s. nadeem peristaltic flow of a jeffrey fluid in arshad … · 2013-10-01 · s. nadeem, a. riaz,...

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Chemical Industry & Chemical Engineering Quarterly 19 (3) 399−409 (2013) CI&CEQ

399

S. NADEEM1

ARSHAD RIAZ2

R. ELLAHI2 1Department of Mathematics,

Quaid-i-Azam University, Islamabad, Pakistan

2Department of Mathematics and Statistics, FBAS, IIU Islamabad,

Pakistan

SCIENTIFIC PAPER

UDC 5/6:519:51-3

DOI 10.2298/CICEQ120402075N

PERISTALTIC FLOW OF A JEFFREY FLUID IN A RECTANGULAR DUCT HAVING COMPLIANT WALLS

In this article, the theoretical and mathematical study of peristaltic transport of a Jeffrey fluid in a rectangular duct with compliant walls is discussed. The constitutive equations are simplified under the implementation of low Reynolds number and long wavelength approximations. The analytical solution of the resulting equations is evaluated by Eigen function expansion method. The gra-phical aspects of all the parameters of interest are also analyzed. The graphs of velocity for two and three dimensional flow are plotted. The trapping bolus phenomenon is also discussed though streamlines.

Keywords: peristaltic flow, Jeffrey fluid, rectangular duct, compliant walls.

The study of peristaltic flows is quite useful in physiology and industry because of its large number of applications and in mathematics due to its com-plicated geometries and solutions of nonlinear equa-tions. In physiology, it is used by many systems in the living body to propel or to mix the contents of a tube. The peristaltic mechanism usually occurs in urine transport from the kidney to the bladder, swallowing food through the esophagus, chyme motion in the gastrointestinal tracts, vasomotion of small blood vessels, movement of Spermatozoa and the human reproductive tract. Theoretically and mathematically, the complete exact solutions of peristaltic flow prob-lems are quite difficult to determine even in viscous fluid theory. However, after using certain physical simplifications such as long wavelength and low Rey-nolds number approximations, the authors success-fully calculate only limited exact and analytical solu-tions. Some interesting studies are given in the refe-rences [1-11]. The study of peristaltic flows of New-tonian and non-Newtonian fluids in two-dimensional symmetric and asymmetric channels is also very useful in a number of applications, specially the study of inter-uterine fluid flow in a nonpregnant uterus [12-21]. Recently, Reddy et al. [22] have given the idea that the sagittal cross-section of the uterus may be

Correspondence: Arshad Riaz, Department of Mathematics and Statistics, FBAS, IIU Islamabad 44000, Pakistan. E-mail: [email protected] Paper received: 2 April, 2012 Paper revised: 9 July, 2012 Paper accepted: 17 July, 2012

better approximated by a tube of rectangular cross section than a two dimensional channel and pre-sented the influence of lateral walls on peristaltic flow in a rectangular duct. More recently, this idea has been extended by Nadeem and Akram [23] for non-Newtonian fluids. More studies on the peristaltic flow in three-dimensional rectangular channel are cited in the references [24-25]. A large number of analytical and numerical studies on the peristaltic flow of New-tonian and non-Newtonian fluids in different flow geo-metries are discussed by Tripathi [26-33]. However, the peristaltic flows of three dimensional non-New-tonian fluids in a rectangular duct having compliant walls have to the best of our knowledge not been explored. The aim of the present work is to discuss the peristaltic flow of a Jeffrey fluid in a rectangular duct with compliant walls. The governing equations of a Jeffrey fluid for three dimensional flows are sim-plified under the assumptions of long wavelength and low Reynolds number approximation. The exact solu-tions of the reduced equations having the compliant wall properties are found with the help of the Eigen function expansion method. The physical features of the pertinent parameters are measured with the help of graphs. The circulating bolus scheme is also described with the help of streamlines graphs.

MATHEMATICAL FORMULATION

Consider the peristaltic flow of an incompres-sible non-Newtonian Jeffrey fluid in a cross section of

S. NADEEM, A. RIAZ, R. ELLAHI: PERISTALTIC FLOW OF A JEFFREY FLUID… CI&CEQ 19 (3) 399−409 (2013)

400

rectangular channel having the width 2d and height 2a. In the present geometry, the Cartesian coordinate system is taken in such a way that the x-axis is taken along the axial direction, the y-axis is taken along the lateral direction and the z-axis is along the vertical direction of rectangular channel (Figure 1). The walls of the channel are assumed to be flexible and are taken as compliant, on which waves with small ampli-tude and long wave length are considered.

