Volume 2012, Article ID 860239,26pages doi:10.1155/2012/860239
Research Article
Slip Effects on the Unsteady
MHD Pulsatile Blood Flow through Porous Medium in an Artery under the Effect
of Body Acceleration
Islam M. Eldesoky
Basic Engineering Sciences Department, Faculty of Engineering, Menoufia University, Egypt
Correspondence should be addressed to Islam M. Eldesoky,eldesokyi@yahoo.com Received 30 March 2012; Accepted 28 June 2012
Academic Editor: R. H. J. Grimshaw
Copyrightq2012 Islam M. Eldesoky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Unsteady pulsatile flow of blood through porous medium in an artery has been studied under the influence of periodic body acceleration and slip condition in the presence of magnetic field considering blood as an incompressible electrically conducting fluid. An analytical solution of the equation of motion is obtained by applying the Laplace transform. With a view to illustrating the applicability of the mathematical model developed here, the analytic explicit expressions of axial velocity, wall shear stress, and fluid acceleration are given. The slip condition plays an important role in shear skin, spurt, and hysteresis effects. The fluids that exhibit boundary slip have important technological applications such as in polishing valves of artificial heart and internal cavities. The effects of slip condition, magnetic field, porous medium, and body acceleration have been discussed. The obtained results, for different values of parameters into the problem under consideration, show that the flow is appreciably influenced by the presence of Knudsen number of slip condition, permeability parameter of porous medium, Hartmann number of magnetic field, and frequency of periodic body acceleration. The study is useful for evaluating the role of porosity and slip condition when the body is subjected to magnetic resonance imagingMRI.
1. Introduction
The investigations of blood flow through arteries are of considerable importance in many cardiovascular diseases particularly atherosclerosis. The pulsatile flow of blood through an artery has drawn the attention of researchers for a long time due to its great importance in medical sciences. Under normal conditions, blood flow in the human circulatory system depends upon the pumping action of the heart and this produces a pressure gradient throughout the arterial network. Chaturani and Palanisamy 1 studied pulsatile flow of
blood through a rigid tube under the influence of body acceleration as a Newtonian fluid.
Elsoud et al.2studied the interaction of peristaltic flow with pulsatile couple stress fluid.
The mathematical model considers a viscous incompressible couple stress fluid between infinite parallel walls on which a sinusoidal travelling wave is imposed. El-Shehawey et al. 3 investigated the pulsatile flow of blood through a porous medium under periodic body acceleration. The arterial MHD pulsatile flow of blood under periodic body acceleration has been studied by Das and Saha4. Assuming blood to be an incompressible biviscous fluid, the effect of uniform transverse magnetic field on its pulsatile motion through an axi- symmetric tube was analyzed by Sanyal and Biswas5. Rao et al.6analyzed the flow of combined two phase motion of viscous ideal medium through a parallel plate channel under the influence of an imposed pressure gradient and periodic body acceleration.
During recent years, the effect of magnetic field on the flow of viscous fluid through a uniform porous media has been the subject of numerous applications. The red blood cell RBC is a major biomagnetic substance, and the blood flow may be influenced by the magnetic field. In general, biological systems are affected by an application of external magnetic field on blood flow, through human arterial system. The presence of the stationary magnetic field contributes to an increase in the friction of flowing blood. This is because the anisotropic orientation of the red blood cells in the stationary magnetic field disturbs the rolling of the cells in the flowing blood and thereby the viscosity of blood increases.
The properties of human blood as well as blood vessels and magnetic field effect were the subjects of interest for several researchers. Mekheimer7investigated the effect of a magnetic field on peristaltic transport of blood in a non-uniform two-dimensional channel. The blood is represented by a viscous, incompressible, and electrically conducting couple stress fluid.
A mathematical model for blood flow in magnetic field is studied by Tzirtzilakis8. This model is consistent with the principles of ferrohydrodynamics and magnetohydrodynamics and takes into account both magnetization and electrical conductivity of blood. Jain et al.9 investigated a mathematical model for blood flow in very narrow capillaries under the effect of transverse magnetic field. It is assumed that there is a lubricating layer between red blood cells and tube wall. Fluid flow analysis of blood flow through multistenosis arteries in the presence of magnetic field is investigated by Verma and Parihar10. In this investigation, the effect of magnetic field and shape of stenosis on the flow rate is studied. Singh and Rathee 11studied the 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.
