Slip effects on the peristaltic transport of MHD fluid with variable viscosity

Physics Letters A 372 (2008) 1477–1489

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Slip effects on the peristaltic transport of MHD fluid with variable viscosity

N. Ali a,∗, Q. Hussain a, T. Hayat a, S. Asghar b

a Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistanb Department of Mathematical Sciences, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan

Received 27 June 2007; accepted 23 September 2007

Available online 29 September 2007

Communicated by F. Porcelli

Abstract

This Letter concerns with the peristaltic analysis of MHD viscous fluid in a two-dimensional channel with variable viscosity under the effectof slip condition. A long wavelength and low Reynolds number assumption is used in the problem formulation. An exact solution is presented forthe case of hydrodynamic fluid while for magnetohydrodynamic fluid a series solution is obtained in the small power of viscosity parameter. Thesalient features of pumping and trapping phenomena are discussed in detail through the numerical integration. It is noted that an increase in theslip parameter decreases the peristaltic pumping region. Moreover, the size of trapped bolus decreases by increasing the slip parameter.© 2007 Elsevier B.V. All rights reserved.

PACS: 47.15.G-; 47.15.Rq; 47.27.nd; 47.45.Gx; 47.63.mf; 47.65.-d

Keywords: Peristaltic transport; Slip condition; Variable viscosity; MHD fluid

1. Introduction

The peristaltic flows of viscous and non-Newtonian fluids have been studied due to their application in urine transport from thekidney to bladder, swallowing food through the esophagus, chyme motion in the gastrointestinal tract, transport of spermatozoa,movement of ovum in the female fallopian tube and vasomotion of small blood vessels. The importance of such flows has alsobeen recognized in transport of slurries, corrosive fluids, sanitary fluid and noxious fluids in the nuclear industry. Further, rollerand finger pumps are widely operated under such mechanism. The literature on the topic is quite extensive. Some recent attemptsdealing with such flows are made in the references [1–22]. In all these investigations one or more assumption(s) of long wavelength,low Reynolds number, small amplitude ratio etc are used.

To the best of our knowledge, no investigation has been made yet to analyze the peristaltic flow with variable viscosity whenno-slip condition is inadequate. The no-slip condition is inadequate when one considers fluids exhibiting macroscopic wall slipand that in general is governed by relation between the slip velocity and traction. Due to such fact in mind, the main purposeof the present investigation is to examine the peristalsis of a magnetohydrodynamic (MHD) fluid with variable viscosity and slipcondition. To the authors’ best knowledge, this problem has not been even studied for the hydrodynamic fluid. In Section 2, thenon-dimensional problem is formulated in wave frame under the long wavelength and low Reynolds number approximation. Thecorresponding boundary conditions are prescribed in Section 3. In the current study, regular perturbation method has been usedto solve the flow problem in Section 4. The pumping and trapping phenomena have been discussed in Section 5. A comparisonof the present analysis with hydrodynamic and no-slip cases has also been made in the same section. The main conclusions aresummarized in Section 6.

* Corresponding author. Tel.: +92 51 90642172.E-mail address: [email protected] (N. Ali).

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1478 N. Ali et al. / Physics Letters A 372 (2008) 1477–1489

2. Problem formulation

Consider the two-dimensional channel of uniform thickness 2a, which is filled with an incompressible viscous fluid with variableviscosity. The walls of the channel are flexible and non-conducting. The sinusoidal wave trains propagate on the channel walls withconstant speed c and propel the fluid along the walls. In rectangular coordinate system (X, Y ), the geometry of the wall surface isdescribed by

(1)h(X, t) = a + b cos

[2π

λ(X − ct )

].

In above equation b is the wave amplitude, λ is the wave length, c is the velocity of propagation, t is the time and X is the directionof wave propagation. A uniform magnetic field B0 is applied in the transverse direction to the flow. The electric field is taken zero.The magnetic Reynolds number is taken small so that the induced magnetic field is neglected.

