1

Military Technical College

Kobry Elkobbah, Cairo, Egypt

April 19-21,2016

8th International Conference on Mathematics and Engineering

Physics (ICMEP-8)

EM-1

Series Solution For APeristaltic Flow Of a Magneto-fluid

Through ATaperEccentric Cylinders Khaled S. Mekheimer1* , Mohamed S. Mohamed1 ,

Shurouq Abdulaziz A. Alnufiai2

1Dept of Mathematics and statistics, Faculty of Science, Taif University, Hawia 888, Taif, Saudi Arabia & Dept of Mathematics,Faculty of Science, Al-Azhar University,

Nasr City, Cairo, EGYPT. UKh_Mekheimer@yahoo.comU 2 Dept of Mathematics, Faculty of Science, Taif University, Hawia 888, Taif, Saudi

Arabia Abstract: The magneto-hydrodynamic peristaltic flow of an incompressible Newtonian fluid was investigated between two eccentric tubes. The problem is measured in cylindrical coordinates. Geometrically, we considered two eccentric tubes in which the inner tube is rigid while the outer tube is tapered and have a sinusoidal wave generated on its walls. The governing equations are observed nonlinear second order partial differential equations. Under the conditions of long wavelength approximation the problem has been solved with the help of the homotopy perturbation method (HPM). The obtained results are then plotted to see the influence of the different physical parameters on the velocity profile, pressure gradient , pressure rise and frictionless force expressions. The velocity profile is drawn in the two and three dimensions. The trapping bolus phenomena is also discussed through the contour plot of the streamlines. Keywords Peristaltic Flow ; Magneto Fluid; Taper Eccentric Cylinders; HPM.

3. 1 Introduction Peristalsis pumping is a phenomenon in which fluid transport happens when a

gradual wave of area contraction orexpansionpropagates along the length of distensible duct. Peristalsis is oneof thetopic highlyimportant in applied mathematics, engineering, physiological world and its have many applications in real life. It is an automatic and vital process that moves food through the digestive tract, transport from kidney to bladder, in the vasomotion ofsmall blood vessels such as venues, capillaries and arterioles. And also the mechanism of peristaltic transport has beenexploited for industrial applications like sanitary fluid transport, corrosive fluids, a toxicliquid transport in the nuclear industry.The peristaltic transport with long wavelength at low Reynolds number [1,2] with all Reynoldnumbers [3] or for long and short wavelength [4] have been analyzed.Tsui, Yeng-Yung, et al.[5] Pumpingflow in a Channel with a peristaltic wallRACHID, Het al.[6]The effect of a pulsatile flow onthe Peristaltic Output: Case of a Newtonian Fluid.And with through a porous mediumby Afifi NAS et

2

al.[7].Mekheimer [8] have shown the effect of the induced magnetic fieldon peristaltic flow of acouple stressfluid.N.Nagendra[9]MHD flow of a Newtonian fluid througha porousmedium inan asymmetricchannel with peristalsis.The non-Newtonian effects ofdifferenttypes offluids [1--13]or of the Nonlinear and curvature effects [14] on the peristaltic flow have been studied.Investigations are available in the literature to study the effectof an endoscope on peristaltic motion of Newtonian and non-Newtonianfluids [15-16] . Theeccentric annulus is normally not easy to discuss even without peristalsis. There are only a fewstudiestake intoattentiontheeffect of the eccentricitywith peristalsis [16].In the present work we discussthe peristaltic flow of Newtonian fluidsunder the effect of the magnetic fieldin an eccentriccylinders.which the inner tube is rigid while theouter have asinusoidal waves generated on it taperedwalls.Peristalticpumping characteristics are discussed in detail.The flow analysis is developed in the unsteady state by using the long wavelength approximation.Theproblem is firstmodeledand then solved analyticall for the axial velocity, axialpressuregradient andpressurerise.This analysisgivesa betterjudgement forthe speed of injection andthefluid flowcharacteristics within the syringe. Also,theinjection canbecarried out more proficientlyandpain of thepatient can be extenuated.