Porous medium is defined as a material volume consisting of solid matrix with an interconnected void. It is mainly characterized by its porosity, ratio of the void space to the total volume of the medium. Earlier studies in flow in porous media have revealed the Darcy law which relates linearly the flow velocity to the pressure gradient across the porous medium. The porous medium is also characterized by its permeability which is a measure of the flow conductivity in the porous medium. An important characteristic for the combination of the fluid and the porous medium is the tortuosity which represents the hindrance to flow diffusion imposed by local boundaries or local viscosity. The tortuosity is especially important as related to medical applications12. Flow through porous medium has been studied by a number of workers employing Darcy’s law. A mathematical modeling of blood flow in porous vessel having double stenosis in the presence of an external magnetic field has been investigated by Sinha et al.13. The magnetohydrodynamics effects on blood flow through a porous channel have been studied by Ramamurthy and Shanker14. Eldesoky and Mousa15investigated the peristaltic flow of a compressible non-Newtonian Maxwellian fluid through porous medium in a tube. Reddy and Venkataramana 16investigated the
peristaltic transport of a conducting fluid through a porous medium in an asymmetric vertical channel.
No slip boundary conditions are a convenient idealization of the behavior of viscous fluids near walls. The inadequacy of the no-slip condition is quite evident in polymer melts which often exhibit microscopic wall slip. The slip condition plays an important role in shear skin, spurt, and hysteresis effects. The boundary conditions relevant to flowing fluids are very important in predicting fluid flows in many applications. The fluids that exhibit boundary slip have important technological applications such as in polishing valves of artificial heart and internal cavities 17. The slip effects on the peristaltic flow of a non- Newtonian Maxwellian fluid have been investigated by El-Shehawy et al.18. The influence of slip condition on peristaltic transport of a compressible Maxwell fluid through porous medium in a tube has been studied by Eldesoky 19. Many recent researches have been made in the subject of slip boundary conditions20–25.
In situations like travel in vehicles, aircraft, operating jackhammer, and sudden movements of body during sports activities, the human body experiences external body acceleration. Prolonged exposure of a healthy human body to external acceleration may cause serious health problem like headache, increase in pulse rate and loss of vision on account of disturbances in blood flow6. Many mathematical models have already been investigated by several research workers to explore the nature of blood flow under the influence of external acceleration. Sometimes human being suffering from cardiogenic or anoxic shock may deliberately be subjected to whole body acceleration as a therapeutic measure4. El-Shahed 26studied pulsatile flow of blood through a stenosed porous medium under periodic body acceleration. El-Shehawey et al.3,27–30studied the effect of body acceleration in different situations. They studied the effect of MHD flow of blood under body acceleration. Also, studied Womersley problem for pulsatile flow of blood through a porous medium. The flow of MHD of an elastic-viscous fluid under periodic body acceleration has been studied. The blood flow through porous medium under periodic body acceleration has been studied.
In the present paper, the effect of slip condition on unsteady blood flow through a porous medium has been studied under the influence of periodic body acceleration and an external magnetic field. The analysis is carried out by employing appropriate analytical methods and some important predictions have been made basing upon the study. This investigation can play a vital role in the determination of axial velocity, shear stress, and fluid acceleration in particular situations. Since this study has been carried out for a situation when the human body is subjected to an external magnetic field, it bears the promise of significant application in magnetic or electromagnetic therapy, which has gained enough popularity. The study is also useful for evaluating the role of porosity and slip condition when the body is subjected to magnetic resonance imagingMRI.
2. Mathematical Modeling of the Problem
Consider the unsteady pulsatile flow of blood in an axisymmetric cylindrical artery of radius Rthrough porous medium with body acceleration. The fluid subjected to a constant magnetic field acts perpendicular to the artery as in Figure 1. Induced magnetic field and external electric field are neglected. The slip boundary conditions are also taken into account. The cylindrical coordinate systemr, θ, zare introduced withz-axis lies along the center of the
Bo
z r
Bo Bo
Bo Bo
Homogenous porous medium
R
Figure 1: schematic diagram for the flow geometry.
artery andr transverse to it. The pressure gradient and body acceleration are respectively given by
−∂p
∂z AoA1cos ωpt
, Gaocosωbt,
2.1
where Ao andA1 are pressure gradient of steady flow and amplitude of oscillatory part respectively,aois the amplitude of the body acceleration,ωp 2πfp, ωb 2πfbwith fp is the pulse frequency, andfb is the body acceleration frequency andtis time.