Figure 1. Schematic diagram for the peristaltic flow in a rectangular duct.

The geometry of the channel wall is given by:

( )2( , ) cosz h x t a b x ct

πλ

= = ± ± − (1)

where b is the amplitude of the wave, λ is the wavelength, c is the velocity of propagation, t is the time and x is the direction of wave propagation. The walls parallel to the xz-plane remain undisturbed and do not measure any peristaltic wave motion. We assume that the lateral velocity is zero as there is no change in lateral direction of the duct cross section. Let (u,0,w) be the velocity for a rectangular duct. The stress tensor for the Jeffrey model is defined by [31-34]:

21

. ..

1S

μ γ λ γλ

= + +

(2)

In the above equation, λ1 is the ratio of relaxation to retardation times, λ2 is the delay time, γ is shear stress and double dots denote the diffe-rentiation with respect to time. Under the assumption of long wave length and low Reynolds number, the governing equations in non-dimensional form for the considered flow problem are stated as [23]:

0u wx z

∂ ∂+ =∂ ∂

(3)

2 2 2

2 21 1

1

1 1

p u ux y z

βλ λ

∂ ∂ ∂= +∂ + +∂ ∂

(4)

Here β = a/d is the aspect ratio. The corres-ponding non-dimensional boundary conditions for compliant walls are stated as:

( , , , ) 1 at 1u x y z t y= − = ± (5)

( )( , , , ) 1 at ( , ) 1 ,u x y z t z h x t x tη= − = ± = ± ± (6)

where η(x,t) = ϕcos2π(x-t), ϕ = b/a (amplitude ratio) and 0≤ ϕ ≤1. The governing equation for the flexible wall may be specified as:

( ) 0L p pη = −

where L is an operator, which is used to represent the motion of stretched membrane with viscosity damping forces such that [22]:

2 4 2

2 4 2L m D B T K

tt x x∂ ∂ ∂ ∂= + + − +

∂∂ ∂ ∂ (7)

In the above equation, m is the mass per unit area, D is the coefficient of the viscous damping membrane, B is the flexural rigidity of the plate, T is the elastic tension in the membrane, K is spring stiff-ness and p0 is the pressure on the outside surface of the wall due to tension in the muscle, which is assumed to be zero here. The continuity of stress at z=±1±η and using the x-momentum equation yield:

3 2 5 3

1 2 3 4 52 5 3

pE E E E E

x t x xt x x xη η η η η∂ ∂ ∂ ∂ ∂ ∂= + + − +

∂ ∂ ∂ ∂∂ ∂ ∂ ∂(8)

3 2 5 3

1 2 3 4 52 5 3

2 2 2

2 21 1

1

1 1

E E E E Et x xt x x x

u uy z

η η η η η

βλ λ

∂ ∂ ∂ ∂ ∂+ + − + =∂ ∂ ∂∂ ∂ ∂ ∂

∂ ∂= ++ +∂ ∂

(9)

at 1z η= ± ± , in which E1 = ma3c/λ3µ, E2 = Da3/λ2µ, E3 = Ba3/cλ5µ, E4 = Ta3/cλ3µ and E5 = Ka3/cλµ are the non-dimensional elasticity parameters. Now we diffe-rentiate Eq. (4) with respect to z as follows:

2 3 3

2 31 1

10

1 1

u uz y z

βλ λ

∂ ∂+ =+ +∂ ∂ ∂

(10)

The expressions for stream function satisfying Eq. (3) are defined as (u = ∂ψ/∂z, w = -∂ψ/∂x).

Solution of the problem

The solution of Eq. (10) with boundary con-ditions (5), (6) and (9) is computed by the eigen-function expansion method and is directly defined as:


401

( )( ) ( )α π

α π β

= − +−

+ − − − 3

3 2

1

16 1cosh1 cos 2 1

cosh 2 1 2

n

n

n

u

Czn y

h n

(11)

where:

( )2 12n

nπα β= − (12)

( ) ( )( )( ) ( )

1 2

2 2

5 1 4 3

2 1 [2 cos 2

4 4 sin2 ]

C E x t

E E E E x t

π λ ϕ π π

π π π

= + − −

− + − + + − (13)

The detailed calculation is given in the appendix. It is noted that limiting λ1→0 results in reversing the present problem to the viscous fluid case. It is also observed from the above analysis that employing β→0 and β→1 reduces the discussed geometry to the two-dimensional channel and square duct, respectively.

