ∆ Ψ

nb

nb

nb

i 2, j

i 1, j i 1, j

Rhs Rhs( x, y, H ,T ,

, J

, J

J

)

+

∆ ∆

Ψ

the notation nb denotes the neighboring notes to the point (i,j).

Equation (16) constitutes a scalar tridiagonal system along each x grid line (i constant) and we solve it using the

Thomas algorithm. It can be seen that this scheme is implicit, considering each i line constant, and therefore is

called line by line implicit method



. The solution of equation (12) is achieved iteratively, by solving, for all i

lines, the arising tridiagonal systems until the unknown function at all the grid points of the computational

domain, has been evaluated up to an accuracy e. Equation (13) can be solved in an exactly similar manner.

We can now proceed to the solution of the system of equations (11)~(13). The first step is to make initial

guesses for Ψ, T and J at the interior points. Hereafter, considering J constant we obtain a new estimation of Ψ

by solving (11). The next step is to construct the boundary conditions for J according to the corresponding

paragraph 3.1. Considering now Ψ known, a new estimation of the vorticity J and temperature T, from equations

(12) and (13), respectively, is calculated using the line by line implicit method described above. After this cycle

we compare the new estimations of Ψ, T and J with the old ones, and we seek the difference to be less than a

quantity ε. If this criterion of convergence is not satisfied we set the new estimations as old and we calculate

once more the new ones starting again from (11) and so on.

E.Tzirtzilakis, N. Kafoussias, P. Hatzikonstantinou

5  RESULTS AND DISCUSSION

In order to study the biomagnetic fluid flow, in the rectangular duct, under the influence of an applied

magnetic field, the above described numerical technique was applied to solve the system of equations (11)~(13),

under the appropriate boundary conditions. For the numerical solution it is necessary to assign values in the

dimensionless parameters entering into the problem under consideration. For this purpose we consider a realistic

case in which the fluid is the blood (density

ρ

=1050kgr/m

3

µ

=3.2x10

-3

kgr/msec)



velocity

r

u

= 1.22 x10

-2

m/sec and the plates are located at a distance h=1.0x10

-2

m. In this case the Reynolds

number Re is equal to 40.

The new dimensionless parameter appearing in the problem is the magnetic number Mn and it can written as

2

o

o

u

w

o

o

o

u

w

o

o

2

2

r

r

H K(T

T )

H KH (T

T )

B M

Mn

u

u

µ

µ

=

=

ρ

ρ

2

r

u

=

ρ

, (17)

where

o

B

and

o

M

are the magnetic induction and the magnetization at the point (3,0), respectively. For

magnetic field 8 Tesla, the blood has reached magnetization of 60A/m



. From the definition of Reynolds

number also, it can be obtained that

r

e/ h

= µ

ρ

u

R

and substitution of this relation to (17), gives

2

o

o

2

M B h

Mn

Re

ρ

=

µ

(18)

From (18) it is derived that the corresponding Mn for magnetic field strength 8 Tesla at the pole is Mn

≈ 3000.

We also consider the temperature of the plates to be

u

T

=50

o

C whereas

w

T

=10

o

C. For these values of plate

temperatures the temperature number ε is equal to 8.

Although the viscosity

µ

, the specific heat under constant pressure

p

c  and the thermal conductivity

k

of

any fluid, and hence of the blood, are temperature dependent, Prandtl number can be considered constant. Thus,

for the temperature range considered in this problem, the value of

p

c   and

k

is equal to 14.65 Joule/kgr

o

K and

2.2x10

-3

Joule/m sec

o

K, respectively,

[14,15]

and hence we can consider Pr=20.  For these values of the parameters

it is also derived that the Eckert number Ec=2.536x10

-7

. For these values of the dimensionless parameters the

obtained results are shown in Figures 3~7 concerning the velocity field, the temperature field and the coefficients

of skin friction and rate of heat transfer.