2 Mathematical formulation of the problem: Consider a MHDflow of anincompressible Newtonian fluid through eccentric

tubes. The inner tube is rigid(endoscope or catheter) andtheouter havea sinusoidalwaves generated on the walls of tapered. The radius ofthe inner tube is δ ′but we need to reference the fluidmotion to the center of the outer tube. The center of the inner tube isnowat position 0, =′′=′ yz ε where and are coordinates in the cross-section of the pipe asshown in figure 1.b Then the boundary of the inner tube isdescribed toorderε ′by ].cos[1 θεδ ′′+′=′r (obtained by using the cosine rule) where( δε ′<<′ ) is theparameter that controls theeccentricity of the inner tube position. Thegeometry of the walls surface is described in fig.(1)

3

The equations for the radii are

)z(ξza(z)a

)],tcz(λπ2cos[b)z(ar

],θcos[εδr

1

12

1

′′′++=′

′−′′+′′=′

′′+′=′

(1)

where )(1 za′ is the radius of the tapered tube segment in theannulus region, a is the radiu of tubeofthe non-tapered wall, ξ is the tapering parameter,δ ′ is the radius of the inner tube, b′ is the amplitude of the wave,λ′ is the wavelength, c′ is the propagation velocity and t ′ is the time. The problem has beenstudiedin cylindrical coordinate system ),,( zr ′′ θ radial, azimuthal and axial coordinatesrespectively. The equation of motion of the flow in the gap between the inner and the outer tubes are

0θw

r1

rv

rv

zu

=′∂′∂

′+

′′

+′∂′∂

+′∂′∂

(2)

][][ 2

2

22

2

2

2

θu

r1

ru

r1

ru

zuμ

zp

θu

rw

ruv

zuu

tuρ

′′′′ ∂′∂

+′∂′∂

′+

∂′∂

+∂

′∂+′∂′∂

−=′∂′∂

′′

+′∂′∂′+

′∂′∂′+

′∂′∂ u

ρ(r)Bσa 2

02

′−

(3)

][

][

2

2

222

2

22

2

2

2

2

θw

r2

rv

θv

r1

rv

r1

rv

zvμ

rp

rw

θv

rw

rvv

zvu

tvρ

′′′′′′′

′

∂′∂

−′

−∂

′∂+

′∂′∂

′+

∂′∂

+∂

′∂+

′∂′∂

−

=′

−′∂′∂

′′

+′∂′∂′+

′∂′∂′+

′∂′∂

(4)

][

][

θv

r2

rw

θw

r1

rw

r1

rw

zwμ

θp

rvw

θw

rw

rwv

zwu

twρ

222

2

22

2

2

2

′∂′∂

+′

−∂

′∂+

′∂′∂

′+

∂′∂

+∂

′∂+′∂′∂

−

=′′′

+′∂′∂

′′

+′∂′∂′+

′∂′∂′+

′∂′∂

′′′′′′

(5)

Fig 3.1 The simplified model of geometry of the problem

4

Where u,w,v ′′′ are the velocity components in r′ ,θ′ and z ′ -directions respectively, ρ is the density, p′ is the pressure and µ is theviscosity, )(0 rB is the magnetic field, σ is the electrical conductivity. The boundary conditions are:

(6)

It isconvenient to non-dimensionalize thevariables appearing inequations(2-5)andintroducingReynoldsnumber Re ,wave number ratio δ as follows:

(z)ξaλξ(z),

μa(r)σB

(r)M

t)),π(zcos(2ab1

ar

r ,λzz ,

λaδ,

aεε ,

aδδ ,

abφ

)θcos(aε

aδ

ar

r,μ

ρcaRe, tλct,θθ,

cVV,

cuu,p

μcλap

2202

220

*

11

2

′==

−′

+==′

==′

=′

==

′′

+′

===′=′=′

=′

=′=

′

′

(7) where Re is the Reynolds number, M is the Hartmannnumber, where the parameter of taper is(ξ = tan φ ), φ is calledtaperangle andfor the converging tapering (ξ< 0), non-tapered artery (ξ = 0) and the diverging tapering (ξ> 0)as shown in( fig.3.1) . 0δ is thedimensionless wave number and ε is eccentricityparameter. *φ is the amplitude ratio. After using the above assumption andthe long wavelength approximation )0( 0 →δ then

taking r

MM = , the equations of motion in the dimensionless form become :