The governing equation of the motion for flow in cylindrical polar coordinates is given by
ρ∂u
∂t −∂p
∂zμ∇2uρG− μ
k
uJ×B. 2.2
Maxwell’s equations are
∇ · B0, ∇ × BμoJ, ∇ ×E−∂B
∂t. 2.3
Ohm’s law is
Jσ
EV×B
, 2.4
whereV 0,0, uis the velocity distribution,ρthe blood density,μomagnetic permeability, B 0, Bo,0 the magnetic field, Ethe electric field, J the current density, k is the permeability parameter of porous medium,μthe dynamic viscosity of the blood, andσ the
electric conductivity of the blood. For small magnetic Reynolds number, the linearlized magnetohydrodynamic forceJ×Bcan be put into the following form:
J×B −σBO2u, 2.5
whereur, trepresents the axial velocity of the blood.
The shear stressτis given by13as
τ −μ∂ u
∂ r. 2.6
Under the above assumptions the equation of motion is
ρ∂u
∂t AoA1cos ωpt
μ ∂2u
∂r2 1 r
∂u
∂r ρaocosωbt− μ
k
u−σB2Ou. 2.7
The boundary conditions that must be satisfied by the blood on the wall of artery are the slip conditions. For slip flow the blood still obeys the Navier-Stokes equation, but the no-slip condition is replaced by the slip conditionutAp∂ut/∂n,whereutis the tangential velocity,nis normal to the surface, andAp is a coefficient close to the mean free path of the molecules of the blood31. Although the Navier condition looked simple, analytically it is much more difficult than the no-slip condition, and then the boundary conditions on the wall of the artery are
u0, tis finite atr 0, uR, t Ap∂ur, t
∂r
rR,
Slip condition
. 2.8
Let us introduce the following dimensionless quantities:
u∗ u
ωR, r∗ r
R, t∗tω, A∗o R μω Ao, A∗1 R
μω A1, a∗o ρR
μω ao, z∗ z
R, k∗ k
R2, b ωb
ωp.
2.9
The Hartmann number Ha, the Womersley parameterα, and the Knudsen number kn, are defined respectively by
HaBoR σ
μ, αR ρω
μ , kn A
R. 2.10
Under the above assumptions2.7and2.8can be rewritten in the non-dimensional form after dropping the stars as
α2∂u
∂t AoA1cost aocosbt ∂2u
∂r2 1 r
∂u
∂r −
Ha2 1 k
u. 2.11
Also the boundary conditions are
u0, tis finite atr 0 2.12a u1, t kn∂ur, t
∂ r
r1. 2.12b And the initial condition is
ur,0 1 att0 2.12c
3. Solution of the Problem
Applying Laplace Transform to2.11, we get
α2su∗r, s − u∗r, o Ao 1
s
A1 s
s21
ao s
s2b2
d2u∗ dr2 1
r du∗
dr −
Ha21 k
u∗,
3.1
whereu∗r, s ∞
0 ur, te−stdt,s >0.
Substituting by the I.C. equation2.12cinto3.1and dropping the stars, we get
r2d2u dr2 rdu
dr −λ2r2u −r2G, 3.2 where
λ2α2sHa2 1 k α2
sHa2 1/k α2 , Gα2Ao
1 s
A1
s s21
ao
s s2b2
.
3.3
Homogenous solution is as follows:
r2d2u dr2 rdu
dr −λ2r2u0. 3.4
This equation is modified Bessel differential equation so the solution is
uhC1IOλr C2KOλr, 3.5
whereIO andKO are modified Bessel functions of order zero. Since the solution is bounded atr0, then the constantC2 equals zero, then
uhC1IOλr. 3.6
We can get the particular solution using the undetermined coefficients as the following:
upβ1β2r, dup
dr β2, d2up dr2 0.
