mathematical models for biomagnetic fluid flow and applications

MATHEMATICAL MODELS FOR BIOMAGNETIC FLUID FLOW AND
APPLICATIONS
E. E. Tzirtzilakis and N. G. Kafoussias
Department of Mathematics, Section of Applied Analysis,
University of Patras, 26500 Patras, Greece
1. SUMMARY
In this work a mathematical model governing the biomagnetic fluid flow is presented.
Expressions describing the variation of the saturation magnetization of the fluid with
temperature or the magnetic field intensity are also given. After proper simplifications of the
above-mentioned mathematical model the flow in a rectangular channel of a biomagnetic
fluid (blood) under the action of an applied magnetic field is studied. The results obtained
from the numerical solution of this problem, showed that the fluid flow is appreciably
influenced by the applied magnetic field.
2. INTRODUCTION
During the last decades an extensive research work has been done on the fluid dynamics of
biological fluids in the presence of magnetic field due to bioengineering and medical
applications [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. 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 [4] and also that the blood possesses the property of diamagnetic material
when oxygenated and paramagnetic when deoxygenated [5].
In order to examine the flow of a biomagnetic fluid under the action of an applied magnetic
field, Haik et. all [6] developed a mathematical model for the Biomagnetic Fluid Dynamics
(BFD) in which the saturation or static magnetization is given by the Langevin magnetization
equation. BFD differs from MagnetoHydroDynamics (MHD) in that it deals with no electric
current and the flow is affected by the magnetization of the fluid in the magnetic field. In
MHD, which deals with conducting fluids, the mathematical model ignores the effect of
polarization and magnetization.
The behavior of a biomagnetic fluid when it is exposed to magnetic field (magnetized) is
described by the magnetization property M. Magnetization is the measure of how much the
magnetic field is affecting the magnetic fluid and is a function of the magnetic field intensity
H and the temperature T.
In the present work, the mathematical model, describing the biomagnetic fluid flow, is
presented and relations are given, expressing the dependence of the saturation magnetization
M o on the temperature and the magnetic field intensity. A simplification of this mathematical

model is used to obtain numerical solution of the differential equations describing the fluid
flow (blood) in a rectangular channel under the action of a magnetic field. The obtained
numerical results, presented graphically, showed that the flow is appreciably influenced by
the magnetic field. These results indicate that application of a magnetic field, in the flow of a
biomagnetic fluid, could be useful for medical and engineering applications.
3. MATHEMATICAL FORMULATION
The mathematical model for the Biomagnetic Fluid Dynamics 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 [6]. Under these
assumptions, the governing equations for incompressible fluid flow are similar to those
derived for FerroHydroDynamics (FHD) [7], and are given by :
Continuity Equation
∇⋅ V=
0
(1)
Linear Momentum
DV
2 2
ρ =−∇ p+ρ F+η∇ V+ξ( ∇ V+ 2∇×ω ) +µ
o ( M⋅∇ ) H+ ⎡∇× H⎤×
Dt
⎣ ⎦
B (2)
Angular Momentum
Dω
2
ρ I =µ
oM× H+η∇ ′ ω+ 2ξ( ∇× V−2ω )
(3)
Dt
Magnetization
DM 1
=ω× M− ⎡M−M
⎤
o
Dt τ ⎣ ⎦
(4)
Maxwell Equations (Amperes law and Gauss second law)
∇× H= J =σ ( V×
B)
(5)
∇⋅ B=∇⋅ H+ M = 0
( )
Magnetization Equations
These equations describe the dependence of saturation or static magnetization M o
on the
applied magnetic field intensity H and temperature T and the appropriate expressions are
given below.
In the above equations V
is the velocity field, ρ is the fluid density, P is the pressure, F is
the body force per unit volume, η and ξ are the coefficients of dynamical and rotational
viscosity, respectively, ω is the angular velocity, µ is the magnetic permeability, M
is the
magnetization, H is the magnetic field intensity, B is the magnetic induction, I is the
moment of inertia, per unit mass, η ′ is the shear spin viscosity, τ is the magnetic relaxation
time, M
is the saturation magnetization and σ is the electrical conductivity of the fluid.
o
3.1 Equilibrium Flow
The above set of equations is a very complicated system and simplifications must be made in
order to solve it for a specific problem.
As a first approximation we can consider that biofluids are poor conductors and the induced
current is negligibly small. Thus, unlike MHD, Lorentz forces (the last term in eq.(2)) are
much smaller in comparison to the magnetization force (the last but one term in eq.(2)). Also,
ο

