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TITLE OF PROJECT
LAYOUT OF PRESENSTATION
 Title
 Summary of Study
 Flow Assumptions
 Equations of Motion
 Results and Discussion
 Literature Review
 References
SUMMARY OF WORK
 We studied the bio-magnetic fluid flow in a parallel plate
channel when both the plates are shrinking and the lower
plate is porous. A magnetic field is applied perpendicular
to the plane of the channel. The convective heat transfer
with viscous dissipation is also analyzed.
 Numerical computations have been made for several values
of the physical parameters namely magnetic parameter ,
suction parameter , Reynolds’ number R, Prandtl number
Pr, Eckert number and local Eckert number .
 The effects of these parameters have been observed on
fluid velocity and temperature distribution. The results are
presented in graphical form.
FLOW ASSUMPTIONS
 The biomagnetic fluid flow is laminar, steady, two




dimensional and incompressible.
The fluid flow is confined between two parallel nonconducting plates. Both the plates are shrinking sheets.
A uniform magnetic field of strength 𝑩𝟎 is applied in the
positive y-direction normal to the stretching sheet.
Fluid with constant velocity 𝒗𝟎 flows out of the lower
porous plate
The blood temperature is T
The lower plate is kept at fixed temperature Tw and
temperature of the upper plate is TC.
RESEARCH QUESTIONS
EQUATIONS OF MOTION
𝜕𝑢 𝜕𝑣
+ =0
𝜕𝑥 𝜕𝑥
𝜕𝑢 𝜕𝑣
u +v
𝜕𝑥 𝜕𝑦
𝜕𝑣 𝜕𝑣
u +v
𝜕𝑥 𝜕𝑦
𝜕𝑇 𝜕𝑇
u +v
𝜕𝑥 𝜕𝑦
(1)
+
𝜕2 𝑢
𝑣( 2
𝜕𝑥
+
𝜕2 𝑣
𝑣( 2
𝜕𝑥
𝐾 𝜕2 𝑇
⍴𝑐𝑝 𝜕𝑦 2
+ µФ
=
1 𝜕𝑝
⍴ 𝜕𝑥
=
1 𝜕𝑝
⍴ 𝜕𝑦
=
Where Ф is dissipation
+
𝜕2 𝑢 σ 2
)- 𝐵
𝜕𝑦 2 ⍴
+
𝜕2 𝑣
)
𝜕𝑦 2
u
(2)
(3)
(4)
and u, v are velocity components and T is
fluid temperature, 𝑣 is kinematic viscosity
The boundary conditions are:
u=-cv, v=𝑉0 , T= 𝑇𝑤
at y=0
u=-cv, v=0, T= 𝑇𝑐
at y=h
THE ORDINARY DIFFERENTIAL
EQUATIONS
𝑓 𝑖𝑣 −R(𝑓 ′ 𝑓 ′′ − 𝑓𝑓 ′′′ ) - 𝑀2 𝑓 ′′ =0
(5)
θ′′ + Pr[R(fθ′ + 4𝐸𝑐𝑓′2 + 4𝐸𝑐𝑥 𝑓′′2 ]=0 (6)
THE BOUNDARY CONDITIONS
𝑓 0 = λ , 𝑓 ′ 0 = −1, θ 0 = 1
𝑓 1 = 0 , 𝑓 ′ 1 = −1, θ 1 =0
(7)
RESULTS AND DISCUSSION
• The non linear ordinary differential equations
(9) and (10) have been solved numerically
with Mathematica software version 6.
• The effect of the parameters namely , R, , Pr,
EC and ECX have been computed and
presented in graphical form for velocity and
temperature distribution.
Fig.1,It is noticed that 𝑓 ′ increases near the two plates
but decreases in the center of the channel with
increasing values of 𝑀2
1.0
f'
0.5
0.0
M2
0.5
1.0
0.0
0.2
0.4
1, 30, 60
0.6
0.8
1.0
Fig.2,It is noticed that 𝑓 ′ decreases with increasing values of 𝑅
1.0
f'
0.5
R
0.0
0.1 , 10, 20
0.5
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig.3 shows that 𝑓 ′ increases with increasing values of
suction parameter .
1.5
1.0
f'
0.5
0.0
0.1 ,
0.2 ,
0.3 ,
0.4
0.5
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig.4 show that the temperature decreases with increase
in the values of 𝑀2
1.4
M2
1.2
1.0
1
15
30
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig.5 shows that temperature increases with increasing
values of suction parameter .