3.7
Substituting into3.2and comparing the coefficients ofrand r2we get
up G
λ2. 3.8
The general solution is
uguhupC1IOλr G
λ2. 3.9
Substituting from2.12binto3.9to calculate the constantC1we get
C1 − G/λ2
−knλI1λ Ioλ. 3.10
Then the general solution can obtained on the following form:
ugr, s G λ2
1− IOλr IOλ−knλI1λ
. 3.11
For the sake of analysis, the part 1−IOλr/IOλ−knλI1λwhich represents an infinite convergent series as its limit tends to zero whenrtends to one and kn tends to zero has been approximated32,33.
The final form of the general solution as a function ofrandsis ugr, s
16
1−r2−2kn
×
α2Ao1/sA1
s/
s21 ao
s/
s2b2
6416α2sHa21/k/α21−2knα2sHa21/k/α221−4kn
1−r4−4kn
×
⎛
⎜⎝ α2
s
Ha21/k
/α2
α2Ao1/sA1
s/
s21 ao
s/
s2b2 6416α2sHa21/k/α21−2knα2s Ha21/k/α221−4kn
⎞
⎟⎠.
3.12
Rearranging the terms and taking the inversion of Laplace Transform of3.12which gives the final solution as
ugr, t 16
1−r2−2kn −1/16M0 Aok2M1 A1k2M2 aok2M3
1−r4−4kn α2kM4 AokM5 A1kM6 aokM7 .
3.13
The expression for the shear stress is given by
τr, t μ162r
−1/16M0 Aok2M1 A1k2M2 aok2M3 μ
4r3 α2kM4 AokM5 A1kM6 aokM7 .
3.14
The expression for the fluid acceleration is given by:
Fr, t ∂u
∂t. 3.15
4. Numerical Results and Discussion
We studied unsteady pulsatile flow of blood through porous medium in an artery under the influence of periodic body acceleration and slip condition in the presence of magnetic field considering blood as an incompressible electrically conducting fluid. The artery is considered a circular tube. We have shown the relation between the different parameters of motion such as Hartmann number Ha, Knudsen number kn, Womersley parameterα, frequency of the body accelerationb, the permeability parameter of porous mediumk, and the axial velocity, shear stress, fluid acceleration to investigate the effect of changing these parameters on the flow of the fluid. Hence, we can be controlling the process of flow.
r 0
0.2 0.4 0.6 0.8 1
Axial velocity
Ha=0.5
0 0.2 0.4 0.6 0.8 1
=2 Ha
=1.5 Ha
=1 Ha
Figure 2: Effect of Hartmann number on the axial velocityb2,α3,ao3, Ao2, A14,t1, kn 0.001, andk0.5.
r
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
kn=0.001
=0.02
=0.04
=0.06
Axial velocity
kn kn kn
Figure 3: Effect of Knudsen number on the axial velocityb2,α3,ao3,Ao2,A14,t1, Ha 1.0, andk0.5.
r 0
0.2 0.4 0.6 0.8 1
Axial velocity
0 0.2 0.4 0.6 0.8 1
1.2
k=0.5
=1
=2
=5 k k k
Figure 4: Effect of permeability parameter on the axial velocityb2,α3,ao 3,Ao2,A14,t 1, kn0.001, and Ha1.0.
r
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
α=1
Axial velocity
=7 α=5 α
=3 α
Figure 5: Effect of Womersley parameter on the axial velocityb2, Ha1,ao 3,Ao 2,A1 4,t 1, kn0.001, andk0.5.
r 0
0.2 0.4 0.6 0.8 1
Axial velocity
0 0.2 0.4 0.6 0.8 1
1.2
b=1 kn=0.001
=4 b
=2 b
=3 b
Figure 6: Effect of frequency of body acceleration on the axial velocity at kn0.001,α3, Ha1,ao 3,Ao2,A14,t1, kn0.001, andk0.5.
r
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
Axial velocity
b=1 kn=0.1
−0.2
=4 b=3 b=2 b
Figure 7: Effect of frequency of body acceleration on the axial velocity at kn 0.1,α 3, Ha 1,ao 3,Ao2,A14,t1, kn0.1, andk0.5.