,0zu=

∂∂

(8)

ur

Murr

urr

uzu

zp

2

2

2

2

22

2

2

220

11−

∂∂

+∂∂

+∂∂

+∂∂

=∂∂

θδ (9)

,0rp=

∂∂

(10)

,0p=

θ∂∂

(11)

Eqs. (10, 11) showsthat p is not a function of r and θ The corresponding boundary conditions in non-dimensional form are

,0=u at )],(2cos[1 *2 tzrr −++== πφξ z (12)

,Vu = at ]cos[1 θεδ +== rr (13) 3 Solution of the problem Solution of the above boundary value problem is obtained by series solution method [21]. The deformation equation for thegiven problem is defined as

0)zp

θu~

r1]u~[f(]u~[]u~[f)((1f)H(u, 2

2

20 =∂∂

−∂∂

++−−=

(14)

,rrat ,Vu,rrat 0,u

1

2

′=′′=′′=′=′

5

where , the linear operator is assumed to berrr ∂∂

+∂∂

=1

2

2

2

2

rM

− We define the

following initial guess satisfying the boundary conditions

])],2r1r(Log[ [MCsch ]))

2rr(Log[ Sinh(M V 0u~ = (15)

Now we describe ...uffuuf)t,z,θ,(r,u~ 2

210 +++= (16)

Using the above equation into Eq. (14) and then finding theterms of first two orders of embedding parameter f ,we get thefollowing problems including boundary conditions Zeroth order system:

0]u~[][u 00 =− (17)

20 rrat 0,u == (18)

10 rrat V,u == (19) The solution of the above zeroth order system can be obtainedby using Eq. (15) and is simply found as

=f)t,z,θ,(r,u 0 ])]rr(Log[ Csch[M]))

rr(Log[ Sinh(M Vu~

2

1

20 = (20)

First order system:

0dzdp

θu

r1]u~[][u 2

02

201 =+∂∂

−−

or

dzdpu

rM

θu

r1u

rM

ru

r1

ru

12

2

20

2

212

21

21

2

=−∂∂

+−∂∂

+∂∂

(21) 21 rrat 0,u == (22)

11 rr at 0,u == (23)

The solution of the above linear ordinary differential equation is given in appendix

]]) rrr[Sinh[M(Log ])

rr)(Log[M+4g(+

])]rr[Sinh[M(Log

zp M r4 ])]

rr[Sinh[M(Log ])

rrr)(Log[M+4g(

zp M r ])](4

rr[Sinh[M(Log r

zp M4])](

rrCsch[MLog[

) M+(-4 M41=u

2

112

221

1

2

12

22

2

12

2

121

−

∂∂

−−−

∂∂

∂∂

−

Where

][θ Sin) ]rr(M(Log[ [M Csch 2 +ε(1 M

+]])] rr(Log[ Coth[M r ]))]

rrM(Log[ Csch[ ε V M

r1 = g

2

2

12

2

11

2

12

1

Finally, for 1f → , we approach the final solution. So from Eq. (16),

6

we get 10),,,(~ uutzru +=θ (24)

where 0u and 1u are defined in Eq. (20). The instantaneous volume flow rate ),( tzQ is given by

∫= 2

1

,2),(r

rrudrtzQ π (25)

])]rr[Cosh[M(Log2(r+)

zp M)r(r+r g)(2M+((4

2)M4M(4

1π2

Q

2

121

42

41

22

2

2

∂∂

−+−

=

])))rrM((Log[ VCsch[)M+4M(+])]

rrLog[ [M(Coth M(g2+)M+g(4 (

1

22

2

12 −−

])2r1r)(Log[2M +4(2

2r g+zpM)4

2r412(r])](

2r1rog[M(Coth[M(L+ −

∂∂

+−

]))))) rr[Sinh[M(Log r 2M+r (

])]rr g[VCsch[M(Lo)2M+2(-4 -])]

rr [Sinh[M(Log r g2 +

])rr)((Log[2M+4g(+

zpMr 4g+ 2])](

rr[Csch[M(Logr

2

121

22

1

2

2

121

1

222

2

121 −

∂∂

−+

26)( The mean volume flow rate Q over one period is given as[22]