1.5
1.0
0.1
0.3
0.5
0.5
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig.6 shows that with the increasing values of R, temperature
function initially decreases near the lower the plate and then
increases onward in the channel.
.
1.4
1.2
1.0
0.8
0.6
R
0.4
1
10
20
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig.7 shows that temperature increases with increasing
values of Pr.
6
Pr
0.7 , 4, 7
5
4
3
2
1
0
0.0
0.2
0.4
0.6
0.8
1.0
Fig.8 shows that temperature increases with increasing
values of Ec.
2.0
1.5
1.0
0.5
0.0
0.0
Ec
0.2
0.5 , 4, 8
0.4
0.6
0.8
1.0
Fig.5 shows that temperature increases with increasing
values of Ecx.
4
3
2
1
Ecx
0
0.0
0.5 , 2, 4
0.2
0.4
0.6
0.8
1.0
LITERATURE SURVEY
Dynamics of biological fluids (like animal blood) in the
presence of magnetic field is important because of its
applications in bioengineering and medical science.
Theoretical analysis and experimental observations of
blood flow are very useful for the diagnosis of a number
of cardiovascular diseases and development of
pathological patterns in animal or human physiology.
Blood is considered as an electrically conducting fluid as
reported in [1, 2].
• Tzirtzilakis and Tanoudis [7] studied the biomagnetic fluid flow over a stretching sheet.
• Eldesoky [8] presented mathematical analysis
of unsteady MHD blood flow through parallel
plate channel with heat source.
• Tzirtzilakis [11] presented a mathematical
model for blood flow in magnetic field.
• Misra and Sinha [12] presented a theoretical
analysis for magnatohydrodynamic flow of
blood in a capillary, with lumen being porous
and wall permeable.
The analysis of blood flow through parallel plate channel can
be employed in biomedical research to examine the effects of
fluid shear stress on the structure and function of endothelial
cells.
The concept has been attributed to wide spread usage because
of its simplicity and its various designs have been developed for
different applications of biomedical research.
Khalil et al. [14] and Krizanac-Bengez et al. [15] studied the
effects of ischemia/reperfusion on inflammatory gene
expression of endothelial cells.
Li et al. [16] remarked that the blood shear stress causes
cardiovascular diseases by regulating inflammatory reactions in
the vascular endothelium.
A large quantity of heat is carried by the blood to different
parts of the body, while it flows through the arterial tree. The
Pennes [17] bioheat equation is the most common method
that is available to describe blood perfusion in the tissue.
Chen and Homes [18] presented a bioheat transfer model that
accounts for the thermally significant blood vessels. Brink and
Werner [19] presented a three-dimensional thermal and
vascular model in which the convective heat exchange
between the feeder vessels and tissue was computed by the
values for the Nusselt number and the temperatures in and
near individual vessels were predicted.
Wang [20] investigated heat transfer to blood flow in a small
tube. Tzirtzilakis and Tanoudis [21] obtained numerical
solution of biomagnetic fluid flow over a stretching sheet
with heat transfer.

Pavlov (1974) studied the boundary layer flow of
an electrically conducting fluid due to stretching of a
plane elastic surface in the presence of a uniform
transverse magnetic field.
 Kumaran et al. (2009) reported that magnetic field
makes the streamlines steeper which results the
boundary layer thinner.
 Sajjad and Kamal (2012) studied boundary layer flow
for micro polar electrically conducting fluid on a
rotating disk in the presence of magnetic field.
LITERATURE REVIEW
 El-Hakiem et al. (1999) analyzed the effect of viscous
and Joule heating on the flow of an electrically
conducting and micro polar fluid past a plate whose
temperature varies linearly with the distance from the
leading edge in the presence of a uniform transverse
magnetic field.
 Khidir (2013) investigated the effects of viscous
dissipation and Ohmic heating on steady MHD
convective flow due to a porous rotating disk taking
into account the variable fluid properties in the
presence of Hall current and thermal radiation.
 Ibrahim et al. (2013) studied radiation and mass
transfer effects on MHD free convection flow of a
micro polar fluid past a stretching surface embedded
in a non-Darcian porous medium with heat
generation.
 The effect of thermal radiation and magnetic field on
unsteady mixed convection flow and heat transfer over
a porous stretching surface was discussed by
Elbashbeshy et al.(2010).
 Ashraf et al (2009) obtained numerical simulation for
two dimensional flow of a micro polar fluid between
an impermeable and a permeable disk.
 Hady (1996) studied the solution of a heat transfer to a
micro polar fluid from a non isothermal stretching
sheet with injection.
