EFFECT OF INDUCED MAGNETIC FIELD ON PERISTALTIC …...Effect Of Induced Magnetic Field On Peristaltic Transport Of A Micropolar Fluid Int. J. of Appl. Math and Mech. 10 (6): 81-107,

Int. J. of Appl. Math and Mech. 10 (6): 81-107, 2014.

EFFECT OF INDUCED MAGNETIC FIELD ON PERISTALTIC TRANSPORT OF A MICROPOLAR FLUID IN THE PRESENCE OF

SLIP VELOCITY

G. C. Shit, N. K. Ranjit, A. Sinha and M. Roy

Department of Mathematics, Jadavpur University, Kolkata - 700032, India Email: [email protected]

Received 14 February 2014; accepted 5 April 2014

ABSTRACT This paper concern with the investigation of peristaltic transport of a micropolar fluid in an asymmetric channel in the presence of induced magnetic field. The effect of slip velocity on the boundaries of the flow field is taken into account. The analysis is carried out by using long wave length and low Reynolds number assumptions. Mathematical expressions for axial velocity, stream function, magnetic force function and axial induced magnetic field are constructed. The above said quantities are computed for a specific set of values of the different parameters involved in the model analysis. The results estimated on the basis of the computation are presented graphically. The results for different values of the parameters involved in the problem presented here, show that the flow is appreciably influenced by the presence of a magnetic field. The role of slip velocity has similar effect as the induced magnetic field. Keywords: Slip velocity, Peristaltic Flow, Micropolar Fluid, Induced Magnetic Field 1 INTRODUCTION In recent years, there has been a growing interest in peristaltic transport of physiological fluids. This is perhaps due to the fact that fluid transport through a tube by peristaltic motion as fundamental physiological process has key importance in biomechanical and engineering sciences. To be more specific it encounters in urine transport from the kidney to the bladder, chyme transport in the gastrointestinal tract, the movement of spermatozoa in the ductus afferent of the the male reproductive tract, the movement of the ovum in the fallopian tube and the vasomotion of small blood vessels. The importance of such flows have been recognized in transport of slurries, corrosive fluids, sanitary fluid and noxious fluids in the nuclear industry. Furthermore, roller and finger pumps are widely operated under such mechanism. Latham (Latham 1966) first attempted the study of the mechanism of peristaltic transport. Later on several theoretical and experimental studies (Fung and Yih 1968; Burns and Perkes 1967; Chow 1970; Hung and Brown 1976; Takabatakes and Ayukava 1982; Srivastava and Srivastava 1989; Misra and Pandey 1994; Misra and Pandey 2002) have been made by others to understand the clear idea of peristaltic action in different situations. Literature survey

mailto:[email protected]

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indicates that most of the contributions on peristaltic motion deal with the blood and other physiological fluids behaves as a Newtonian fluid. This approach is satisfactory for peristalsis in ureter but it is not adequate when the peristaltic mechanism involved in small blood vessels, lymphatic vessel, intestine, ductus afferent of the male reproductive tract, and the transport of spermatozoa in the cervical canal is considered. The models of Shapiro (Shapiro 1987) and Shapiro et al. (Shapiro et al. 1969) were highly idealized and represented the peristalsis by an infinite train of sinusoidal waves in a two-dimensional channel. Their models focused themselves in part with offering an explanation of the important phenomenon of reflux. One manifestation of this reflux is that bacteria sometimes travel from the bladder to the kidneys against the mean urinal flow. Srivastava and Srivastava (Srivastava and Srivastava 1995) mathematically analyzed the effects of suspension of small spherical rigid particles in an incompressible Newtonian viscous fluid in a channel with flexible walls. The studies on peristaltic motion in electrically conducting physiological fluids have become a subject of growing interest for researchers. This is due to the fact that such studies are useful particularly for having a proper understanding of different machines used by clinicians for pumping blood and magnetic resonance imaging (MRI). The related studies also contribute to updating the information that is required to operate machines like MHD peristaltic compressors. Li et al. (Li et al. 1994) pointed out that an impulsive magnetic field can be used for therapeutic treatment of patients who have stone fragments in their urinary tract. Hakeem et al. (Hakeem et al. 2003) investigated the effects of magnetic field on peristaltic motion in a uniform tube having variable fluid viscosity. Misra et al. (Misra et al. 2008) theoretically analyzed the peristaltic transport of a physiological fluid in a porous asymmetric channel under the influence of magnetic field. Wang et al. (Wang et al. 2011) carried out the peristaltic transport of a magneto micropolar fluid in a symmetric tube under the assumptions of long wave length and low Reynolds number approximation. However, all these studies have neglected the effect of induced magnetic field even in the case of low magnetic Reynolds number. Owing to the above mentioned assumption, Shit et al. (Shit et al. 2010) considered the effect of induced magnetic field on peristaltic transport of a micropolar fluid in an asymmetric uniform channel in their study. They pointed out that the consideration of induced magnetic field have significant impact on velocity profile, pressure rise per wave length as well as on streamlines. Mekheimer (Mekheimer 2008a; Mekheimer 2008b) studied the effect of induced magnetic field on the peristaltic transport in a symmetric channel by considering couple stress fluid and micropolar fluid model respectively. Although the no-slip condition is known as the central tends of the Navier-Stokes theory, there are some problems where it does not hold. Such problems are then described by defining a partial slip between the fluid and boundary, e.g., the fluid may be particulate or it could be a rarefield gas with a suitable Knudsen number value. Beavers and Joseph (Beavers and Joseph 1967) proposed the slip condition in terms of the tangential components of the velocity and the stress at the boundary. The slip condition is important in the polishing of artificial heart valves and internal cavities in a variety of manufactured parts, microchannels or nanochannels and same applications where a thin film of light oil is attached to the moving plates or when the surface is coated with a special coating such as a thick monoplayer or hydrophobic octadecyltrichlorosilane (Tretheway and Meinhart 2002). Many investigators (Ali et al. 2009; Ali et al. 2008; Kumari and Radhakrishnamacharya 2012; Saleem et al. 2012) have studied the effect of slip velocity at the wall on the peristaltic motion of physiological fluids under different situations.