Axial velocity
r kn=0.2
b=1
0 0.2 0.4 0.6 0.8 1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8
=3 b
=4 b
=2 b
Figure 8: Effect of frequency of body acceleration on the axial velocity at kn 0.2,α 3, Ha 1,ao 3,Ao2,A14,t1, kn0.2, andk0.5.
r
0 0.2 0.4 0.6 0.8 1
−1.5
−0.5 0
kn=0.3
Axial velocity −1
b=1
=2 b
=4 b=3 b
Figure 9: Effect of frequency of body acceleration on the axial velocity at kn 0.3,α 3, Ha 1,ao 3,Ao2,A14,t1, kn0.3, andk0.5.
r
0 0.2 0.4 0.6 0.8 1
0 1 2 3 4 5 6 7
Ha=1
Shear stress
=5 Ha
=7 Ha
=3 Ha
Figure 10: Effect of Hartmann number on the shear stressα3,b2,ao3,Ao2,A14,t1, kn 0.01, andk0.5.
r
0 0.2 0.4 0.6 0.8 1
0 1 2 3 4 5 6
k=0.01
Shear stress
=0.5 k
=0.1 k
=0.05 k
Figure 11: Effect of permeability parameter on the shear stressα3, Ha1,ao3,Ao2,A14,t1, kn0.01, andb2.
r
0 0.2 0.4 0.6 0.8 1
0 1 2 3 4 5
Shear stress
α=1
=5 α
=7 α
=3 α
Figure 12: Effect of Womersley parameter on the shear stressb3, Ha 1,ao 3,Ao 2,A1 4,t 1, kn0.001, andk0.5.
0 2 4 6 8 10 12
r
0 0.2 0.4 0.6 0.8 1
Shear stress
kn=0.001
=0.2 kn
=0.3 kn
=0.1 kn
Figure 13: Effect of Knudsen numberon on the shear stressα3, Ha1,ao3,Ao2,A14,t1,b 2, andk0.5.
r
0 0.2 0.4 0.6 0.8 1
Shear stress
0 1 2 3 4 5 6 7
b=1
=3 b
=2 b b=4
Figure 14: effect of frequency of body acceleration on the shear stress Ha1,α3,ao3,Ao2,A14, t1, kn0.2, andk0.5.
Ha=0.5
t
0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1 0
Blood acceleration
=2 Ha
=1 Ha
Figure 15: Effect of Hartmann number on the blood acceleration kn0.001,α3,ao3,Ao2,A14, t1,b1, andk0.5.
0 2 4 6 8 10
t
0 0.1 0.2 0.3 0.4 0.5
Blood acceleration
kn=0.001
=0.1 kn
=0.3 kn
Figure 16: Effect of Knudsen number on the Blood acceleration Ha1, α3, ao 3, Ao 2, A1 4, t1,b2 andk0.5.
k=0.01 0 1 2 3 4
t
0 0.1 0.2 0.3 0.4 0.5
Blood acceleration
=5 k=1 k
Figure 17: Effect of permeability parameter on the blood accelerationα 3, Ha 1, ao 3, Ao 2, A14,t1,b2, and kn0.01.
0 0.2 0.4 0.6 0.8 1 t
Blood acceleration
α=1
=3 α
=5 α
−0.2
−0.4
−0.6
−0.8
−1
−1.2
−1.4
−1.6
−1.8
−2
Figure 18: Effect of Womersley parameter on the blood accelerationb2, Ha1, ao3, Ao2, A14, t1, kn0.01, andk0.5.
A numerical code has been written to calculate the axial velocity, shear stress, and fluid acceleration according to3.13–3.15, respectively. In order to check our code, we run it for the parameters related to a realistic physical problem similar to the ones used by other authors 9,33–36. For instance, forb 2,α3, ao 3, Ao 2, A1 4,t1, k 0.5,r 0.5, and kn0.0 we obtain the axial velocityu 0.88340, which equalsif we keep five digits after the decimal pointto the result of the authors of34. The same confirmation was made with the references1,26,33.
