The flow of a visco-elastic fluid in a channel with stretching walls under the action of an externally applied magnetic field generated by a magnetic dipole was studied in this paper. As per an experimental report, the variation in magnetization $M$ of the fluid with temperature $T$ was approximated as a linear equation of state $M=K1T$, where $K1$ is a constant called the pyromagnetic coefficient. In this investigation the model used is that of Walter’s liquid B fluid, which includes the effect of fluid visco-elasticity. By introducing appropriate nondimensional variables, the problem is reduced to solving a coupled nonlinear system of ordinary differential equations subject to a set of boundary conditions. The problem is solved by developing a suitable numerical technique based on finite difference approach. Computational results concerning the variation in the velocity, pressure and temperature fields, skin friction and the rate of heat transfer with magnetic field strength, Prandtl number, and blood visco-elasticity are presented graphically. The results presented reveal that the velocity of blood in the normal physiological state can be lowered by applying a magnetic field of sufficient intensity. The study bears the promise of important applications in controlling the flow of blood during surgery and also during treatment of cancer by therapeutic means when it involves magnetic drug targeting (hyperthemia).

1.
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
,”
Blood
0006-4971,
82
, pp.
1328
1334
.
2.
Pauling
,
L.
, and
Cryell
,
C. D.
, 1936, “
The Magnetic Properties and Structure of Hemoglobin
,”
0027-8424,
22
, pp.
210
216
.
3.
Haik
,
Y.
,
Pai
,
V.
, and
Chen
,
C. J.
, 1999, “
Development of Magnetic Device for Cell Separation
,”
J. Magn. Magn. Mater.
0304-8853,
194
, pp.
254
261
.
4.
Haik
,
Y.
,
Pai
,
V.
, and
Chen
,
C. J.
, 2001, “
Apparent Viscosity of Human Blood in a High Static Magnetic Field
,”
J. Magn. Magn. Mater.
0304-8853,
225
, pp.
180
186
.
5.
Rosensweig
,
R. E.
, 1985,
Ferrohydrodynamics
,
Cambridge University
,
Cambridge
.
6.
Neuringer
,
J. L.
, and
Rosensweig
,
R. E.
, 1964, “
Ferrohydrodynamics
,”
Phys. Fluids
0031-9171,
7
, pp.
1927
1937
.
7.
Haik
,
Y.
,
Chen
,
J. C.
, and
Pai
,
V. M.
, 1996, “
Development of Biomagnetic Fluid Dynamics
,”
Proceedings of the Ninth International Symposium on Transport Properties in Thermal Fluid Engineering
,
S. H.
,
Winoto
,
Y. T.
,
Chew
, and
N. E.
,
Wijeysundera
, eds., HI, Jun. 25–28, pp.
121
126
.
8.
Perry
,
M. P.
, 1972, “
Thermomechanics
,”
Proceedings of the International Advanced Courses and Workshop
, pp.
219
.
9.
Hiergeist
,
R.
,
Andra
,
W.
,
Buske
,
N.
,
Hergt
,
R.
,
Hilger
,
I.
,
Richter
,
U.
, and
Kaiser
,
W.
, 1999, “
Application of Magnetic Ferrofluids for Hyperthemia
,”
J. Magn. Magn. Mater.
0304-8853,
201
, pp.
420
422
.
10.
Voltairas
,
P. A.
,
,
D. I.
, and
Michalis
,
L. K.
, 2002, “
Hydrodynamics of Magnetic Drug Targeting
,”
J. Biomech.
0021-9290,
35
, pp.
813
821
.
11.
Tzirtzilakis
,
E. E.
, and
Kafoussias
,
N. G.
, 2003, “
Biomagnetic Fluid Flow Over a Stretching Sheet With Non-Linear Temperature Dependent Magnetization
,”
ZAMP
0044-2275,
54
, pp.
551
565
.
12.
Loukopoulos
,
V. C.
, and
Tzirtzilakis
,
E. E.
, 2004, “
Biomagnetic Channel Flow in Spatially Varying Magnetic Field
,”
Int. J. Eng. Sci.
0020-7225,
42
, pp.
571
590
.
13.
,
P. K.
, and
Tzirtzilakis
,
E. E.
, 2004, “
Biomagnetic Flow in a Curved Square Duct Under the Influence of an Applied Magnetic Field
,”
Phys. Fluids
1070-6631,
16
, pp.
2952
2962
.
14.
Tzirtzilakis
,
E. E.
,
Xenos
,
M.
,
Loukopoulos
,
V. C.
, and
Kafoussias
,
N. G.
, 2006, “
Turbulent Biomagnetic Fluid Flow in a Rectangular Channel Under the Action of a Localized Magnetic Field
,”
Int. J. Eng. Sci.
0020-7225,
44
, pp.
1205
1224
.
15.
Tzirtzilakis
,
E. E.
, 2005, “
A Mathematical Model for Blood Flow in Magnetic Field
,”
Phys. Fluids
1070-6631,
17
, p.
077103
.
16.
,
H. I.
, and
Valnes
,
O. A.
, 1998, “
Flow of a Heated Ferrofluid Over a Stretching Sheet in the Presence of a Magnetic Dipole
,”
Acta Mech.