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To the best of our knowledge, no investigation has been made yet to analyze the influence of slip velocity on peristaltic transport of a micropolar fluid in the presence of induced magnetic field in an asymmetric channel. The no-slip condition is inadequate when one considers fluid exhibiting macroscopic wall slip and that in general is governed by relation between the slip velocity and traction. Due to such fact in mind, the main purpose of the present investigation is to examine the peristaltic motion of a magnetohydrodynamic (MHD) micropolar fluid by considering the effect of induced magnetic field with wall slip condition. 2 FORMULATION OF THE PROBLEM Let us consider the peristaltic motion of an incompressible viscous fluid through an asymmetric two dimensional channel in the presence of an applied magnetic field in which the effect of induced magnetic field is considered. Let us assume that 1= hY ′′ and 2= hY ′′ be respectively the upper wall and lower wall of the channel (cf. Figure 1). The medium is considered to be induced by a sinusoidal wave train propagating with a constant speed c along the channel wall, such that

,boundaryupper on the )(2cos=),( 111

′−′+′′′ tcXadtXhλπ (1)

,boundarylower on the )(2cos=),( 222

+′−′−−′′′ φλπ tcXadtXh (2)

Figure 1: A Physical sketch of the problem where 1a and 2a are the amplitudes of waves, λ is the wave length, )(0 πφφ ≤≤ the phase difference, X ′ and Y ′ are the rectangular co-ordinates with X ′ measures the axis of the channel and Y ′ the traverse axis perpendicular to X ′ . A constant magnetic field of strength 0H is applied in the transverse direction. This gives rise to an induced magnetic field ,0),( yx hhH ′′′ and therefore the total magnetic field will be

,0),( 0 yx hHhH ′+′′+ , where xh′ and yh′ are the components of induced magnetic field along the co-ordinate axes.

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The governing equations for magneto-micropolar fluid, neglecting body couples are represented by the followings set of equations

0=v

′⋅∇′ (3)

( ) +++ ′∇′⋅′−′×∇′+′∇′++

′+′∇′−

′∇′⋅′+

′∂′∂ HHwkvkHpvv

tv

ee

µµµρ 22 )()(21=)( (4)

( ) )()(2=)( wwvkwkwvtwj

′⋅∇′∇′+++′×∇′×∇′−′×∇′+′−

′∇′⋅′+

′∂′∂ γβαγρ (5)

The Maxwell’s equations,

′∂′∂

−′×∇′′′×∇′+

+

tHEJH e

µ= ,= (6)

along with the Ohm’s law

( ))(= +′×′+′′ HvEJ e

µσ (7) in addition, it should be noted that

0= and 0= EH

′⋅∇′′⋅∇′ (8) Now, combining equations (6) - (8) we get the induction equation,

+++

′∇′+′×′×∇′′∂′∂ HHv

tH

e

21)(=σµ

(9)

where σµe

1 (=η ) is the magnetic diffusivity, 2

2

2

22

YX ′∂∂

+′∂

∂≡∇′ and v′ is the velocity vector,

ω′ is the microrotation vector, p′ is the fluid pressure, ρ the fluid density, j the micro-gyration parameter, σ the electrical conductivity, eµ is the magnetic permeability, E′ is an induced electrical field. Also the material constants (or viscosity coefficients of the micro-polar fluid ) µ and k are satisfies the following inequalities 0)(2 ≥+ kµ , 0≥k . The governing equations (5)(3)− and the magnetic induction equation (9) will be solved subject to the boundary conditions defined in the next section.