 Na and Pop (1997) investigated the boundary layer
flow of micro polar fluid due to a stretching wall.
 Hassanien et al. (1998) studied a numerical solution
for heat transfer in a micro polar fluid over a stretching
sheet.
 Chakrabarti and Gupta (1979) extended Pavlov's work
to study the heat transfer when a uniform suction is
applied at the stretching surface.
REFERENCES
 J. Singh and R. Rathee, Analytical Solution of Two-
Dimensional Model of Blood Flow with Variable
Viscosity through an Indented Artery Due to LDL
Effect in the Presence of Magnetic Field, International
Journal of Physical Sciences, 5 (12), 2010, 1857-1868.
 N. Verma and R. S. Parihar, Effects of MagnetoHydrodynamic and Hematocrit on Blood Flow in an
Artery withMultiple Mild Stenosis, International
Journal of AppliedMathematics and Computer
Science, 1(1), 2009, 30-46.
 Tzirtzilakis E.E., Tanoudis GB, Numerical study of
biomagnetic fluid flow over a stretching sheet with
heat transfer. Int J Numer Methods Heat Fluid Flow, 13,
2003, 830–848.
 Islam M. Eldesoky, Mathematical Analysis of Unsteady
MHD Blood Flow through Parallel Plate Channel with
Heat Source, World Journal of Mechanics, 2, 2012, 131137.
 E. E. Tzirtzilakis, A mathematical model for blood
flow in magnetic field, PHYSICS OF FLUIDS, 17, 2005,
077103.
 J. C. Misra, A. Sinha, Effect of thermal radiation on MHD flow of
blood and heat transfer in a permeable
capillary in stretching motion, Heat Mass Transfer, 49,
2013,
617–628.
 Khalil, A. A., Aziz, F. A., Hall, J. C., Reperfusion injury. Plastic
and Reconstructive Surgery 117, 2006. 1024-1033.
 Krizanac-Bengez, L., Mayberg, M. R., Cunningham, E., Hossain,
M., Ponnampalam, S., Parkinson, F. E., Janigro, D., Loss of shear
stress induces leukocyte-mediated cytokine release and bloodbrain barrier failure in dynamic in vitro blood-brain barrier
model. Journal of Cellular Physiology 206, 2006, 68-77.
 [16] Li, Y. J., Haga J. H., Chien, S., Molecular basis of the effects
of shear stress on vascular endothelial cells. Journal of
Biomechanics 38, 2005, 1949-1971.
 Elbashbeshy, E.M.A. and Aldawody, D.A., 2010. Effect
of thermal radiation and magnetic field on unsteady
mixed convection flow and heat transfer over a porous
stretching surface.International Journal of Nonlinear
Sciences. Vol. 9 (4), pp. 448-454.
 Ashraf, M., Kamal, M. A. and Syed, K.S.,
2009.Numerical simulation of a micropolar fluid
between a porous disk and a non- porous disk, Appl.
Math. Modell, Vol.33, pp. 1933-1943.
 [17] Pennes, H. H., 1948, Analysis of tissue and arterial
blood temperatures in the resting human forearm, J.
Appl. Physiol., 1 (2), 93-122.
 Hassanien, I., Gorla,R. S. R. and A. A. Abdulah.,1998.
Numerical solution for heat transfer in micropolar
fluids over a stretching sheet, Applied Mechanics
ForEngineers, vol. 3, pp. 3-12.
 Chakrabarti, A.andGupta,A. S., 1979.Hydromagnetic
flow and heat transfer over a stretching sheet,
Quarterly of Applied Mathematics, vol. 37(1), pp. 7378.
[18]Chen, M. M. and Homles, K. R., Microvascular contributions in tissue
heat transfer, Annals of the New York
Academy of Sciences, 335, 1980, 137-150.
[19] Brinck, H. and Werner, J., Estimation of the thmermal effect of blood
flow in a branching countercurrent
network using a three-dimensional vascular model, Trans. ASME, J.
biomech. Eng., 164, 1994, 324-330.
[20 ] C. Y. Wang, Heat Transfer to Blood Flow in a Small Tube," Journal of
Biomechanical Engineering, 130 (2),
2008, 024501. doi:10.1115/1.2898722
[21] E. E. Tzirtzilakis and G. B. Tanoudis, Numerical study of biomagnetic
fluid flow over a stretching sheet
with heat transfer, International Journal of Numerical Methods for Heat
& Fluid Flow, 13(7), 2003, 830 -848.