)](2[cos)](2cos[2)](2[2

),( 22**2*

tztztzzQtzQ −+−+−+−= πφπφπφξφπ

Cos

(27)

Now pressure gradient dzdp

will be evaluated by using Eqs.(26) and (27) and is defined

in appendix

7

(28)

The pressure rise )(tp∆ in non-dimensional form is defined as

∫ ∂∂

=∆1

0)( dz

zptp (29)

4 Results and discussion The analytical and numerical results obtained above for the given analysis are

discussed graphicallyThe graphical treatment for the data of pressure rise ∆p, pressure

gradient and velocity profileu(r, θ, z, t) with thevariation of all emerging

dimensionlessparameters like time t, flow rate Q, the taper parameterζ , the velocity of the inner tube V, the eccentricity parameter ε and the MHD parameter M has been analyzed. In the end, the stream lines observing the peristaltic flow aredrawn forthe parametersM, Q and ζ while other parameters remain fixed. The comparison graph for the values obtained in present work with the results of R. Ellahi et. al. [68] is displayed in figures. The graphs for the pressure rise∆ p(t) versus flow rate Q under the effects of givenparameters are drawn in figs2-6. These graphs showthepumpingregions, that is, the peristaltic (Q > 0,∆ p > 0) , the augmented pumping (Q > 0,∆ p < 0) and

theretrogradepumping(Q < 0,∆ p > 0)The pressure gradient against the the coordinatez with the variation of pertinent parameters are shown in figs7-10. The velocity field u(r,θ, z, t) versus the radial coordinate r is plotted in fig.s 11-15 for both two and three dimensions. Thestream line graphs are shown in fig16-18 fig2 is plotted to seethe variation of pressure risefor differentvaluesof theeccentricityparameter ε andthe angle θ while all other parameters are keptfixed. It isobserved that peristaltic pumping region isinbetweenQ ∈ [0, 0.7] , augmented pumping is in Q ∈ [0.7, 2] and retrograde pumping part is Q∈ [−1, 0]. Itis also observed from this graph that the pressure rise increases with thevariation of ε but decreases with the angle θ in between the region Q ∈ [−1, 0.7] and opposite behavior is seen in the remaining part. The graph of pressure rise for theparameter M and δ is plotted in fig3. Fig4. shows that theperistalticpumping partisQ

8

∈[0, 0.3]when 1.0* =φ and varies valuesof ε , while augmented and retrograde pumping regions are Q∈ [0.3, 2] and Q∈ [−1, 0] , respectively. that the pressure rise in fig 5. decreases with parameterζ in case(ζ > 0 and ζ <0) when it is increases with thetaperparameterζ in case (ζ = 0), The variation of pressure rise∇p for V is

similar to that of M (See fig6)The pressure gradient for the parameters M and δ is drawn in fig 3.6. It is measured from this figurethat pressure gradient is in linear relation with both of the parameters in narrowest parts of thecylinders but inverse relation is seen in thewider parts.The variation of pressure gradient with the parameters φ and ε is very much similar to that of the parameters M and δ and is shown in fig8. The only difference is that the pressure gradient is minimum on the left and right sides of the cylinder while appears maximum at the centre. It means that flow can easily pass without imposition of large pressure gradient in the two sides of the channel while much pressure gradient is required to maintain the flux in thecentral part near z = 0.8. This is in good agreement with the physical condition. Also, for a divergingtapering with angleζ .Figs(8a) extending case ζ >0figs( 8b), the pressure gradient values higher than all other existing results corresponding to converging tapering case ζ <0 .It can be observed from figs9 and 10 that the pressure gradient increases with the parameters Q and V,while when δ is increased the pressure gradient decreaseson the left and right sides but increases at the centre of the cylinders. It is also seen thatthe variation of pressure gradient remains same in the two sides of the channel and become different at the central part with changing V but this variation remains same throughout for the parameter Q.Thefig11 shows that the velocity field is anincreasing function of the parameterδ while decreasing with the parameter M. The velocity field is in inverserelation with Q but have a direct variation with ε (see fig12).It is also observed that the presence of magneticfield for fluid causes to slow down the flow. It is observed from fig13 that the velocity distribution is increasing withδ and *φ while reducing for t. Fig14 shows that the velocity profile is linearlychanging with *φ and V.shows the Fig 15 that velocity in the case ζ > 0 at (15.a) higher than reported in the case ζ ≤ 0 at (3.15.b). Fig 3.16 is drawn to see the stream lines for the parameter M. It is measured from this figure that numbers of bolus are not changing but size is increasing with the increasingeffects of M in the bottom of the cylinder,while bolus are lessened in number when seen in the upper part. The boluses are reduced both in size and number when seen for the parameter Q in both parts of the geometry (see fig17).Fig 16 is drawn to see the stream lines for the parameter M..It is measured from this figure that numbers of bolus are not changing but size is decreasing with the increasing effects of M in upper part of the cylinder, while bolus are disappears when seen in the the bottom part. The boluses are reduced both in size and number when seen for the parameter Q in both parts of the geometry (see fig 17). It is seen from fig 18 that the numbers of bolus are decreasing with different values of the parameterζ in both sides of the cylinder but in the lower half of the tube, the bolus becomes smaller with increasing magnitude of the parameterζ >0 and fade out with less values of ζ <0.