Choosing moving coordinates (x, y), (wave frame) which travel in X direction with the same speed of the wave, the unsteadyflow in the laboratory frame (X, Y ) can be treated as steady in the wave frame (x, y). The two coordinate frames are related by thetransformations

(2)x = X − ct, y = Y , u = U − c, υ = V and p(x) = P (X, t)

in which U , V and u, v are the respective velocity components in the corresponding coordinate systems.In wave frame, the equations which govern the flow are

(3)∂u

∂x+ ∂v

∂y= 0,

(4)ρ

(u

∂

∂x+ v

∂

∂y

)u = −∂p

∂x+ 2

∂

∂x

(μ(y)

∂u

∂x

)+ ∂

∂y

[μ(y)

(∂v

∂x+ ∂u

∂y

)]− σB2

0 (u + c),

(5)ρ

(u

∂

∂x+ v

∂

∂y

)v = −∂p

∂y+ 2

∂

∂y

(μ(y)

∂v

∂y

)+ ∂

∂x

[μ(y)

(∂v

∂x+ ∂u

∂y

)],

where ρ is the density, B0 is the strength of the magnetic field, p is the pressure and μ(y) is the viscosity function.To make the above equations non-dimensional, it is convenient to introduce the following non-dimensional variable and para-

meters:

x = x

λ, y = y

a, u = u

c, υ = υ

cδ, t = ct

λ, δ = a

λ, p = a2p

μ0cλ,

(6)h = h

a, φ = b

a, μ(y) = μ(y)

μ0, Re = ρca

μ0, S = aS

μ0c, M =

√σ

μ0aB0,

in which M is the Hartmann number, Re is the Reynolds number, σ is the electrical conductivity, δ is the wave number and μ0 isthe constant viscosity.

Invoking Eq. (6) into Eqs. (3)–(5), we have

(7)∂u

∂x+ ∂v

∂y= 0,

(8)ρδ

(u

∂

∂x+ v

∂

∂y

)u = −∂p

∂x+ 2δ2 ∂

∂x

(μ(y)

∂u

∂x

)+ ∂

∂y

[μ(y)

(δ2 ∂v

∂x+ ∂u

∂y

)]− M2(u + 1),

(9)ρδ3(

u∂

∂x+ v

∂

∂y

)v = −∂p

∂y+ 2δ2 ∂

∂y

(μ(y)

∂v

∂y

)+ δ2 ∂

∂x

[μ(y)

(δ2 ∂v

∂x+ ∂u

∂y

)].

Introducing the stream function as u = ∂ψ∂y

, v = − ∂ψ∂x

, in above equations and under the assumptions of long wavelength (δ � 1)

and low Reynolds number [1,2,4,6,9–11,13–22], Eqs. (7)–(9) will take the form

(10)0 = −∂p

∂x+ ∂

∂y

(μ(y)

∂2ψ

∂y2

)− M2

(∂ψ

∂y+ 1

),

(11)0 = −∂p

∂y.

Note that the above equation indicates that p �= p(y).

N. Ali et al. / Physics Letters A 372 (2008) 1477–1489 1479

3. Rate of volume flow and boundary conditions

The instantaneous volume flow rate in fixed frame is given by

(12)Q =h(X,t )∫

0

U(X, Y , t ) dY ,

where h is the function of X and t .The above expression in the wave frame becomes

(13)q =h(x)∫0

u(x, y) dy,

where h is the function of x alone. Employing Eqs. (2), (12) and (13), the two volume flow rates can be related through the followingrelation

(14)Q = q + ch(x).

The time mean flow over a period T at a fixed position X can be written as

(15)Q = 1

T

T∫0

Qdt.

Invoking Eq. (14) into Eq. (15) and integrating, we arrive at

(16)Q = q + ac.

Defining the dimensionless time mean flow rate θ in the fixed frame and F in the wave frame one has

(17)θ = Q

ca, F = q

ca

and thus Eq. (16) reduces to

(18)θ = F + 1,

where

(19)F =h∫

0

∂ψ

∂ydy = ψ(h) − ψ(0).