Figure 3 shows the stream function contours for the values of the above mentioned parameters and for

magnetic numbers Mn=1000, 1500 and 3000. It is observed that a vortex is arising at the area where the

magnetic dipole is located. As the magnetic number increases, which means increment of the magnetic field

strength at the pole as long as the temperature difference is fixed, the vortex is extended. It should be remarked

that in the absence of the magnetic field (Mn=0) the stream function contours are straight lines and the velocity

profile u is the same with the one shown at the entrance of the duct in Figure 6. The velocities profiles u along

specific locations in the duct are shown in Figure 6. At the point x=2.8 the flow is separated from the wall and at

x=3.0, where the dipole is located, has already reversed near the lower plate. Finally, the flow at the exit x=7 is

again reverted to fully developed.

The temperature contours for the same values of the magnetic number Mn are shown in Figure 4. It is

observed an increment in the temperature of the biofluid near the lower plate where the dipole is located. For

Mn=3000, this variation of temperature, at different positions in the duct as well as the corresponding contours,

are shown in Figure 7. It should be remarked, once more, that in the absence of magnetic field the contours of

temperature are straight lines.

The most important flow and heat transfer characteristics are the local skin friction coefficient and the local

rate of heat transfer coefficient. These quantities can be defined by the following relations:

w

f

2

r

2

C

u

τ

=

ρ

,

u

w

qh

Nu

k(T

T )

=

(19)

where

(

)

w

y 0,h

u y

=

τ = µ ∂ ∂

is the wall shear stress and

(

)

y 0,h

q

k T y

=

= − ∂ ∂

is the heat flux. Using (8), (10), the

above mentioned quantities can be written as

f

y 0,1

2

(x, y)

C

Re

=

′′

Ψ

=

y 0,1

T

Nu

y

=

=

(20)

E.Tzirtzilakis, N. Kafoussias, P. Hatzikonstantinou

where Nu is the Nusselt number,

y 0,1

(x, y)

=

′′

Ψ

is the dimensionless wall shear parameter and

y 0,1

T (x, y)

=

is

the dimensionless wall heat transfer parameter (( )′=∂( )/∂y). The variation of these mentioned dimensionless

parameters are shown in Figure 5. The wall shear as well as the heat transfer parameters are more influenced on

the lower plate, below of which the dipole is located. It is remarkable that the wall shear parameter increases

rapidly in the region x=2 to x=2.76, where it reaches its maximum value. Very close to the region where the

dipole is located (x=3), a corresponding decrement take place and at x=3.18 this parameter takes its minimum

negative value. Finally, far downstream, x=5, the wall shear parameter of both plates, reach its original value

corresponding to fully developed flow. The wall shear parameter of the upper plate increases near the area of the

dipole in a smoother way and it does not take negative values as it happens with the lower plate where reverse of

the flow occurs. Analogous variation is observed for the heat transfer parameter for both plates, but in this case,

far downstream the value of  T´(x,0) or T´(x,1) is greater that in the area ahead of the region of the dipole.

6.   CONCLUDING REMARKS

The steady, two dimensional, incompressible, viscous flow of the biomagnetic fluid (blood) in a rectangular

duct under the influence of an applied magnetic field is studied. The stream function-vorticity formulation is

adopted and an efficient numerical technique is developed, based on known numerical methods. The obtained

results showed that near the area were the dipole is located a vortex is arising and the temperature is increasing.

This phenomenon is extended with the increment of the magnetic field strength, whereas in the absence of the

magnetic field these disturbances in the flow field are not present. The skin friction and heat transfer coefficients

are also appreciably influenced particularly on the lower plate. These results indicate that the application of a

magnetic field, in the flow of a biomagnetic fluid should be further studied for possible useful medical and

engineering applications

Acknowledgment

This work was supported by the program K. Karatheodoris No. 2439 of Research Committee, University of

Patras

REFERENCES



Haik, Y., Pai, V. and Chen C.J. (1999), “Development of magnetic device for cell separation”, Journal of

Magnetism and Magnetic Materials, Vol. 194, pp.254-261.

