A mathematical model for the peristaltic flow of chyme ...

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Author's Personal Copy Mathematical Biosciences 233 (2011) 90–97

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A mathematical model for the peristaltic flow of chyme movement in small intestine Dharmendra Tripathi Department of Mathematics, Birla Institute of Technology Pilani, Hyderabad Campus, Hyderabad 500078, India

a r t i c l e

i n f o

Article history: Received 6 January 2011 Received in revised form 23 April 2011 Accepted 27 June 2011 Available online 23 July 2011 Keywords: Peristaltic flow Fractional Oldroyd-B model Inclined tube Small intestine Homotopy analysis method

a b s t r a c t A mathematical model based on viscoelastic fluid (fractional Oldroyd-B model) flow is considered for the peristaltic flow of chyme in small intestine, which is assumed to be in the form of an inclined cylindrical tube. The peristaltic flow of chyme is modeled more realistically by assuming that the peristaltic rush wave is a sinusoidal wave, which propagates along the tube. The governing equations are simplified by making the assumptions of long wavelength and low Reynolds number. Analytical approximate solutions of problem are obtained by using homotopy analysis method and convergence of the obtained series solution is properly checked. For the realistic values of the emerging parameters such as fractional parameters, relaxation time, retardation time, Reynolds number, Froude number and inclination of tube, the numerical results for the pressure difference and the frictional force across one wavelength are computed and discussed the roles played by these parameters during the peristaltic flow. On the basis of this study, it is found that the first fractional parameter, relaxation time and Froude number resist the movement of chyme, while, the second fractional parameter, retardation time, Reynolds number and inclination of tube favour the movement of chyme through the small intestine during pumping. It is further revealed that size of trapped bolus reduces with increasing the amplitude ratio whereas it is unaltered with other parameters. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction The peristaltic transport of non-Newtonian fluids has received considerable attention in recent years in engineering as well as in physiological sciences. The physiological phenomena like movement of food bolus through oesophagus, movement of chyme in the small intestine, urine transport from kidney to bladder through the ureter, the movement of spermatozoa in the ducts afferents of the male reproductive tract and the ovum in the female fallopian tube are some examples of peristaltic transport. An intestinal infection like Gastroenteritis has become a rather common today’s world predominantly due to unhygienic lifestyle and water contamination. This intestinal infection causes distension and other complication in the intestine. Due to such infectious conditions, a strong wave called peristaltic rush develops which travels relatively long distances in few minutes in the small intestine. The small intestine is a convoluted tube of about 6–7 m in length and has average radius of about 1.25 cm lying in the central and lower parts of abdomen. The study of peristaltic motion in both mechanical and physiological situations has been studied in Refs. [1–4]. Shapiro et al. [4] have investigated the peristaltic pumping under assumptions of long wavelength and low Reynolds number. They have considered two-dimensional and axisymmetric flows of Newtonian fluids,

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and they discussed the mechanical efficiency and some important phenomena of peristaltic pump such as reflux and trapping. Their investigation is focused only about Newtonian fluids and does not cover the peristaltic flow of other fluids such as non-Newtonian fluids. Applications involving viscoelastic fluid jets are quite broad, and include such areas as microdispensing of bioactive fluids through high throughput injection devices, creation of cell attachment sites, scaffolds for tissue engineering, coatings and drug delivery systems for controlled drug release, and viscoelastic blood flow past valves. Some workers [5–7] have reported the peristaltic transport of viscoelastic fluid with Maxwell model and they discussed the effect of relaxation time on the peristaltic transport. Jeffery model [8–10] is used to study the peristaltic flow of viscoelastic fluids through the different geometries of wall surface. Fractional calculus is a collection of relatively little-known mathematical results concerning generalizations of differentiation and integration to noninteger orders. While these results have been accumulated over centuries in various branches of mathematics, they have recently found little appreciation or application in physics and other mathematically oriented sciences. As described by Wheatcraft and Meerschaert [11], a fractional conservation of mass equation is needed when the control volume is not large enough compared to the scale of heterogeneity and when the flux within the control volume is non-linear. Fractional Oldroyd-B model is one of the models of viscoelastic fluids if fractional element is in-