9

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

-4

-2

0

2

4

6

8

ε=0.3

ε=0.2

θ=500 θ=750

θ=500 θ=750

∆p

Q

Fig 2pressure rise versus flow rate for fixed parameters

5.0,5.0,1.0,1.0,0,2.0* ====== VMt δζφ

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0-4

-3

-2

-1

0

1

2

3

4

5

δ=0.2

δ=0.1

M=0.1 M=0.7 M=0.1 M=0.7

∆p

Q

Fig 3pressure rise versus flow rate for fixed parameters

5.0,50,01.0,1.0,0,2.0 0* ====== Vt θεζφ

10

Fig 4pressure rise versus flow rate for fixed parameters

5.0,5.0,1.0,1.0,0,500 ====== VMt δζθ

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

-2

0

2

4

6

8

10

ξ=-0.1 ξ=0 ξ=0.1

∆p

Q

Fig 5pressure rise versus flow rate for fixed parameters

5.0,5.0,1.0,1.0,01.0,500 ====== VMt δεθ

11

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

-3

-2

-1

0

1

2

3

4

5

6

7

V=0.5

V=0.1

δ=0.1 δ=0.2δ=0.1 δ=0.2

∆p

Q

Fig 6pressure rise versus flow rate for fixed parameters

0* 50,5.0,01.0,1.0,0,2.0 ====== θεζφ Mt

0.4 0.6 0.8 1.0 1.2

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

δ=0.2

δ=0.1

M=0.5 M=1 M=0.5 M=1

dp/d

z

z

Fig 7Variation of pressure gradient with z for fixed

parameters 5.0,3.0,01.0,50,3.0,0,1.0 0* ======= QVt εθζφ

12

0.2 0.4 0.6 0.8 1.0 1.2

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

ε=0 ξ=0.1 ε=0 ξ=0.2 ε=0.3 ξ=0.1 ε=0.3 ξ=0.2

dp/d

z

z

a

0.2 0.4 0.6 0.8 1.0 1.2

-2

0

2

4

6

8

10

12

14

ξ= -0.1 ξ= 0 ξ= 0.1

dp/d

z

z

b Fig8Variation of pressure gradient with z

forfixedparameters 5.0,3.0,01.0,50,3.0,5.0 0 ====== QVtM εθ

13

0.2 0.4 0.6 0.8 1.0 1.2

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

V=0.6

V=0.3

δ=0.1 δ=0.2δ=0.1 δ=0.2

dp/d

z

z

Fig10Variation of pressure gradient with z

forfixedparameters 3.0,5.0,01.0,50,3.0,0,1.0 0* ======= VMt εθζφ

0.2 0.4 0.6 0.8 1.0 1.2-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Q=0.6