If we select the zero value of the stream line at the centre line (y = 0):

ψ(0) = 0,

then the wall at (y = h) is a stream line of value

ψ(h) = F.

The boundary conditions in the dimensionless stream function will now take the following form

(20a)ψ = 0,∂2ψ

∂y2= 0, at y = 0,

(20b)ψ = F,∂ψ

∂y+ βμ(y)

∂2ψ

∂y2= −1, at y = h = 1 + φ cos(2πx),

where β is the slip parameter, μ(y) is the dimensionless viscosity function, φ(= ba

< 1) is the amplitude ratio and h is the dimen-sionless form of the peristaltic wall.

In the forthcoming analysis, we will use

(21)μ(y) = e−αy or μ(y) = 1 − αy for α � 1,

where α is the viscosity parameter. The choice of μ here is justified physiologically because normal person or animal of similar sizetakes 1 − 2L of the fluid every day. Also 6 − 7L of the fluid is received by the small intestine as secretions from salivary glands,stomach, pancreas, liver and small intestine itself. This indicates the dependence of fluid concentration upon y and hence the choiceof μ in the present analysis is appropriate [17–21].


4. Solution of the problem

4.1. Case 1 (M = 0)

For M = 0, Eqs. (10) and (11) yield

(22)0 = −∂p

∂x+ ∂

∂y

(μ(y)

∂2ψ

∂y2

),

(23)0 = −∂p

∂y.

The compatibility equation is

(24)∂2

∂y2

(μ(y)

∂2Ψ

∂y2

)= 0.

Inserting Eq. (21) in above equation and then solving the resulting equation subject to the boundary conditions (20a) and (20b)we have

(25)ψ = −y(2h3 + αh4 − hy2(2 + αy) + F(6h2 + 4αh3 − y2(2 + αy) + 12hβ))

h2(4h + 3αh2 + 12β).

The expression of the axial pressure gradient is

(26)dp

dx= −3(F + h)(4h − 3αh2 + 12β)

4h2(h + 3β)2.

4.2. Case 2 (M �= 0)

Compatibility equation here is of the following form

(27)∂2

∂y2

(μ(y)

∂2ψ

∂y2

)− M2 ∂2ψ

∂y2= 0.

For small parameter α we can write

(28)ψ = ψ0 + αψ1 + O(α2),

(29)dp

dx= dp0

dx+ α

dp1

dx+ O

(α2),

(30)F = F0 + αF1 + O(α2).

Using Eq. (28) into Eqs. (27), (10), (20a) and (20b) and then comparing the coefficients of like powers of α, we have the followingzeroth order system

(31)∂2

∂y2

(∂2ψ0

∂y2

)− M2 ∂2ψ0

∂y2= 0,

(32)dp0

dx= ∂

∂y

(∂2ψ0

∂y2

)− M2

(∂ψ0

∂y+ 1

),

(32a)ψ0 = 0,∂2ψ0

∂y2= 0, at y = 0,

(32b)ψ0 = F0,∂ψ0

∂y+ β

∂2ψ0

∂y2= −1, at y = h.

The first order system is

(33)∂2

∂y2

(∂2ψ1

∂y2

)− M2 ∂2ψ1

∂y2= ∂2

∂y2

(y

∂2ψ0

∂y2

),

(34)dp1

dx= ∂

∂y

(∂2ψ1

∂y2− y

∂2ψ0

∂y2

)− M2 ∂ψ1

∂y,

(34a)ψ1 = 0,∂2ψ1

2= 0 at y = 0,

∂y


(34b)ψ1 = F1,∂ψ1

∂y+ β

∂2ψ1

∂y2= βy

∂2ψ0

∂y2at y = h.

4.2.1. Solution of the zeroth order systemAt this order, the expressions of stream function and pressure gradient are

(35)ψ0 = y{F0M cosh[hM] + (1 + F0M2β) sinh[hM]} − (F0 + h) sinh[My]

hM cosh[hM] + (−1 + hM2β) sinh[hM] ,

(36)dp0

dx= − (F0 + h)M3(cosh[hM] + Mβ sinh[hM])

hM cosh[hM] + (−1 + hM2β) sinh[hM] .