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3 ANALYTICAL SOLUTION It is further noticed that the flow field in laboratory frame ),( YX ′′ and wave frame ).( yx ′′ are treated as the unsteady and steady motion respectively. Considering the wave frame ),( yx ′′ moving with a velocity c away from a fixed frame ),( YX ′′ that follows from the following transformations

.=),( ,=),( ,= ,= VyxvcUyxuYytcXx ′′′′−′′′′′′′−′′ In which ),( vu ′′ and ),( VU ′′ are the respective velocity components in the laboratory and wave frames. The equations that govern the fluid motion for unsteady flow of an incompressible magneto-micropolar fluid described in the previous section are derived in a cartesian co-ordinate system as

0=YV

XU

′∂′∂

+′∂′∂ (10)

Ywk

YU

XUk

Xp

YUV

XUU

tU

′∂′∂

+′∂′∂

+′∂′∂+

+′∂′∂

−′∂′∂′+

′∂′∂′+

′∂′∂

ρρµ

ρ)(1 = 2

2

2

2

YhH

Yhh

Xhh xex

yx

xe

′∂′∂

−′∂′∂′+

′∂′∂′− 0)(

ρµ

ρµ (11)

Xwk

YV

XVk

Xp

YVV

XVU

tV

′∂′∂

+′∂′∂

+′∂′∂+

+′∂′∂

−′∂′∂′+

′∂′∂′+

′∂′∂

ρρµ

ρ)(1 = 2

2

2

2

Yh

HYh

hXh

h yeyy

yx

e

′∂

′∂−

′∂

′∂′+

′∂

′∂′− 0)(

ρµ

ρµ (12)

)()(2=)( 2

2

2

2

Yw

Xw

YU

XVkwk

YwV

XwU

twj

′∂′∂

+′∂′∂

+′∂′∂

−′∂′∂

+′−′∂′∂′+

′∂′∂′+

′∂′∂ γρ (13)

Let us introduce the following non-dimensional variables

cxpap

axhxh

ca

acvv

cuu

ayyxx

λµωωλ

λ)(= ,)(=)( ,= ,= ,= ,= ,=

2 ′′′′′′′′′

0002 = = ,= ,= ,=,=

Hh

handHhh

aHcaajJtct y

yx

x

′′′′′ ζζψψλ

(14)

where ψ and ζ represent the dimensionless stream function and magnetic force function respectively. Using the dimensionless variables defined in equation (14) into the equations (13)(11)− , we obtain

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86

yyeym

yxye SRyN

NNx

Pyx

R ζωψψψψδ 22

111= +

∂∂

−+∇

−+

∂∂

−

∂∂

−∂∂

∂∂

−∂∂

+ yxye yxSR ζζζδ2 (15)

xyexm

xyxe SRxN

NNy

Pxy

R ζδωδψδψψψδ 222

22

3

11= −

∂∂

−−∇

−−

∂∂

−

∂∂

−∂∂

∂∂

−∂∂

− xxye yxSR ζζζδ 32 (16)

ωψωωψψδ 22

2 22=1∇

−+∇−−

∂∂

−∂∂−

mN

yxNNJR xye (17)

ERm

xyyxy =1)( 2ζζψζψδψ ∇+−− (18)

with

. ,= ,= ,= ,= 2

2

2

222

yxxh

yh

xv

yu yx ∂

∂+

∂∂

≡∇∂∂

−∂∂

∂∂

−∂∂ δζδζψψ (19)

The following dimensionless parameters that appear in (15) - (19) are defined as

µρcaRe = the Reynolds number,

λδ a= the Wave number, )(= 0

ρµe

cHS the Strommer’s

number is also known as magnetic force number, acR em σµ= the magnetic Reynolds number,

µ+kkN = , 1)(0 ≤≤ N the coupling parameter and

))(()(2=

22

kkkam

++

µγµ is the micropolar or

microrotation parameter. The total pressure in the fluid, which is equal to the sum of the ordinary and magnetic

pressure given by ))((21= 2

2

cHRpP em ρ

δδ+

+ , and )(=0cH

EEeµ′

− is defined as the electrical

field strength in non-dimensional form. Eliminating the total pressure from equations (15) and (16) , the resulting equation can be written as

ωψψψψδ 242

111= ∇

−+∇

−

∇

∂∂

−∂∂

NN

NyxR xye



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∇

∂∂

−∂∂

+∇+ ζζζδζ 2222

yxSRSR xyeye (20)

The instantaneous volumetric flow rate in the fixed frame is given by

ydtYXUQh

h′′′′′∫

′

′),,(= 1

2 (21)

where 1h′ and 2h′ are functions of X ′ and t′ . The rate of volume flow in the wave frame is found to be given by

ydyxuqh

h′′′′∫

′

′),(= 1

2 (22)

where 1h′ , and 2h′ are function of X ′ alone. We note that )(1 xh and )(2 xh represent the dimensionless form of the peristaltic channel walls given by the equations of the form

)(2cos=)( ),(2cos1=)( 21 φππ +−−+ xbdxhxaxh (23)

where, 1

1=daa ,

1

2=dab ,

1

2=ddd .