Q=0.3

δ=0.1 δ=0.2δ=0.1 δ=0.2

dp/dz

z

Fig9Variation of pressure gradient with z

forfixedparameters 3.0,01.0,50,3.0,0,1.0 0* ====== Vt εθζφ

14

0.2 0.4 0.6 0.8 1.00.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

δ=0.25

δ=0.18 M=0.1 M=0.5 M=0.9 M=0.1 M=0.5 M=0.9

u

r

Fig11Variation of velocity profile uwith rfor fixedparameters

1.0,5.0,5.0,1.0,75,0,1.0,0 0* ======== εθζφ tQVz

0.2 0.4 0.6 0.80.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

ε=0.1

ε=0.0

Q=0.1 Q=0.2Q=0.1 Q=0.2 Q=0.3

u

r

Fig12Variation of velocity profile uwith rfor fixed parameters

1.0,5.0,5.0,1.0,75,0,1.0,0 0* ======== εθζφ tQVz

15

0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

0.20

t=0.5 φ*=0.1 t=0.55 φ*=0.12

t=0.5 φ*=0.12

δ=0.2

δ=0.1 t=0.5 φ*=0.1 t=0.55 φ*=0.1 t=0.5φ*=0.12

u

r

Fig13Variation of velocity profile u with r for fixed

parameters 5.0,01.0,1.0,1.0,75,0,1.0,0 0* ======== MQVz εθζφ

0.2 0.4 0.6 0.80.0

0.1

0.2

0.3

0.4

V=0.2

V=0

φ*=0.1 φ*=0.15 φ*=0.2φ*=0.1 φ*=0.15 φ*=0.2

u

r

Fig14Variation of velocity profile u with r for fixed parameters

5.0,1.0,2.0,5.0,1.0,50,0,0 0 ======== MtQz εδθζ

16

Fig 15Variation of velocity profile uwith rfor fixedparameters 5.0,1.0,2.0,5.0,1.0,2.0,50 *0 ======= MtQ εδφθ

0.2 0.4 0.6 0.8 1.0-0.1

0.0

0.1

0.2

0.3

ξ=0.1 z=0.08 ξ=0.3 z=0.08 ξ=0.1 z=0.15 ξ=0.3 z=0.15

u

r

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-0.1

0.0

0.1

0.2

0.3

0.4

ξ=-0.3 z=0.08ξ=0 z=0.08ξ=-0.3 z=0.2ξ=0 z=0.2

u

r

17

Fig 16:Stream lines for different values of M for The other

parameters are 0,4.0,05.0,2.0,1,3.0,02.0,50 *0 ======== ζεδφθ tQV

Fig 17: Stream lines for different values of Q for The other parameters are 0,4.0,05.0,5.0,1,3.0,05.0,50 *0 ======== ζεδφθ tMV

M=4

M=3

M=6

M=5

Q=0.7 Q=0.6

Q=0.9 Q=0.8

18

Fig 18: Stream lines for different values of a for (ζ >0), b for (ζ ≤0.0) The other parameters are 3.0,4.0,05.0,5.0,6.0,3.0,2.0,50 *0 ======== MtQV εδφθ

15.0=ζ

3.0=ζ

02.0=ζ

05.0=ζ

15.0−=ζ

0.0=ζ

3.0−=ζ

05.0−=ζ

a

b

19

5 Concluding remark In the present investigation Homotopy Perturbation method solutions are presented

for the peristaltic flow of MHD fluid between two eccentric tubes.the inner tube is rigid and the outer havea sinusoidal waves generated on the walls of tapered. The problem is measured under the assumptions of long wave length and low Reynolds number. The following observationshave been found:

• It is observed that pressure rise is a decreasing function of taper parameterζ and θ while increasing function of radius δ , magnetic field M

• pressure rise is a increasing function of eccentricity parameterε ,amplitude ratio *φ andthe velocity of the inner tube V.

• The pressure gradient increases with radius δ , eccentricity parameterε andthe velocity of the inner tube V while decreas with flow rate Q, taperparameterζ and magnetic field M .

• The velocity profile is increase with the increase in radius δ ,amplitude ratio *φ and eccentricity parameterε while decreas with theflow rate Q, magnetic field M an d taper parameter ζ ,

• The velocity in the case of eccentric cylinders higher than concentric one • It is depicted that number of bolus is changing inversely with taper parameterζ

and flow rate Q while not changing increasing effects of magnetic field M but dimensions.

References

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[10] Mekheimer, K.S., Abdelmaboud, Y., & Abdellateef, A.I. (2012). Peristaltic transport through an eccentric cylinder: mathematical model. Appl Bionics Biomech. doi: 10.3233. ABB-2012-0071.

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