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D. Tripathi / Mathematical Biosciences 233 (2011) 90–97

cluded in the classical Oldroyd-B model. This model plays an important role as a valuable tool to study viscoelastic properties. Some authors [12–21] have investigated unsteady flows of viscoelastic fluids with fractional Maxwell model, fractional generalized Maxwell model, fractional second grade fluid, fractional Oldroyd-B model, fractional Burgers’ model and fractional generalized Burgers’ model through different geometries of wall surface. Solutions for velocity field and the associated shear stress are obtained by using Laplace transform, Fourier transform, Weber transform, Hankel transform and discrete Laplace transform. Recently, Tripathi et al. [22] have incorporated the application of fractional element models of viscoelastic materials in the study of bio-fluids flow. They obtained the solutions by using the Adomian decomposition method (ADM) and homotopy perturbation method (HPM) and discussed the effects of fractional parameters and relaxation time on the peristaltic flow. This result is again extended for generalized fractional Maxwell model, fractional second grade model, fractional Oldroyd-B model and fractional Burgers’ model in other investigations [23–28] in different geometries of peristaltic flow pattern. ADM, HPM and variational iteration method have been used to obtain the approximate analytical solution and numerical solution. None of these investigations have reported the peristaltic flow pattern through the inclined tube which is relevant in physiological vessels since most of the vessels are neither vertical nor horizontal. Motivated from the reported literature, a mathematical model is prepared to study the peristaltic flow of viscoelastic fluids with fractional Oldroyd-B model through the inclined cylindrical tube under the assumptions of long wavelength and low Reynolds number. Homotopy analysis method (HAM) is applied to find approximate analytical solution of the problem and numerical results of the problem for different cases are depicted graphically. The influences of fractional parameters, relaxation time, retardation time, Reynolds number, Froude number and inclination of tube on the peristaltic flow pattern are discussed numerically. In small intestine, it is shown that there are many inclinations in gastrointestinal tract from stomach to large intestine. On basis of this, the inclined cylindrical tube is considered. The physical properties of chyme are found to be viscoelastic nature. Therefore, this model is suitable to study the movement of chyme through the small intestine. 2. Mathematical formulation When the wall of the tube is brought under the influence of a periodic radial contraction wave, a part of the wall begins to contract initially at the inlet, which then relaxes and the portion lying ahead of this begins to contract showing that the contraction wave progresses towards the outlet. Relaxation culminates at the natural boundary without expanding further beyond it. This process continues until complete transportation takes place. Such a motion (cf. Fig. 1) may be mathematically modeled as

~ x~; ~tÞ ¼ a 0:5/ ~ 1 þ cos 2p ð~x c~tÞ ; hð k

ð1Þ

~ are respectively axial coordinate, time, ~ k; c and h where ~ x; ~t; a; /; radius of the tube, amplitude of wave, wavelength, wave-speed and radial displacement of the walls from the center line. The constitutive equation for viscoelastic fluid with fractional Oldroyd-B model is given by

@a @b 1 þ ~ka1 a e S ¼ l 1 þ ~kb2 b c_ ; @~t @~t

ð2Þ

where e S; c_ and ~ k1 ; ~ k2 are shear stress, rate of shear strain and material constants (relaxation time and retardation time), l is viscosity, and a,b are the fractional parameters such that 0 < a 6 b 6 1. This

Fig. 1. Geometry of wall surface with inclination of angle A.

model reduces to fractional Maxwell, fractional second grade models respectively when, ~ k2 ¼ 0; ~ k1 ¼ 0 and with a = b = 1, these models reduce to Maxwell, second grade models. Classical Navier Stokes k2 ¼ 0. model is obtained by substituting ~ k1 ¼ ~ The governing equations of the motion of viscoelastic fluids with fractional Oldroyd-B model for inclined tubular flow are given by

n 9 o a ~ > ¼ 1 þ ~ka1 @@~ta @@p~x qg sin A > > > > n o > > b 2 b @ ~ ~ 1 @ ~ @u @ u > ~ > þl 1 þ k2 @~tb ~r @~r r @~r þ @~x2 ; > > = n o a a ~ @ p ~ q 1 þ ~ka1 @@~ta DDv~t ¼ 1 þ ~ka1 @@~ta @~r þ qg cos A > > n > @2 v~ o > > b @b @ 1 @ ~~ ~ > þl 1 þ k2 @~tb @~r ~r @~r ðr v Þ þ @ ~x2 ; > > > > > ; ~ @u 1 @ð~r v~ Þ þ ¼ 0; @ ~x r @~r a

q 1 þ ~ka1 @@~ta

~ Du D~t

ð3Þ

~ @@~x þ v~ @@~r, and q; u ~ ; g; A; v ~ ; ~r ; p ~ are the fluid denwhere DD~t @@~t þ u sity, axial velocity, acceleration due to gravity, inclination angle of tube, radial velocity, radial coordinate and pressure. For carrying out further analysis, we introduce the following non-dimensional parameters:

9 ~ ~ ~ v~ ; = x ¼ ~xk ; r ¼ a~r ; k1 ¼ ckk1 ; k2 ¼ ckk2 ; t ¼ ckt ; u ¼ u~c ; v ¼ cd ~ ~ e ~ 2 c2 ; h ¼ ha ; d ¼ ak ; / ¼ /a ; p ¼ lpack ; S ¼ alSc ; Re ¼ qlca ; Fr ¼ ga