RESULTS AND DISCUSSIONS

In this section, the effects of different physical parameters of a Jeffrey fluid model on the velocity profile of the fluid under discussion are examined graphically and the trapping phenomenon is also illus-trated by plotting streamlines for different pertinent parameters. Figures 2-7 are plotted to see the vari-ation of the velocity profile with the emerging para-meters β, λ1, E1, E2, E3 and E4. The streamlines are sketched in Figures 8-13, which show the flow behavior with various values of all the observing para-meters. In Figures 2, 4 and 5, the velocity profile is plotted with different values of the parameters β, E1 and E2. From these figures, we can observe that the magnitude of the velocity profile is a decreasing func-tion of the above three parameters. The effects of

(a) (b)

Figure 2. Velocity profile for different values of β for fixed ϕ = 0.2, x = 0.5, t = 0.4, λ1 = 0.5, E1 = 0.1, E2 = 0.2, E3 = 0.01, E4 = 0.2, E5 = 0.3.a) For 2-dimensional, b) for 3-dimensional.

(a) (b)

Figure 3. Velocity profile for different values of λ1 for fixed ϕ = 0.2, x = 0.5, t = 0.4, β = 2.5, E1 = 0.1, E2 = 0.1, E3 = 0.05, E4 = 0.2, E5 = 0.5.a) For 2-dimensional, b) for 3-dimensional.


402

(a) (b)

Figure 4. Velocity profile for different values of E1 for fixed ϕ = 0.2, x = 0.5, t = 0.4, β = 1.5, λ1 = 0.5, E2 = 0.1, E3 = 0.05, E4 = 0.2, E5 = 0.5. a) For 2-dimensional, b) for 3-dimensional.

(a) (b)


(a) (b)



403

(a) (b)

Figure 7. Velocity profile for different values of E4 for fixed ϕ = 0.2, x = 0.5, t = 0.4, β = 3, λ1 = 0.5, E1 = 0.1, E2 = 0.1, E3 = 0.2, E5 = 0.5. (a) For 2-dimensional, (b) For 3-dimensional.

(a) (b)

(c) (d)

Figure 8. Streamlines for different values of β. a) For β = 0.4, b) for β = 0.6, c) for β = 0.8 d) for β = 1. The other parameters are y = 0.5, λ1 = 1, ϕ = 0.2, t = 0.5, E1 = 1, E2 = 0.2, E3 = 0.05, E4 = 0.1, E5 = 0.3.


404

(a) (b)

(c) (d)

Figure 9. Streamlines for different values of λ1. a) For λ1 = 0.5, b) for λ1 = 1, c) for λ1 = 1.5, d) for λ1 = 2. The other parameters are y = 0.5, β = 1, ϕ = 0.2, t = 0.5, E1 = 1, E2 = 0.2, E3 = 0.01, E4 = 0.2, E5 = 0.3.

(a) (b)

(c) (d)

Figure 10. Streamlines for different values of ϕ. a) For ϕ = 0.1, b) for ϕ = 0.2, c) for ϕ = 0.3, d) for ϕ = 0.4. The other parameters are y = 0.5, β = 1, λ1 = 1, t = 0.5, E1 = 1, E2 = 0.2, E3 = 0.01, E4 = 0.2, E5 = 0.3.


405

(a) (b)

(c) (d)

Figure 11. Streamlines for different values of E1. a) For E1 = 1, b) for E1 = 2, c) for E1 = 3, d) for E1 = 4. The other parameters are y = 0.5, β = 1, λ1 = 1, t = 0.5, ϕ = 0.2, E2 = 0.2, E3 = 0.05, E4 = 0.2, E5 = 0.3.

(a) (b)

(c) (d)

Figure 12. Streamlines for different values of E2. a) For E2 = 0.5, b) for E2 = 1, c) for E2 = 1.5, d) for E2 = 2. The other parameters are y = 0.5, β = 1, λ1 = 1, t = 0.5, ϕ = 0.2, E1 = 0.2, E3 = 0.05, E4 = 0.2, E5 = 0.3.