Using the transformation into the equations (21) and (22), the relation between Q and q can be obtained as

)(= 21 hhcqQ ′−′+ (24) The time mean flow over a period T at a fixed position X ′ is defined as

QdtT

QT

∫′0

1= (25)

Using (24) in (25) the flow rate Q′ has the form

21210=)(1= cdcdqhhcqdt

TQ

T++′−′+′ ∫ (26)

The non-dimensional form of equation (26) is given by

dF ++1=θ (27)

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where 1

=cdQ′θ and

1

=cdqF ,

such that

)()(== 211

2hhdy

yF

h

hψψψ

−∂∂

∫ (28)

The boundary conditions for the dimensionless stream function ),( yxψ and magnetic force function ),( yxζ in the wave frame can be put mathematically as,

12

2

= on 1= hyyy

−∂∂

+∂∂ ψβψ

22

2

= on 1= hyyy

−∂∂

−∂∂ ψβψ

21 = and = on 0= hyhyω

1=on 2

= hyFψ

2= on 2

= hyF−

21 = and = on 0== and hyhyy∂

∂ζζ (29)

where β is consider as the slip parameter. Applying long wave length approximation 1)( =δ and assuming the Reynolds number to be small, the dimensionless equations (20), (17) and (18) becomes

0=)(1 3

32

2

2

4

4

yNSR

yN

y e ∂∂

−+∂∂

+∂∂ ζωψ (30)

2

2

2

2

2 2=)(2yym

N∂∂

+∂∂− ψωω (31)

)(=2

2

yER

y m ∂∂

−∂∂ ψζ (32)

By operating 2

2

y∂∂ and

y∂∂ on both sides of (31) and (32) respectively and substituting the

expressions for 4

4

y∂∂ ψ and 3

3

y∂∂ ζ in the equation (30) we obtain



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0=)(2

)(12})(1{22

2

222

4

4

ωωωN

NHmy

HNmy −

−+

∂∂

−+−∂∂ (33)

Again, substituting the expression of 2

2

y∂∂ ω from (31) and 3

3

y∂∂ ζ from (32) after differentiating

with respect to y , in the expression (30) the stream function ψ may be written as

++−

∂∂−

− 212

2

2

22 }{)(2)(1

1= CyCmym

NNH

ωωψ (34)

where me RSRH 22 = i.e, µσµ aHH e )(= 0 is the Hartmann number and 1C and 2C are two

integrating constants which are to be determined. The general solution of equation (33) takes in the form,

),(sinh)(cosh)(sinh)(cosh= 2211 yDyCyByA θθθθω +++ (35) where

)(2)(18}){(1}){(1

21=

2222222

1 NHNmmHNmHN

−−

−+−++−θ

and

)(2)(18}){(1}){(1

21=

2222222

2 NHNmmHNmHN

−−

−+−−+−θ

together with four integrating constants ,A ,B C and .D Using the corresponding boundary conditions for ω in equations (34) and (35), the stream function ψ has the form

))(sinh)(cosh)({()(2[)(1

1= 1122

122 yByAmm

NNH

θθθψ +−−

−

]))}(sinh)(cosh)(( 212222

2 CyCyDyCm +++−+ θθθ (36) Applying the boundary conditions given in the equation (29) into the expressions ω and ψ the values of all the constants have been determined and are given in the appendix.

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Thus, the expressions for the axial velocity u and stream function ψ are obtained as

[ ]12221110

)}(cosh)(sinh{)}(cosh)(sinh{1== CyDyCZyByAZZy

u ++++∂∂ θθθθψ (37)

and

+++++ 2122

2

211

1

1

0

)}(sinh)(cosh{)}(sinh)(cosh{1=),( CyCyDyCZyByAZZ

yx θθθ

θθθ

ψ (38)

Now solving the equation (32) with the corresponding equation (38) , we get the magnetic force function ζ as

+− )}(cosh)(sinh{

2=),( 112

1

1

0

2

yByAZZRyERyx m

m θθθ

ζ

432

2

12222

2

2)}(cosh)(sinh{ CyCCyCyDyCZ

++

++++ θθ

θ (39)

where the constants 3C and 4C are obtained so far using the boundary conditions (29) are also included in the appendix. The expressions of axial-induced magnetic field and current density distribution across the channel are respectively given by

3== CREyRy

h mmx +−∂∂ ψζ (40)

and

.= uRERJ mmz − (41) The electric field strength E can be determined by integrating (32) and using the boundary conditions on ζ and ψ across the wall surface as

)()(=

21 xhxhFE−

(42)

Now using the equation (15) , we obtain the pumping characteristics by means of axial pressure gradient can be written as

).()(1)(1

1= 23

3

yEH

yNN

yNxp

∂∂

−+∂∂

−+

∂∂

−∂∂ ψωψ (43)