   Ruuge, E. K. and Rusetski, A.N., (1993), “Magnetic fluid as Drug Carriers: Targeted Transport of Drugs

by a Magnetic Field”, Journal of Magnetism and Magnetic Materials, Vol. 122, pp.335-339.

     Plavins, J. and Lauva, M. (1993), “Study of Colloidal Magnetite Binding Erythrocytes: Prospects for Cell

Separation”, Journal of Magnetism and Magnetic Materials, Vol. 122, pp.349-353.



Haik, Y.,Chen, J.C. and Pai V.M., In: Winoto, S.H., Chew Y.T. (1996), “Development of bio-magnetic

fluid dynamics”, Proceedings of the IX International Symposium on Transport Properties in Thermal

Fluids Engineering, Singapore, (June 25-28), Pacific Center of Thermal Fluid Engineering

, Wijeysundera

N.E. (eds.),  Hawaii, U.S.A., pp.121-126.

 Rosensweig,

R.E.

(1985),

Ferrohydrodynamics

, Dover Publications, Mineola, New York.



Higashi, T., Yamagishi, A., Takeuchi, T., Kawaguchi, N., Sagawa, S., Onishi, S. and Date, M. (1993),

“Orientation of Erythrocytes in a strong static magnetic field”, J. Blood, Vol. 82, Issue 4, pp.1328-1334.

   Pauling, L. and Coryell, C. D. (1936), “The magnetic Properties and Structure of Hemoglobin,

Oxyhemoglobin and Carbonmonoxy Hemoglobin”, Proceedings of the National Academy of Science

USA, Vol. 22, pp.210-216.



Matsuki, H. Yamasawa, K. and Murakami, K. (1977), “Experimental Considerations on a new Automatic

Cooling Device Using Temperature Sensitive Magnetic Fluid”, IEEE Transactions on Magnetics, Vol.

Mag-13, No. 5, pp.1143--1145.

  L. Quartapelle, F. Valz-Gris (1981), “Projection conditions on the vorticity in viscous incompressible

flows”, Int. J. Numer. Methods Fluids, Vol 1, pp. 129-144,

  C. Y. Chow (1979), “An Introduction to Computational Fluid Mechanics”, John Wiley & sons, New

York.

  Fletcher C. A. J. (1991), Computational Techniques for Fluid Dynamics 1, Second Edition, Springer-

Verlag , Berlin

  Patankar S. (1980), Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation.

  Pedley T. L. (1980), The fluid mechanics of large blood vessels, Cambridge University press

  Valvano, J.W. Nho, S. and Anderson, G.T. (1994), “Analysis of the Weinbaum-Jiji model of blood flow

in the Canine Kidney cortex for self-heated thermistors”. J. Biomechanical Engineering, Vol.116, No.2,

pp.201-207.

  Chato J.C. (1980), “Heat tranfer to blood vessels”, J. Biomechanical Engineering, Vol.102, pp.110-118.

E.Tzirtzilakis, N. Kafoussias, P. Hatzikonstantinou

Figure 2 Contours of Magnetic Field strength H

Figure 3.   Stream Function contours for Re=40

and Mn=1000, 1500 and 3000

Figure 4.   Temperature contours for Re=40

and Mn=1000, 1500 and 3000

Figure

5 Variation of skin friction

coefficient of lower and upper

plate (above-left) and heat

transfer coefficient of the upper

(left) and lower (above) plate,

respectively, with x direction

E.Tzirtzilakis, N. Kafoussias, P. Hatzikonstantinou

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## Document Outline

• BIOMAGNETIC FLUID FLOW IN A RECTANGULAR DUCT
• E. E. Tzirtzilakis†, N. G. Kafoussias† and P. M. H