406

(a) (b)

(c) (d)

Figure 13. Streamlines for different values of E3. a) for E3 = 0.01, b) for E3 = 0.05, c) for E3 = 0.09, d) for E3 = 0.13. The other parameters are y = 0.5, β = 1, λ1 = 1, t = 0.5, ϕ = 0.2, E1 = 0.2, E2 = 0.05, E4 = 0.2, E5 = 0.3.

different values of the physical parameters λ1, E3 and E4 are mentioned in Figures 3, 6 and 7. From these plots, it is seen that velocity profile rises directly with increasing the magnitude of λ1, E3 and E4. From Fi-gures 2-7, it can also be seen that the velocity attains its maximum value at the centre of the channel and remains symmetric throughout the channel. The streamlines for different values of the emerging para-meters are drawn in Figures 8-13 to lookout for the trapping bolus phenomenon. From Figure 8, it can be seen that number of the trapped bolus is reduced when increasing the value of parameter β. Figure 9 is plotted to show the streamlines with the λ1 being increased. From this plot, it is clear that the size of the trapping bolus rises with increasing magnitude of λ1. The streamlines for different values of the parameter ϕ are shown in Figure 10. It is clear from this graph that the number of boluses is decreasing monotoni-cally with increasing ϕ, but the size of the bolus is increasing with ϕ. Figure 11 reveals that the number of trapped boluses is decreasing with E1. In Figure 12, the number of trapped boluses remains unchanged, but increases in size with increasing values of E2 on the left side of the channel and has the opposite

behavior on the other side. The streamlines for E3 are shown in Figure 13. It is easy to see from this figure that the boluses decrease in number, but their size changes with the increase of E3.

CONCLUDING REMARKS

In the present study, the mathematical and graphical results of the peristaltic flow of a Jeffrey fluid in a compliant rectangular duct were discussed. The governing equations were simplified by employ-ing the long wavelength and low Reynolds number approximations. The resulting equations were then solved by using the method of Eigen function expan-sion. The following main results were observed:

• The profile of the velocity is decreasing function of the parameters β, E1 and E2.

• The influence of the pertinent parameters λ1, E3 and E4 is totally opposite to that of β, E1 and E2.

• The fluid flows more rapidly at the central part of the channel.

• The number of boluses is reduced with the increasing effects of the parameters β, ϕ, E1 and E3, while increased in case of λ1.


407

• The size of the bolus changes randomly with the variation of all the physical parameters.

• The results for the viscous fluid case can be obtained by taking λ1→0.

Nomenclature

u, w velocity components b amplitude of the wave a height of the channel d width of the channel x, y, z Cartesian coordinates λ wavelength μ viscosity p pressure c velocity of propagation t time λ1 relaxation time λ2 delay time γ shear stress γ derivative of shear stress β aspect ratio ϕ amplitude ratio ψ stream function m mass per unit area D coefficient of the viscous damping membrane B flexural rigidity of the plate T elastic tension in the membrane K spring stiffness p0 pressure on the outside surface

REFERENCES

[1] S. Nadeem, S. Akram, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 312-321

[2] S. Nadeem, N.S. Akbar, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 3844-3855

[3] M.A. Abd Elnaby, M.H. Haroun, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 752-762

[4] K.S. Mekheimer, Phys. Lett., A. 372 (2008) 4271-4278

[5] J.J. Lozano, M. Sen, Chem. Eng. Process. 49 (2010) 704-715


[7] D. Tripathi, Math. BioSci. 233 (2011) 90-97

[8] A.M. Siddiqui, W.H. Schwarz, J. Non-Newton Fluid Mech. 53 (1994) 257-284

[9] N.S. Akbar, S. Nadeem, Int. J. Heat Mass Tran. 55 (2012) 375-383

[10] N.S. Akbar, S. Nadeem, Int. Commun. Heat Mass Tran. 38 (2011) 154-159

[11] T. Hayat, S. Abelman, E. Momoniat, F. M. Mahomed, Math. Comput. Appl. 15 (2010) 638-657

[12] S. Nadeem, S. Akram, Math. Comput. Model. 52 (2010) 107-119

[13] S. Nadeem, S. Akram, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 1705-1716

[14] A. Ebaid, Phys. Lett., A 372 (2008) 4493-4499


[16] S. Srinivas, V. Pushparaj, Commun. Nonlinear Sci. Numer. Simul., 13 (2008) 1782-1795

[17] E.F. Elshehawey, N.T. Eladabe, E.M. Elghazy, A. Ebaid, App. Math. Comput. 182 (2006) 140-150

[18] S. Nadeem, S. Akram, Z. Naturforsch, A 64 (2009) 559– –567

[19] S. Tsangaris, N.W. Vlachakis, J. Fluid. Eng-T. ASME 125 (2003) 382-385

[20] S. Nadeem, S. Akram, Arch. Appl. Mech. 81 (2011) 97- –109

[21] S. Nadeem, S. Akram, Int. J. Numer. Methods Fluids 63 (2010) 374-394

[22] M.V. Subba Reddy, M. Mishra, S. Sreenadh, A. R. Rao, J. Fluid. Eng. 127 (2005) 824-827