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Now using the equations (35) and (38) , the axial pressure gradient written in the form

))}(cosh)(sinh())(cosh)(sinh({)(1

1= 2222211

211

0

yDyCZyByAZNzx

p θθθθθθ +++−∂

∂

)}(cosh)(sinh)(cosh)(({)(1 22221111 yDyCyBysinhA

NN θθθθθθθθ +++−

+

)(2 uEH −+ (44) The pressure rise per wave length p∆ in the non-dimensional form is given by

dxxpp∂∂

∆ ∫1

0= (45)

Interestingly we note that the stress tensor in micropolar fluid is not symmetric. Therefore, the dimensionless form of the shear stress involved in the present problem under consideration are given by

ωτ)(1

=N

Nyu

xy −−

∂∂ (46)

ωτ)(1)(1

1=N

Nyu

Nyx −+

∂∂

− (47)

The numerical computations for the shear stresses xyτ and yxτ are obtained at both the upper and lower walls of the channel and whose graphical representation is presented in the next section. 4 COMPUTATIONAL RESULTS AND DISCUSSION The analytical expressions for the axial velocity, pressure rise, axial induced magnetic field, current density, wall shear stress, stream function and magnetic force function are derived in the previous section. The numerical results corresponding to the above mentioned analytical expressions have been computed subject to the following data available in the literatures (Ali et al. 2008; Ali et al. 2009; Shit et al. 2010): 0.5== ba , 1.0=d , 0.1=m , 0.01=mR ,

2.4=θ , 168,4,2,0.001,=H , 0.80.6,0.4,0.2,=N , πππφ ,4

3,2

0,= and

.1.51.0,0.5,0.0,=β Figures 42− represent the variation of axial velocity u for different values of the Hartmann number ,H Slip parameter β and the micropolar coupling parameter N . Figure 2 shows that the axial velocity decreases near the central line of the channel with increasing Hartmann number ,H whereas the accelerating effect is observed at the channel walls, but it retains constant flow rate for any values of .H From Figure 3 we observe that the axial velocity

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decreases in the central region and increases at the wall for increasing slip parameter β . Therefore the slip effects has a significant impact on the axial velocity.

Figure 2: Variation of axial velocity u for different values of H with 0.5=0.0,=0.4,= βφN

Figure 3: Variation of axial velocity u for different values of β with 0.0=0.4,=2.0,= φNH



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Figure 4: Variation of axial velocity u for different values of N with 0.5=0.0,=2.0,= βφH

Figure 5: Variation of pressure rise P∆ with θ for different values of φ with

0.5=2.0,=0.4,= βHN

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Figure 6 Variation of pressure rise P∆ with θ for different values of β with 0.0,=0.4,=2.0,= φNH

However, the trend is reversed in the case micropolar coupling parameter N as shown in Figure 4 . The presence of micro particles also significantly affects the flow characteristics. The variation of pressure rise p∆ versus dimensionless flow rate θ are shown in Figure 5 and Figure 6 . We observe that the pressure rise and volumetric flow rate has a linear relation. The pressure rise decreases linearly with the increase of the volumetric flow rate θ . The pumping phenomena can be decomposed into three regions, where the variation of pressure rise p∆ takes place. The region for which 0>p∆ is known as pumping, also classified as positive and negative pumping when 0>θ and 0<θ respectively. The region occupied for

0<p∆ is known as co-pumping and 0=p∆ corresponds to the free pumping. It is observed from Figure 5 that the pressure rise p∆ decreases with the increase of the phase difference φ at both the pumping and free pumping regions, whereas the transition takes place at the co-pumping region. Similar observation is found in Figure 6 for different values of the slip parameter β . However, it is interesting to note from this figure that, in the absence of slip parameter 0)(=β , the magnitude of the pressure rise is high enough.

Figure 7: Distribution of pressure gradient

xpP∂∂=δ for different values of φ with

0.5=0.4,=2.0,= βNH



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Figure 8: Distribution of pressure gradient

xpP∂∂=δ for different values of β with

0.0,=0.4,=2.0,= φNH

Figure 7 and Figure 8 show the variation of axial pressure gradient xp∂∂ along the length of

the channel in one wave length. Figure 7 indicates that the peak value of the magnitude of the axial pressure gradient decreases with increasing phase difference φ . One can note from this figure that when the phase difference is 0180 , no change in the pressure gradient is observed. The peak value of the pressure gradient shifted onwards along the phase difference φ . Figure 8 depicts the variation of axial pressure gradient with various slip parameters β . It is shown that the magnitude of the pressure gradient gradually decreases with the increase of the slip parameters β .