The axial velocity profile computed by using the velocity expression3.13for different values of Hartmann number Ha, Knudsen number kn, Womersley parameterα, frequency of the body accelerationb, the permeability parameter of porous mediumkand have been shown through Figures2to13. It is observed that fromFigure 2that as the Hartmann number increases the axial velocity decreases.Figure 3shows that by increasing the Knudsen number the axial velocity decreases with small amount.
InFigure 4the axial velocity of the blood increases with increasing the permeability parameter of porous mediumk. The effect of Womersley parameterαon the axial elocityu has been showed inFigure 5. We can see that the axial velocity increases with increasing the Womersley parameter.
Figures6,7,8, and9present the effect of the frequency of the body acceleration b on the axial velocity distribution for various values of Knudsen number kn. We note that the axial velocity decreases with increasing the frequency of body accelerationb. InFigure 6we note that there is no reflux at kn 0.001negative values of the axial velocity. The reflux appears inFigure 7at kn0.1 the negative values begin atr0.9near to the wall of artery
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
0 0.5 1 1.5
t
Blood acceleration
b=1
=3 b
=2 b
Figure 19: Effect of frequency of body acceleration on the blood accelerationα3, Ha1,ao3,Ao2, A14,t1, kn0.01, andk0.5.
With increasing the value of Knudsen number knkn0.2as inFigure 8the reflux occurs atr0.6. Whereas the reflux occurs atr 0kn0.3as shown inFigure 9.
The blood acceleration profile is computed by using 3.15 for different values of Hartmann number Ha, Knudsen number kn, permeability parameter of porous medium k, the Womersley parameter, and the frequency of the body accelerationb. It is observed from Figure 15that the blood acceleration decreases with increasing the Hartmann number Ha up tot0.2 and then increases with increasing the Hartmann number Ha up tot1. The blood acceleration increases with increasing each of Knudsen number kn, permeability parameter of porous mediumkand Womersley parameterαup tot 0.3 as shown in Figures16,17, and18.
The effect of Hartmann number Ha on the shear stressτ is presented inFigure 10. In all our calculations the dynamic viscosity of the blood is takenμ 2.5 ref. to9. We note that the shear stress equals zero at the center of the artery and decreases with increasing the Hartmann number Ha. Also the shear stressτdecreases with increasing the frequency of the body accelerationbas shown inFigure 14. Figures11,12, and13show that the shear stress τincreases with increasing the permeability parameter of porous mediumk, the Womersley parameterαand the Knudsen number kn.
Figure 19 represents the effect of the frequency of body acceleration on the blood acceleration. We note that there is no effectapproximately up tot 0.4 then the blood acceleration decreases with increasing the frequency of body acceleration.
5. Conclusions
In the present mathematical model, the unsteady pulsatile blood flow through porous medium in the presence of magnetic field with periodic body acceleration through a rigid straight circular tubeartery has been studied. The slip condition on the wall artery has been considered. The velocity expression has been obtained in an approximation way. The corresponding expressions for shear stress and fluid acceleration are also obtained. It is of interest to note that the axial velocity increases with increasing of the permeability parameter of porous medium and Womersley parameter whereas it decreases with increasing the Hartmann number, frequency of body acceleration, and Knudsen number. Also, the shear stress increases with increasing the permeability parameter of porous medium, Womersley parameter, and Knudsen number whereas decreases with increasing Hartmann number and the frequency of body acceleration. Finally, the blood acceleration increases with increasing the permeability parameter of porous medium, Womersley parameter, and Knudsen number whereas decreases with increasing Hartmann number and the frequency of body acceleration.
The present model gives a most general form of velocity expression from which the other mathematical models can easily be obtained by proper substitutions. It is of interest to note that the result of the present model includes results of different mathematical models such as:
1The results of Megahed et al.34have been recovered by taking Knudsen number kn0.0no slip condition.
2The results of Kamel and El-Tawil 33have been recovered by taking Knudsen number kn 0.0, the permeability of porous mediumk → ∞without stochastic and no body acceleration.
3The results of El-Shahed 26 have been recoverd by taking Knudsen number kn0.0 and Hartmann number Ha0.0no magnetic field.
4The results of Chaturani and Palanisamy 1 have been recovered by taking Knudsen number kn 0.0, the permeability of porous medium k → ∞ and Hartmann number Ha0.0no magnetic field.