for medium shear rates, like blood flow in artery, the diffusion of the spin term is much
smaller than that of the magnetic torque or the exchange between internal and external
momentum.
The major simplification however, can take place if we consider that the magnetic fluid has
either achieved instantaneous magnetization or time has elapsed beyond the relaxation time,
after the flow has exposed to the magnetic field. In this situation the flow can be considered
as equilibrium flow and once the particle reaches saturation magnetization it will not have
addition magnetization even if the magnetic field is further increased. Under the equilibrium
assumption the fluid magnetization vector, M , at any given instant is parallel to the vector of
the magnetic field intensity, H , and the property of magnetization is determined by the fluid
temperature, density and magnetic field intensity M=M(T,ρ,H). Although the equilibrium
flow is an idealization for the physical behavior of the biomagnetic fluid, it provides a good
insight to the biomagnetic fluid flow since the governing equations are much more simpler
than the complete set of equations derived in the previous sections for non equilibrium case.
The equations of motion for the equilibrium flow can be written now as:
Continuity Equation
∇⋅ V=
0, (6)
Linear Momentum
DV
2
ρ =−∇ p+ρ F+η∇ V+µ oMo∇H, (7)
Dt
1/2
2 2
where H= ⎡
⎣Hx
+ H ⎤
y⎦
.
3.2 Saturation Magnetization Equations
In equilibrium situation the magnetization property is generally determined by the fluid
temperature, density and magnetic field intensity and various equations, describing the
dependence of M o on these quantities, are given in bibliography [6], [7]. The simplest relation
is the linear equation of state, given in [8]:
M = K( T −T), (8)
o
c
where K is a constant called pyromagnetic coefficient and T
c
is the Curie temperature.
Above the Curie temperature the biofluid does not subjected to magnetization.
Another equation for magnetization, below the Curie temperature is given in [9]
M
c
o
=
1⎜ ⎟
⎝ T1
⎠
)
β
⎛T
− T⎞
M , (9)
where β is the critical exponent for the spontaneous or saturation magnetization. For iron
β=0.368, M 1 =54 Oe and T 1 =1.45 K.
A linear equation involving the magnetic intensity H and temperature T is given in [10]
M = KH T −T
. (10)
o
(
c
Finally, Higashi et. all [4], found that the magnetization process of red blood cells behaves
like the following function, known as Langevin function,
⎡ ⎛µ omH
⎞ κT
⎤
Mo
= mN⎢coth⎜ ⎟− ⎥
⎣ ⎝ κ T ⎠ µ
omH
, (11)
⎦
where m is the particle magnetization, N is the number of particles per unit volume and κ the
Boltzman’s constant.
T c

In all the above cases the magnetization M o is dependent on the temperature T of the
fluid. In such a case (non-isothermal case), it is also necessary to consider in the
mathematical model, describing the problem under consideration, the energy equation
containing the temperature T of the fluid. This equation can be written as [8]
DT ∂M
o
2
ρ Cp
+µ
oT ⎡V⋅( ∇ H) ⎤ = k∇ T+
ηΦ
(12)
Dt ∂T
⎣ ⎦
where k is the coefficient of thermal conductivity of the fluid, C p the specific heat and Φ the
dissipation function.
4. APPLICATIONS
Biomagnetic fluid flow in a Channel
As a simple, but representative, application of biomagnetic fluid flow we consider the steady
two-dimensional laminar flow of an incompressible viscous biomagnetic fluid (blood) in a
space between two parallel flat plates (channel). The length of the plates is L and the distance
between them is h (h