Figure 9 Effect of axial induced magnetic field xh for different values of H with

0.5=0.0,=0.4,= βφN

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Figure 9 and Figure 10 illustrate the effect of induced magnetic field for the variations in Hartmann number H and slip parameters β . Figure 9 shows that the magnitude of the axial induced magnetic field xh decreases with the increasing magnetic field meaning increase of Hartmann number .H It is observed from Figure 10 that the effect of induced magnetic field strongly depends on the slip parameter β . The increase of slip parameter is to decrease in axial induced magnetic field. It is also observed from these two figures that the induced magnetic field xh is positive at the lower wall where the magnetic field is applied and xh is negative at the other wall. This is due to the fact that the induced magnetic field recirculating in opposite direction about the central line and creates two lobes. The centre of these two lobes shifted towards the channel wall as the slip effect increases.

Figure 10: Effect of axial induced magnetic field xh for different values of β with

0.0,=0.4,=2.0,= φNH The distribution of current density zJ for the variation of Hartmann number H and the slip parameter β are illustrated in Figure 11 and Figure 12 . Figure 11 shows that the current density becomes flattening and decreases at the central region with increasing the magnetic field strength. Similarly, from Figure 12 , we observed that the current density also decreases at the central region of the channel, whereas it is enhances near the wall with the increasing values of the slip parameter β .



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Figure 11: Distribution of current density zJ for different values of H with

0.5=0.0,=0.4,= βφN

Figure 12: Distribution of current density zJ for different values of β with

0.0,=0.4,=2.0,= φNH Figures 1513− exhibit the distribution of wall shear stress xyτ with the variation of slip parameters β , phase difference φ between the walls of the channel and the micropolar coupling parameter .N We observe from Figure 13 that the magnitude of the wall shear stress decreases with an increase in the slip parameter β . The magnitude of the wall shear stress is always greater in the case of no-slip velocity condition at the wall. It is interesting to seen from Figure 14 that the magnitude of the shear stress on both the walls decreases with the increase of the phase difference φ . However the peak value of the wall shear stress shifted backward direction as the wave train moves on the forward direction. Therefore, the asymmetric channel has a significant impact on the wall shear stress. The non-Newtonian effect for the micropolar fluid due to the coupling parameter N has also significant role in

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controlling the distribution of wall shear stress as shown in Figure 15 . We observe that as the coupling parameter N increases, the magnitude of the wall shear stress decreases.

Figure 13 Variation of xyτ along with the co-ordinate x for different values of β with

2.0=H , 0.4=N , 0.0=φ

Figure 14: Variation of xyτ along with the co-ordinate x for different values of φ with

2.0=H , 0.4=N , 0.5=β



99

Figure 15: Variation of xyτ along with the co-ordinate x for different values of N with

0.5=2.0,= βH , 0.0=φ It is known that the phenomenon of trapping is the formation of an interesting circulating bolus of the fluid is a region of closed streamlines that move with the speed in the wave frame. Owing to the trapping phenomenon, there will exist stagnation points, where both the velocity components of the fluid vanish in the wave frame. To see the effect of various non-dimensional parameters on the streamlines are depicted in Figures 2316− . We observe from Figures 1816− that the formation of trapped bolus decreases in size and vanishes with the increase of the slip parameter β . In the presence of strong slip velocity the trapped bolus eliminated and the streamlines become parallel to the channel walls. Similarly from Figures 19,17,20 and 21 we have seen that trapped bolus decreases significantly as the magnetic field strength increases. It is interesting to note from Figure 17 that when 0=φ , the bolus appears and moves towards left with decreases in size as the phase difference φ increases as shown in Figure 22 . However, when πφ = , the trapped bolus completely disappear and streamlines aligned parallel to one another as presented in Figure 23.

Figure 16: Streamlines pattern ( 0.40945= 0.40945,= −minmax ψψ ) for 0.0=β with 0.0=φ ,

2.0=H , 0.4=N

G. C. Shit et al.


100


2.0=H , 0.4=N


2.0=H , 0.4=N

Figure 19: Streamlines pattern ( 0.29177= 0.29177,= −minmax ψψ ) for 0=H with 0.0=φ ,

0.4=N , 0.5=β



101


0.4=N , 0.5=β


0.4=N , 0.5=β

Figure 22: Streamlines pattern ( 0.23978= 0.23978,= −minmax ψψ ) for /2= πφ with 2.0=H ,

0.4=N , 0.5=β

G. C. Shit et al.


102

Figure 23: Streamlines pattern ( 0.22101= 0.22099,= −minmax ψψ ) for πφ = with 2.0=H ,