4.2.2. Solution of the first order systemThe corresponding expressions of stream function and longitudinal pressure gradient at this order are

ψ1 = 1

8(hM cosh[hM] + K5 sinh[hM]){(

K2 + K3 cosh[2hM] − 2K4M sinh[2hM])y + 2hM cosh[hM]

× (−K1My2 cosh[My] + (−4F1 + K1(y + h2M2β

))sinh[My]) + 2 sinh[hM] × (−K1K5My2 cosh[My]

(37)+ (−4K5F1 + K1(−y + hM2 × (

h2 − hβ + yβ)))

sinh[My])},(38)

dp1

dx= M2(K6 − K3 cosh[2hM] + 2M(2F1 + hK1 − 4F1hM2β) sinh[2hM])

8(hM cosh[hM] + K5 sinh[hM]) ,

where

K1 = F0 + h, K2 = h(−1 + 4F1M

2) + F0(−1 + 2h2M2) + 2M2(h3 + 2F1β − 2F1hM2β2),

K3 = K1 + 4F1hM2 − 4F1M2β + 4F1hM4β2, K4 = hK1 + F1

(2 − 4hM2β

),

K5 = −1 + hM2β, K6 = K1 − 4F1hM2 − 2F0h2M2 − 2h3M2 − 4F1M

2β + 4F1hM4β2.

Summarizing the perturbation results, the expressions of stream function and longitudinal pressure up to order α are

ψ = y{FM cosh[hM] + (1 + FM2β) sinh[hM]} − (F + h) sinh[My]hM cosh[hM] + (−1 + hM2β) sinh[hM]

+ α

8(hM cosh(hM) + (−1 + hM2β) sinh(hM))2

{(F + h)y cosh[2hM] + y

(−1 + 2h2M2

− 2My cosh[My](hM cosh[hM] + (−1 + hM2β)

sinh[hM]) − 2hM sinh[2hM] + 2(hM

(y + h2M2β

)cosh[hM]

(39)+ (−y + hM2(h2 − hβ + yβ

)) × sinh[hM]) sinh[My])},dp

dx= − (F + h)M3(cosh[hM] + Mβ sinh[hM])

hM cosh[hM] + (−1 + hM2β) sinh[hM]

(40)− α

{M2(F + h)(−1 + 2h2M2 + cosh[2hM] − 2hM sinh[2hM])

8(hM cosh(hM) + (−1 + hM2β) sinh(hM))2

}+ O

(α2).

The non-dimensional pressure rise per wavelength �Pλ and friction force Fλ (at the wall) are

(41)�Pλ =h∫

0

(dp

dx

)dx,

(42)Fλ =h∫

0

h

(−dp

dx

)dx,

where dp/dx is defined by Eq. (40) and F = θ − 1.It is worth mentioning here that the expressions (39) and (40) for M = 0 are reduced to the expressions (25) and (26).


Fig. 1. The pressure rise versus flow rate for φ = 0.6, M = 2, α = 0.0 and α = 0.2.

Table 1Critical values of flow rate θ below which the peristaltic pumping occurs and above which the augmented pumping occurs as the functions of β,M and φ

Parameter β Critical value of θ Parameter M Critical value of θ Parameter φ Critical value of θ

0.0α = 0.0,0.3683α = 0.2,0.3697

0.0α = 0.0,0.4522α = 0.2,0.4612

0.0α = 0.0,0.0001α = 0.2,0.0001

0.03α = 0.0,0.3533α = 0.2,0.3550

1.0α = 0.0,0.4188α = 0.2,0.4245

0.5α = 0.0,0.2396α = 0.2,0.2405

0.06α = 0.0,0.3411α = 0.2,0.3429

1.5α = 0.0,0.3881α = 0.2,0.3916

0.6α = 0.0,0.3580α = 0.2,0.3594

5. Numerical results and discussion

In order to discuss the pumping characteristics the integral given by the equalities (41) and (42) are evaluated numerically usingMathematica and the results are displayed graphically. The values of α are taken to be 0 and 0.2 in all the figures of pressure riseand frictional force.