[23] S. Nadeem, S. Akram, Nonlinear Anal. Real World Appl., 11 (2010) 4238-4247

[24] S. Nadeem, S. Akram, T. Hayat, A.A. Hendi, J. Fluids Eng. 134 (2012)

[25] R. Ellahi, A. Riaz, S. Nadeem, M. Ali, Math. Probl. Eng. 329639 (2012) 24 pages

[26] D. Tripathi, Int. J. Therm. Sci. 51 (2012) 91-101

[27] D. Tripathi, O. A. Beg, Proc. Inst. Mech. Eng. H. J. Eng. Med. 226 (2012) 631-644

[28] D. Tripathi, ASME J. Fluid. Eng. 133 (2011) 121104

[29] D. Tripathi, Comput. Math. Appl. 62 (2011) 1116-1126

[30] D. Tripathi, Int. J. Numer. Meth. Biomed. Eng. 27 (2011) 1812–1828

[31] D. Tripathi, N. Ali, T. Hayat, M.K. Chaube, A.A. Hendi, Appl. Math. Mech. -Engl. Ed. 32 (2011) 1231–1244

[32] S.K. Pandey, D. Tripathi, Int. J. Biomath. 3 (2010) 453- –472

[33] D. Tripathi, T. Hayat, N. Ali, S.K. Pandey, Int. J. Modern Phys., B 25 (2011) 3455-3471

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APPENDIX

Eq. (10) can be written as:

2 2 2

2 21 1

1

1 1

u uC

y zβ

λ λ∂ ∂= +

+ +∂ ∂ (14)


408

Now let us introduce a transformation:

( ) ( ) ( )1 1, , , , , ,u x y z t v x y z t w y= + (15)

After using the above equation in Eq. (14) we get system of two equations:

21

2

d0

d

wy

= (16)

with B.Cs:

( )1 1 1w ± = − (17)

and

2 2 21 12 2

1 1

1

1 1

v vC

y zβ

λ λ∂ ∂= +

+ +∂ ∂ (18)

with B.Cs:

( ) ( ) ( )1 1 1, 1, , 0, , , , 1v x z t v x y h t w y± = ± = − − (19)

Now we solve Eq. (18) with B.Cs (19) by Eigen function expansion method. The Eigen functions for the above problem are defined as:

( ) ( )cos 2 1 , 1,2,3...2n y n y nπϕ = − = (20)

Now we define a series solution of the form:

( ) ( )11

n nn

v y zϕ φ∞

=

= (21)

Now using the above equation in Eq. (18) and after using the orthogonality condition we obtained:

( ) ( )( )3 3 2

16 1cosh1

cosh 2 1

n

nn

n

Czz

h n

αφα π β

− = −

− (22)

Using Eqs. (20) and (22), Eq. (21) can be written as:

( ) ( )( )

( )1 3 3 21

16 1cosh, , , 1 cos 2 1

cosh 22 1

n

n

n n

Czv x y z t n y

h n

α πα π β

∞

=

− = − − −

(23)

Now from Eqs. (15), (16) and (23) we have the final solution:

( ) ( )( )

( )3 3 21

16 1cosh, , , 1 1 cos 2 1

cosh 22 1

n

n

n n

Czu x y x t n y

h n

α πα π β

∞

=

− = − + − − −

(24)

where αn and C are defined in Eqs. (12) and (13).


409

S. NADEEM1 ARSHAD RIAZ2

R. ELLAHI2

1Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

2Department of Mathematics and Statistics, FBAS, IIU Islamabad,

Pakistan

NAUČNI RAD

PERISTALTIČKO STRUJANJE JEFFREY-OVOG FLUIDA U PRAVOUGAONOM KANALU SA POPUSTLJIVIM ZIDOVIMA

Rad se bavi teorijskim i matematičkim izučavanjem peristaltičkog strujanja Jeffrey-evog

fluida u pravougaonom kanalu sa popustljivim zidovima. Konstitutivne jednačine su

pojednostavljena uvođenjem pretpostavkama o malom Rejnolds-ovom broju i velikoj

talasnoj dužini. Analitičko rešenje rezultujućih jednačina je dobijeno primenom metode

Eigen-ve funkcije širenja. Takođe, grafički su analizirani svi značajni parametri. Prika-

zani su grafici brzine za dvo- i trodimenzionalno strujanje.

Ključne reči: peristaltičko strujanje, Jeffrey-ev fluid, pravougaoni kanal, popu-stljivi zidovi.

s. nadeem peristaltic flow of a jeffrey fluid in arshad … · 2013-10-01 · s. nadeem, a. riaz,...

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