0.4=N , 0.5=β

Figure 24: Distribution of magnetic force function ),( yxζ for 0.0=β

with 0.4=N , 0.0=φ , 2.0=H


with 0.4=N , 0.0=φ , 2.0=H



103


with 0.4=N , 0.0=φ , 2.0=H The variation of magnetic force function for different values of the slip parameter β are plotted in Figures 2624− . It has been observed from these figures that the number of magnetic force lines decreases as the slip parameter increases. The study of Shit et al. (Shit et al. 2010) showed that in the presence of low magnetic field strength, the magnetic lines strongly appear in the central region of the channel and vanish rapidly as the Hartmann number H increases. Similar phenomenon is also observed in the case of slip velocity. The application of magnetic field and the presence of slip parameter contribute to the same characteristics. 5 CONCLUSIONS Combined effect of slip velocity and induced magnetic field on the peristaltic transport of a micropolar fluid in an asymmetric channel have been derived under the long wave length and low Reynolds number assumptions. The present analysis pays due attention to see effects of slip velocity as well as the induced magnetic field on the peristaltic transport of a physiological fluid. The main findings of the present study are as follows: • The axial velocity at the central region decreases with the increasing values of the Hartmann number and the velocity slip parameter. • The ratio of the viscosity parameter has an enhancing effect on the axial velocity at the central region of the channel. • The asymmetry of the channel due to the phase difference between the walls has a decreasing effect on the magnitude of the axial pressure gradient. • The role of slip velocity and the induced magnetic field have similar impact on the streamlines. Finally, we conclude that our theoretical investigation bears the potential to useful in the biomedical engineering and technology.

G. C. Shit et al.


104

6 ACKNOWLEDGEMENTS One of the authors (G. C. Shit) is thankful to Jadavpur University for the financial support of this investigation through Jadavpur University Research grant. APPENDIX The expressions that appear in section 3 are listed as follows:

)(cosh)](sinh)(cosh)(sinh[=

21

222221

hhDhChBA

θθθθ ++− ,

29

313028=Z

DZCZZB −− , 40

4139=Z

DZZC − , 41373840

39373640=ZZZZZZZZD

−− ,

][= 654301 DZCZBZAZZC ++++− ,

1112122

21111

1

102 )](sinh)(cosh[)](sinh)(cosh[

2= hChDhCZhBhAZFZC −+−+− θθ

θθθ

θ,

,]2

)()))](cosh)(cosh())(sinh)(sinh((

)))(cosh)(cosh())(sinh)(sinh(([2

[1)(

1=

22

21

12212221222

2

2211221121

1

0

22

21

23

hhChhDhhCZ

hhBhhAZZRmhhRmE

hhC

−+−+−+

−+−−

−−

θθθθθ

θθθθθ

,]2

))(cosh)(sinh())(cosh)(sinh([2

=

132

21

1

121222

211112

1

1

0

21

4

hCChC

hDhCZhBhAZZRmhRmEC

−++

++++− θθθ

θθθ

2211 = m−θξ , 22

22 = m−θξ ,

)(1= 20 NHZ − ,

211

1)(2=

mNZ θξ−

, 2

222

)(2=mNZ θξ−

,

)(cosh)(sinh= 11111113 hZhZZ θθβθ + , )(sinh)(cosh= 12111114 hZhZZ θθβθ + ,

)(cosh)(sinh= 12221225 hZhZZ θθβθ + , )(sinh)(cosh= 12221226 hZhZZ θθβθ + ,



105

)(cosh)(sinh= 21112117 hZhZZ θθβθ − , )(sinh)(cosh= 21112118 hZhZZ θθβθ − ,

)(cosh)(sinh= 22222229 hZhZZ θθβθ − , )(sinh)(cosh= 222222210 hZhZZ θθβθ − ,

)(cosh)(cosh= 211111 hhZ θθ − , )(sinh)(sinh= 211112 hhZ θθ − ,

)(cosh)(cosh= 221213 hhZ θθ − , )(sinh)(sinh= 221214 hhZ θθ − ,

0

11

1113

152)(=

Zhh

ZZZZ −

−θ ,

0

11

1214

162)(=

Zhh

ZZZZ −

−θ ,

0

12

1325

172)(=

Zhh

ZZZZ −

−θ ,

0

12

1426

182)(=

Zhh

ZZZZ −

−θ

2)(=

119 hh

FZ−

, 0

11

1117

202)(=

Zhh

ZZZZ −

−θ ,

0

11

1218

212)(=

Zhh

ZZZZ −

−θ ,

0

12

1329

222)(=

Zhh

ZZZZ −

−θ ,

0

12

14210

232)(=

Zhh

ZZZZ −

−θ , )(cosh)(cosh= 11191124 hZhZ θθ + , )(cosh)(sinh= 1116111525 hZhZZ θθ −

)(cosh)(cosh= 1117121526 hZhZZ θθ − , )(cosh)(sinh= 1118121527 hZhZZ θθ − ,

)(cosh)(cosh= 21192128 hZhZ θθ + , )(cosh)(sinh= 2116211529 hZhZZ θθ − ,

)(cosh)(cosh= 2117221530 hZhZZ θθ − , )(cosh)(sinh= 2118221531 hZhZZ θθ − ,

)()(= 152019201532 ZZZZZZ −−− , )(= 2115162033 ZZZZZ − , )(= 2215172034 ZZZZZ − ,

)(= 2315182035 ZZZZZ − , )(= 2825242936 ZZZZZ − , )(= 3025262937 ZZZZZ − ,

)(= 3125272938 ZZZZZ − , )(= 3225243339 ZZZZZ − , )(= 3425263340 ZZZZZ − ,

)(= 3525273341 ZZZZZ − .