Fig. 1 presents the variation of �Pλ with the flow rate θ for various values of β . We note that there exists a critical value ofθ below which the pressure difference �Pλ is positive and above which it is negative. This critical value of θ corresponds to thefree pumping flux. At this critical value the pressure difference is zero. We further note that the free pumping flux (Table 1) andthe maximum pressure against which the peristalsis works as a pump i.e. �Pλ corresponding to θ = 0 decrease with an increasein the slip parameter β . This means that the peristaltic pumping region (θ > 0, �Pλ > 0) decreases with an increase in the slipparameter β . Moreover, the free pumping flux increases by increasing α (Table 1). However, the maximum pressure against whichthe peristalsis works as a pump decreases by increasing α. This means to say that the peristalsis works against the greater pressureto propel the fluid with constant viscosity as compared to the fluid with variable viscosity. This observation remains true for theforthcoming Figs. 2 and 3 for �Pλ and therefore we will not mention this further. The variation of �Pλ with θ for different valuesof Hartman number M is presented in Fig. 2. We observe that similar to previous figure there exists a critical value of θ below whichthe peristaltic pumping occurs and above which the augmented pumping occurs. Table 1 shows that critical value of θ decreases byincreasing M . Further, the maximum pressure against which the peristalsis works as a pump increases by increasing M . This impliesthat the peristaltic has to work against greater pressure to propel the magnetohydrodynamic fluid as compared to the hydrodynamicfluid. This figure also shows that peristaltic pumping region increases by increasing M . The effect of increasing amplitude ratio φ

on �Pλ can be seen through Fig. 3. The free pumping flux in this case increases rapidly (Table 1). Also similar to previous figurethe maximum pressure against which the peristalsis works as a pump increases by increasing φ.

To discuss the behavior of friction force Fλ with θ for various values of β,M and φ, we have plotted Figs. 4–6. Fig. 4 illustratesthe variation of Fλ with θ for different values of β . It reveals that below a critical value of θ , Fλ resists the flow and above whichit assists the flow. This critical value of θ decreases by increasing β (Table 2). Fig. 5 depicts that an increase in M decreases thecritical value of θ . So for magnetohydrodynamic fluid friction force assists the flow above a critical value of θ that is smaller thanthe case of hydrodynamic fluid (Table 2). Fig. 6 is prepared to see the effect of φ on Fλ with θ . Contrary to the previous case thecritical value of θ increases by increasing φ (Table 2). Moreover, regarding the effects of α on Fλ, it is observed that the criticalvalue of θ below which the friction force resists the flow and above which it assists the flow increases by increasing α (Table 2).


Fig. 2. The pressure rise versus flow rate for φ = 0.6, β = 0.02, α = 0.0 and α = 0.2.

Fig. 3. The pressure rise versus flow rate for M = 2, β = 0.02, α = 0.0 and α = 0.2.

Fig. 4. The friction force at the wall versus flow rate for φ = 0.6, M = 2, α = 0.0 and α = 0.2.


Fig. 5. The friction force at the wall versus flow rate for φ = 0.6, β = 0.02, α = 0.0 and α = 0.2.

Fig. 6. The friction force at the wall versus flow rate for M = 2, β = 0.02, α = 0.0 and α = 0.2.

Table 2Critical values of flow rate θ below which the Fλ has a direction opposite to the wave velocity and above which the directions are same as the functions of β,M

and φ

Parameter β Critical value of θ Parameter M Critical value of θ Parameter φ Critical value of θ

0.0α = 0.0,0.2017α = 0.2,0.2035

0α = 0.0,0.3501α = 0.2,0.3675

0.0α = 0.0,0.0000α = 0.2,0.0000

0.03α = 0.0,0.1809α = 0.2,0.1831

1.0α = 0.0,0.2862α = 0.2,0.2959

0.5α = 0.0,0.1167α = 0.2,0.1177

0.06α = 0.0,0.1645α = 0.2,0.1667

1.5α = 0.0,0.2340α = 0.2,0.2390

0.6α = 0.0,0.1873α = 0.2,0.1893

This observation is true for all the figures of Fλ (i.e. from Figs. 4–6). Thus for a fluid with variable viscosity friction force assiststhe flow above a critical value of θ that is greater than the case of fluid with constant viscosity.