It is possible that a proper understanding of interactions of body acceleration with blood flow may lead to a therapeutic use of controlled body acceleration. It is therefore desirable to analyze the effects of different types of vibrations on different parts of the body.
Such a knowledge of body acceleration could be useful in the diagnosis and therapeutic treatment of some health problemsjoint pain, vision loss, and vascular disorder, to better design of protective pads and machines.
By using an appropriate magnetic field it is possible to control blood pressure and also it is effective for conditions such as poor circulation, travel sickness, pain, headaches, muscle sprains, strains, and joint pains. The slip condition plays an important role in shear skin, spurt and hysteresis effects. The fluids that exhibit boundary slip have important technological applications such as in polishing valves of artificial heart and internal cavities.
Hoping that this investigation may have for further studies in the field of medical research, the application of magnetic field for the treatment of certain cardiovascular diseases, and also the results of this analysis can be applied to the pathological situations of blood flow in coronary arteries when fatty plaques of cholesterol and artery clogging blood clots are formed in the lumen of the coronary artery.
Appendix
M0 α2m2mosinm1t kn−1kn , M1 1
m5 1 16
α2m2moHa2sinm1t
m5kn−1kn −α2m2moHa2sinm1t
m5−1kn −α2cosm1t m5
1 2
α2m2mosinm1t m5kn−1kn 1
16
α2m2mosinm1t km5kn−1kn, M216kcost
m4 cost
m4 64k2cost
m4 − m2cosm1t
m4 12α2m2moHa2k2sinm1t m4 kn−1kn
−6α2m2moHa2ksinm1t m4 −1kn 3
16
α2m2moHa2ksinm1t
m4kn−1kn −m2Ha4k2cosm1t m4
2α2ksint
m4 −32α2knk2sint
m4 2α2Ha2k2sint
m4 16α2k2sint m4
−α4k2cost
m4 −32knkcost
m4 Ha2k2cost
m4 16Ha2k2cost m4 2Ha2kcost
m4 32m2knkcosm1t
m4 −16m2kcosm1t
m4 32m2knk2Ha2cosm1t m4
−32knk2Ha2cosm1t
m4 12m2kα2mosinm1t
m4kn−1kn −3m2k2α2moHa4sinm1t m4kn−1kn 32m2k2α2mosinm1t
m4kn−1kn −64m2k2α2mosinm1t
m4−1kn −64m2k2cosm1t m4
−2m2kHa2cosm1t
m4 1
16
m2kα6mosinm1t
m4kn−1kn −32m2k2α2moHa2sinm1t m4−1kn 32m2k2α2moknHa2sinm1t
m4−1kn 1 16
m2k2α6moHa2sinm1t m4kn−1kn
−32m2kα2moHa2sinm1t
m4−1kn 32m2kα2mosinm1t
m4−1kn −m2k2α6mosinm1t m4−1kn 3m2kα2moHa2sinm1t
m4kn−1kn 1 16
m2α2mosinm1t km4kn−1kn 3
16
m2α2moHa2sinm1t m4kn−1kn
−3m2α2mosinm1t m4−1kn 3
2
m2α2mosinm1t
m4kn−1kn m2k2α4cosm1t
m4 −16m2k2Ha2cosm1t m4