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Beavers GS, and Joseph DD (1967). Boundary Condition at a Naturally Permeable Wall. Journal of Fluid Mechanics. 30, pp. 197-207. Burns YC, and Perkes T (1967). Peristaltic Motion. Journal of Fluid Mechanics. 29, pp. 731-743. Chow TS (1970). Peristaltic transport in a circular cylindrical pipe. Trans ASME Journal of Applied Mechanics. 37, pp. 901-905. Fung YC, and Yih CS (1968). Peristaltic transport. Trans ASME Journal of Applied Mechanics. 35, pp. 669-675. Hakeem A Ei, Naby A Ei, Misiery AEM Ei., and Shamy II Ei (2003). Hydromagnetic flow of fluid with variable viscosity in a uniform tube with peristalsis. Journal of physics A: Mathemetical and General. 36, pp. 8535-8547. Hung TK, and Brown TD (1976). Solid-Particle Motion in two dimensional peristaltic flows. Journal of Fluid Mechanics. 73, pp. 77-96. Kumari AVR, and Radhakrishnamacharya G (2012). Effect of slip magnetic field on peristaltic flow in an inclined channel with wall effects. International Journal of Biomathematics. 5, pp. 1250015-1-17. Latham TW (1966). Fluid motion in a peristaltic pumps. M. S. Thesis, MIT, Cambridge. M. A. Li A, Nesterov NI, Malikova SN, and Kiiatkin VA (1994). The use of an impulsive magnetic field in the combined therapy of patients with stone fragnents in the upper urinary tract. Vopr. Kurortol Fizioter Lech Fiz Kult. 3, pp. 22-24. Mekheimer KhS (2008a). Effect of the induced magnetic field on peristaltic flow of a couple stress fluid. Physics Letters A. 372, pp. 4271-4278. Mekheimer KhS (2008b). Peristaltic flow of a magneto-micropolar fluid: Effect of induced magnetic field. Journal of Applied Mathematics. 570825, pp. 23. Misra JC, Maiti S, and Shit GC (2008). Peristaltic transport of a physiological fluid in an asymmetric porous channel in the presence of an external magfnetic field. Journal of Mechanics in Medicine and Biology. 8, pp. 507-525. Misra JC, and Pandey SK (1994). Peristaltic transport of a particle-fluid suspension in a cylindrical tube. Computers & Mathematics with Applications. 28, pp. 131-145. Misra JC, and Pandey SK (2002). Peristaltic transport of blood in small vessels, Study of mathematical model. Computers & Mathematics with Applications. 43, pp. 1183-1193. Saleem N, Hayat T, and Alsaedil A (2012). Effects of induceed magnetic field and slip condition on peristaltic transport with heat and mass transfer in a non-uniform channel. International Journal of Physical sciences. 7, pp. 191-204.



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Shapiro AH, Jaffrin MY, and Weinberg SL (1969). Peristaltic pumping with long wave lengths at low Reynolds number. Journal of Fluid Mechanics. 37, pp. 799-825. Shapiro AH (1987). Pumping and retrograde diffusion in peristaltic walls. Proc. workshop Ureteral Reflux Children, Nat. Acad. Sci., Washington, D. C. Shit GC, Roy M, and Ng EYK (2010). Effect of induced magnetic field on peristaltic flow of a micropolar fluid in an asymmetric channel. International Journal for Numerical Methods in Biomedical Engineering. 26, pp. 1380-1403. Srivastava LM , and Srivastava VP (1995). Effects of poiseuille flow on peristaltic transport of a particulate suspension. Journal of Applied Mathematics and Physics (ZAMP). 46, pp. 655-679. Srivastava LM, and Srivastava VP (1989). Peristaltic transport of a particle-fluid suspension, Trans ASME Journal of Biomechanical Engineering. 111, pp. 157-165. Takabatake S, and Ayukava K (1982). Numerical study of two dimensional peristaltic flows. Journal of Fluid Mechanics. 122, pp. 439-465. Tretheway DC, and Meinhart CD (2002). Apparent Fluid Slip at Hydrophobic Microchannels Walls. Physics of Fluids. 14, pp. L9-L12. Wang Y, Ali N, Hayat T, and Oberlack M (2011). Peristaltic motion of a magnetohydrodynamic micropolar fluid in a tube. Applied Mathematical Modelling. 35, pp. 3737-3750.

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EFFECT OF INDUCED MAGNETIC FIELD ON PERISTALTIC …...Effect Of Induced Magnetic Field On Peristaltic Transport Of A Micropolar Fluid Int. J. of Appl. Math and Mech. 10 (6): 81-107,