of the magnetic source the friction coefficient is increased substantially due to the action of
the magnetic field. The contours of the stream function, for various magnetic numbers, are
shown in Figures 4-6. The magnet affects the fluid flow and two vortices are created near the
magnetic pole, as the magnetic field strength is increased. The contours of vorticity function
are shown in Figures 7-9 and of temperature function in Figures 10-12. Near the pole we
observe a slight increase of the temperature as a result of additional energy from the magnetic
field of the pole. It should be remarked that in the absence of the magnetic field straight lines
represent temperature, stream as well as the vorticity function. Thus, the formation of these
contours is the result of the action of the applied magnetic field on the flow field. These
results show that in the presence of the magnetic field, the flow field is changing drastically,
and especially the skin friction coefficient, which is affected near the area of the magnetic
pole. These conclusions suggest that a careful choice of the imposed magnetic field will
affect the flow characteristics and hence can be utilized for medical and engineering
applications.
Figure 4. Contours for stream function for Mn=2000
Figure 5. Contours for stream function for Mn=4000
Figure 6. Contours for stream function for Mn=6000
Figure 7. Contours for vorticity function for Mn=2000
Figure 8. Contours for vorticity function for Mn=4000
Figure 9. Contours for vorticity function for Mn=6000

Figure 10. Contours for temperature function for Mn=2000
Figure 11. Contours for temperature function for Mn=4000
Figure 12. Contours for temperature function for Mn=6000
Acknowledgements: The program “K. Karatheodoris” No. 2439 - University of Patras-
Research Committee, financially supported this work.
5. References
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Journal of Magnetism and Magnetic Materials, 194,254-261 (1999).
[2] Ruuge, E. K. and Rusetski, A.N., Magnetic fluid as Drug Carriers: Targeted Transport
of Drugs by a Magnetic Field, Journal of Magnetism and Magnetic Materials, 122, 335-
339 (1993).
[3] Plavins, J. and Lauva, M., Study of Colloidal Magnetite Binding Erythrocytes:
Prospects for Cell Separation, Journal of Magnetism and Magnetic Materials, 122, 349-
353 (1993).
[4] Higashi, T., Yamagishi, A., Takeuchi, T., Kawaguchi, N., Sagawa, S., Onishi, S. and
Date, M., Orientation of Erythrocytes in a strong static magnetic field, J. Blood, 82(4),
1328-1334 (1993).
[5] Pauling, L. and Coryell, C. D., The magnetic Properties and Structure of Hemoglobin,
Oxyhemoglobin and Carbonmonoxy Hemoglobin, Proceedings of the National
Academy of Science, USA, 22, 210-216 (1936).
[6] Haik, Y.,Chen, J.C. and Pai V.M., In: Winoto, S.H., Chew Y.T., Development of biomagnetic
fluid dynamics, Proceedings of the IX International Symposium on Transport
Properties in Thermal Fluids Engineering, Singapore, Pacific Center of Thermal Fluid
Engineering, Wijeysundera N.E. (eds.), Hawaii, U.S.A.,121-126 (June 25-28, 1996).
[7] Rosensweig, R.E., Ferrohydrodynamics, Cambridge University Press (1985).
[8] Andersson, H. I. and Valnes, O. A., Flow of a heated ferrofluid over a stretching sheet
in the presence of a magnetic dipole, Acta Mechanica, 128, 39-47 (1998).
[9] Arrott, A.S., Heinrich, B. and Templeton, T.L., Phenomenology of Ferromagnetism: I.
Effects of Magnetostatics on Susceptibility, IEEE Transactions on Magnetics, 25, 4364-
4373 (1989).
[10] Matsuki, H., Yamasawa, K. and Murakami, K., Experimental Considerations on a new
Automatic Cooling Device Using Temperature Sensitive Magnetic Fluid, IEEE
Transactions on Magnetics, 13 (5), 1143-1145 (1977).
[11] Mazumdar J.N., Biofluid Mechanics, World Scientific Publishing Co., Singapore
(1992).