1 2
m2α6k2mosinm1t m4kn−1kn 1
16
m2α2k2moHa6sinm1t m4kn−1kn 3
2
m2k2α2moHa4sinm1t m4kn−1kn , M3 cosbt
m3 64k2cosbt
m3 16kcosbt
m3 −m2cosm1t
m3 −3α2m2mosinm1t m3−1kn
3 2
α2m2mosinm1t
m3kn−1kn −16km2cosm1t
m3 −16k2m2Ha2cosm1t m3
2bk2α2m2Ha2sinbt
m3 1
16
k2α2m2moHa6sinm1t m3kn −1kn 1
16
α2m2mosinm1t km3kn−1kn
−32k2α2m2moHa2sinm1t
m3−1kn 32k2α2m2moknHa2sinm1t m3−1kn
−3k2α2m2moHa4sinm1t
m3−1kn −32kα2m2mosinm1t
m3−1kn 32kα2m2moknsinm1t m3−1kn
−k2α6m2mob2sinm1t
m3−1kn 3kα2m2moHa2sinm1t
m3kn−1kn 12kα2m2mosinm1t m3kn−1kn
−32b k2α2knsinbt m3 3
16
α2m2moHa2sinm1t m3kn−1kn 3
2
k2α2m2moHa4sinm1t m3kn−1kn 12k2α2m2moHa2sinm1t
m3kn−1kn −6kα2m2moHa2sinm1t m3−1kn 3
16
kα2m2moHa4sinm1t
m3kn−1kn 32k2α2m2mosinm1t
m3kn−1kn 32k m2kncosm1t m3 32k2m2knHa2cosm1t
m3 −64k2α2m2mosinm1t
m3−1kn −k2m2 Ha4cosm1t m3
2bkα2sinbt
m3 16k2Ha2cosbt
m3 −32kkncosbt
m3 2kHa2cosbt m3
k2Ha4cosbt
m3 −32k2knHa2cosbt
m3 −k2α4 b2cosbt
m3 16bk2α2sinbt m3
k2α4b2m2cosm1t
m3 1
16
k2α6m2moHa2b2sinm1t m3kn−1kn 1
2
k2α6m2mob2sinm1t
m3kn−1kn −2k m2Ha2cosm1t
m3 1
16
kα6m2mob2sinm1t m3kn−1kn ,
A.1 M4 1
2
m2mosinm1t
kkn−1kn − m2cosm1t
kα2−1knm2kncosm1t
kα2−1kn −m2mosinm1t
k−1kn , A.2 M5 Ha2k
m5 1
m5 m2cosm1t
m5−1kn−m2kncosm1t
m5−1kn −k m2knHa2cosm1t m5−1kn α2m2mosinm1t
m5−1kn k m2Ha2cosm1t
m5−1kn 8km2cosm1t
m5 −16km2kncosm1t m5
8k m2cosm1t m5−1kn −1
2
k α2m2moHa2sinm1t
m5kn−1kn kα2m2moHa2sinm1t m5−1kn
−4kα2m2mosinm1t
m5kn−1kn −24km2kncosm1t
m5−1kn 16km2kncosm1t m5−1kn
−1 2
α2m2mosinm1t m5kn−1kn ,
A.3 M616kcost
m4 α4Ha2k3cost
m4 64k2cost
m4 −m2cosm1t m4 12α2m2moHa2k2sinm1t
m4kn−1kn
−6α2m2moHa2ksinm1t m4−1kn 3
16
α2m2moHa2ksinm1t
m4kn−1kn −m2Ha4k2cosm1t m4
2α2ksint
m4 −32α2knk2sint
m4 2α2Ha2k2sint
m4 16α2k2sint m4
−α4k2cost
m4 −32knkcost
m4 Ha4k2cost
m4 16Ha2k2cost m4 2Ha2kcost
m4 32m2knkcosm1t
m4 −16m2kcosm1t
m4 32m2knk2Ha2cosm1t m4
−32knk2Ha2cosm1t
m4 12m2kα2mosinm1t
m4kn−1kn −3m2k2α2moHa4sinm1t m4kn−1kn 32m2k2α2mo sinm1t
m4kn −1kn −64m2k2α2mosinm1t
m4−1kn −64m2k2cosm1t m4
−2m2 kHa2cosm1t
m4 1
16
m2k α6mosinm1t
m4kn−1kn −32m2k2α2moHa2sinm1t m4−1kn 32m2k2α2moknHa2sinm1t
m4−1kn 1 16
m2k2α6moHa2sinm1t m4kn−1kn
−32m2k α2moHa2sinm1t
m4−1kn 32m2kα2mosinm1t
m4−1kn −m2k2α6mosinm1t m4−1kn 3m2k α2moHa2sinm1t
m4kn−1kn 1 16
m2α2mosinm1t km4kn−1kn 3
16
m2α2moHa2sinm1t m4kn−1kn
−3m2α2mosinm1t m4−1kn 3
2
m2α2mosinm1t
m4kn−1kn m2k2α4cosm1t m4
−16m2k2Ha2cosm1t
m4 ,
A.4 M7 kcosbt
m4 16α4Ha2k3b2cosm1t
m4 8m2cosm1t m4