0001-5970,
128
, pp.
39
47
.
17.
Misra
,
J. C.
,
Shit
,
G. C.
, and
Rath
,
H. J.
, 2008, “
Flow and Heat Transfer of a MHD Viscoelastic Fluid in a Channel With Stretching Walls: Some Applications to Haemodynamics
,”
Comput. Fluids
0045-7930,
37
, pp.
1
11
.
18.
Misra
,
J. C.
,
Pal
,
B.
, and
Gupta
,
A. S.
, 1998, “
Hydromagnetic Flow of a Second-Grade Fluid in a Channel—Some Applications to Physiological Systems
,”
Math. Models Meth. Appl. Sci.
0218-2025,
8
, pp.
1323
1342
.
19.
Misra
,
J. C.
, and
Pal
,
B.
, 1999, “
Hydromagnetic Flow of a Viscoelastic Fluid in a Parallel Plate Channel With Stretching Walls
,”
Indian J. Math.
,
41
, pp.
231
247
.
20.
,
H. I.
, 1992, “
MHD Flow of a Viscoelastic Fluid Past a Stretching Surface
,”
Acta Mech.
0001-5970,
95
, pp.
227
230
.
21.
Datti
,
P. S.
,
,
K. V.
,
Abel
,
M. S.
, and
Joshi
,
A.
, 2004, “
MHD Viscoelastic Fluid Flow Over a Non-Isothermal Stretching Sheet
,”
Int. J. Eng. Sci.
0020-7225,
42
, pp.
935
946
.
22.
Cortell
,
R.
, 2006, “
A Note on Flow and Heat Transfer of a Viscoelastic Fluid Over a Stretching Sheet
,”
Int. J. Non-Linear Mech.
0020-7462,
41
, pp.
78
85
.
23.
Rajagopal
,
K. R.
,
Na
,
T. Y.
, and
Gupta
,
A. S.
, 1984, “
Flow of a Viscoelastic Fluid Over a Stretching Sheet
,”
Rheol. Acta
0035-4511,
23
, pp.
213
215
.
24.
Dandapat
,
B. S.
, and
Gupta
,
A. S.
, 1989, “
Flow and Heat Transfer in a Viscoelastic Fluid Over a Stretching Sheet
,”
Int. J. Non-linear Mech.
,
24
, pp.
215
219
. 0020-7462
25.
,
H. I.
,
Bech
,
K. H.
, and
Dandapat
,
B. S.
, 1992, “
Magnetohydrodynamic Flow of a Power-Law Fluid Over a Stretching Sheet
,”
Int. J. Non-linear Mech.
,
27
, pp.
929
936
. 0020-7462
26.
Mahmoud
,
M. A. A.
, and
Mahmoud
,
M. A.-E.
, 2006, “
Analytical Solutions of Hydromagnetic Boundary Layer Flow of a Non-Newtonian Power-Law Fluid Past a Continuously Moving Surface
,”
Acta Mech.
0001-5970,
181
, pp.
83
89
.
27.
,
H. I.
, and
Dandapat
,
B. S.
, 1991, “
Flow of a Power-Law Fluid Over a Stretching Sheet
,”
Stability Appl. Anal. Continuous Media (SAACM) Italy
,
1
, pp.
339
347
.
28.
Dewhirst
,
M. W.
, and
Samulski
,
T. V.
, 1988, “
Hyperthemia in the Treatment for Cancer
,”
Upjhon
,
Kalanazoo, MI
.
29.
Craciunescu
,
O. I.
, and
Clegg
,
S. T.
, 2001, “
Pulsatile Blood Flow Effects on Temperature Distribution and Heat Transfer in Rigid Vessels
,”
ASME J. Biomech. Eng.
0148-0731,
123
, pp.
500
505
.
30.
Weinbaun
,
S.
,
Jiji
,
L. M.
, and
Lemons
,
D. E.
, 1984, “
Theory and Experiment for the Effect of Vascular Microstructure on Surface Tissue Heat Transfer—Part-I: Anatomical Foundation and Model Conceptualization
,”
ASME J. Biomech. Eng.
0148-0731,
106
, pp.
321
330
.
31.
Beard
,
B. W.
, and
Walters
,
K.
, 1964, “
Elastico-Viscous Boundary Layer Flows
,”
Proc. Cambridge Philos. Soc.
0068-6735,
60
, pp.
667
6674
.
32.
Labropulu
,
F.
, and
Chinichian
,
M.
, 2004, “
Unsteady Oscillatory Stagnation-Point Flow of a Viscoelastic Fluid
,”
Int. J. Eng. Sci.
0020-7225,
42
, pp.
625
633
.
33.
Matsuki
,
H.
,
Yamasawa
,
K.
, and
Murakami
,
K.
, 1977, “
Experimental Considerations on a New Automatic Cooling Device Using Temperature-Sensitive Magnetic Fluid
,”
IEEE Trans. Magn.
0018-9464,
13
, pp.
1143
1145
.
34.
Crane
,
L. J.
, 1970, “
Flow Past a Stretching Plate
,”
Z. Angew. Math. Phys.
0044-2275,
21
, pp.
645
647
.