The formation of an internally circulating bolus of fluid by closed streamlines is called trapping and this trapped bolus ispushed ahead along with the peristaltic wave. The effects of α,β,M and φ on trapping can be seen through Figs. 7–10. Fig. 7shows the effect of β on trapping both for fluids with constant and variable viscosities. We observe that an increase in theslip parameter β decreases the size of trapped bolus. Similarly the size of trapped bolus for the fluid with variable viscosity issmaller when compared with that of fluid with constant viscosity. This observation is true for the rest of the figures for trapping.Fig. 8 is made to see the effects of M on trapping both for the fluid with constant and variable viscosities. Here like the pre-vious case the size of trapped bolus decreases by increasing M . To see the effects of φ on trapping we have prepared Fig. 9.


Fig. 7. Streamlines for three different slip parameter values β = 0.0 (panels (a), (b)), β = 0.02 (panels (c), (d)), β = 0.04 and (panels (e), (f )) with two values ofα = 0.0 (left panels) and α = 0.1 (right panels). The other parameters are chosen as φ = 0.4, M = 1, and θ = 0.65, respectively.

We note that increase in the amplitude ratio increases the size of trapped bolus. The effect of flow rate θ on trapping can beseen through Fig. 10. We observe that by increasing the value of flow rate θ the trapped bolus moves towards the boundarywall.


Fig. 8. Streamlines for three different Hartman number values M = 0 (panels (a), (b)), M = 1 (panels (c), (d)), M = 1.5 and (panels (e), (f )) with two values ofα = 0.0 (left panels) and α = 0.1 (right panels). The other parameters are chosen as φ = 0.4, β = 0.0, and θ = 0.65 respectively.

6. Concluding remarks

We have theoretically analyzed the problem of peristaltic transport of a MHD viscous fluid with variable viscosity under theinfluence of slip condition. The expressions of stream function and longitudinal pressure gradient are constructed analytically.


Fig. 9. Streamlines for three different amplitude ratio values φ = 0.3 (panels (a), (b)), φ = 0.4 (panels (c), (d)), φ = 0.5 and (panels (e), (f )) with two values ofα = 0.0 (left panels) and α = 0.1 (right panels). The other parameters are chosen as M = 1, β = 0.0, and θ = 0.65 respectively.

Numerical integration is used to analyze the novel features of pumping and trapping. The main findings of the present study aregiven in the following points:


Fig. 10. Streamlines for three different flow rate values θ = 0.65 (panels (a), (b)), θ = 1.2 (panels (c), (d)), θ = 1.5 and (panels (e), (f )) with two values of α = 0.0(left panels) and α = 0.1 (right panels). The other parameters are chosen as φ = 0.4, M = 1, and β = 0.0, respectively.

• The free pumping flux for a fluid with variable viscosity is greater as compared to fluid with constant viscosity. Moreover,peristalsis has to work against a greater pressure for a fluid with constant viscosity when compared with that fluid of variableviscosity.

• The peristaltic pumping region narrows down by increasing the slip parameter β . However, it widens by increasing M and φ.


• The critical value of θ below which the friction force resists the flow and above which it assists the flow increases by increasingα and φ. However, it decreases by increasing β and M .

• The size of trapped bolus decreases by increasing α, β and M . However, it increases by increasing φ.• The bolus moves towards the boundary wall when large values of flow rate are taken into account.• The results for hydrodynamic fluid with [22] and without slip can be obtained as the special cases of our analysis by choosing

M = 0 and M = β = 0, respectively.

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Slip effects on the peristaltic transport of MHD fluid with variable viscosity

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