ð4Þ

where Re, Fr and d stand for the Reynolds number, Froude number and wave number respectively. We introduce the non-dimensional parameters, Eq. (1) reduces to

hðx; tÞ ¼ 1 / cos2 pðx tÞ;

ð5Þ

and under the assumptions of long wavelength and low Reynolds number, Eq. (3) reduce to

a

1 þ ka1 @t@ a

@p @r @u @x

@p @x

n 2 b Re sin A ¼ 1 þ kb2 @t@ b @@r2u þ 1r Fr

¼ 0; þ

1 @ðr v Þ r @r

¼ 0:

@u @r

o 9 ;> > = > > ;

ð6Þ



The boundary conditions are

@u ¼ 0 at r ¼ 0; @r

regularity condition; no slip condition;

Z 2 ðx; tÞ ¼ h ð7Þ

ðh þ 1Þ p0 Re sin A ta Fr ð h þ 2ÞG a Cða þ 1Þ k1

ð8Þ

t ab Cða b þ 1Þ 2 Re p sin A h t 2a 0 Fr þ a G a Cð2a þ 1Þ k1 k1

ð9Þ

hð h þ 2ÞGkb2

u ¼ 0 at r ¼ h:

Integrating Eq. (6) with respect to r, and using Eq. (7), we get

r @a @p Re @b @u : 1 þ ka1 a sin A ¼ 1 þ kb2 b 2 @x Fr @r @t @t

2

Further integrating Eq. (9) from h to r and using Eq. (8), yields

1 @a @p Re @b 2 1 þ ka1 a sin A ðr 2 h Þ ¼ 1 þ kb2 b u: 4 @x Fr @t @t The volume flow rate is defined as Q ¼ Eq. (10), reduces to

Rh 0

ð10Þ

Z 3 ðx; tÞ ¼ h

ð11Þ

R ¼ r;

U ¼ u 1;

V ¼ v;

q ¼ Q h2 ;

kb2 t 2ab ka1 Cð2a b þ 1Þ 3 h p Re sin A t 3a þ 2a 0 Fra G Cð3a þ 1Þ k2 k1 2

ð12Þ

where the parameters on left hand side are in the wave frame and the parameters on right hand side are in the laboratory frame. The time-averaged flow rate Q is given by

3/2 3/2 2 Q ¼qþ1/þ ¼Q h þ1/þ : 8 8

ð13Þ

3

h G

Z 4 ðx; tÞ ¼ h

Eq. (11), in view of Eq. (13) gives

@ a @p Re 1 @p Re sin A þ sin A @t a @x Fr ka1 @x Fr ! 2 8 @b Q þ h 1 þ / ð3/2 =8Þ : ¼ a 1 þ kb2 b 4 k1 @t h

ð14Þ

w¼

2h

4

2

ðr4 2r 2 h Þ

r2 2

kb2

t 3ab

k21a

Cð3a b þ 1Þ

ð20Þ

;

! ð h þ 1Þ3 p0 Re sin A 3 2 Fr ð h þ 4 h þ 6 h þ 4ÞG ka1 ta

3

Cða þ 1Þ 2

þ

h ka1

tab

2

hðh þ 4h þ 6h þ 4ÞGkb2

Cða b þ 1Þ

! 3ðh þ 1Þ2 p0 Re sin A 2 Fr ð3 h þ 8 h þ 6ÞG ka1 2

3

t 2a h t2ab h 2 a ð3h þ 8h þ 6ÞGkb2 Cð2a þ 1Þ k1 Cð2a b þ 1Þ k21a 3ðh þ 1Þðp0 Re sin AÞ t 3a Fr Gð3 h þ 4Þ a Cð3a þ 1Þ k1 4 b 3ab k t h p Re sin A 3 h ð3h þ 4ÞG a2 G þ 3a 0 Fra k1 k1 Cð3a b þ 1Þ k1

From Eq. (10), and using Eqs. (11) and (13), the stream function (w) in wave frame U ¼ 1r @w is obtained as @r

! 2 Q þ h 1 þ / ð3/2 =8Þ

! ð h þ 1Þ2 p0 Re sin A ta 2 Fr ð h þ 3 h þ 3ÞG a Cða þ 1Þ k1

h ð2h þ 3ÞG

The transformations between the wave and the laboratory frames in the dimensionless form are given by

X ¼ x t;

ð19Þ

t ab 2 hð h þ 3h þ 3ÞGkb2 Cða b þ 1Þ 2 sin A h 2ðh þ 1Þ p0 Re t 2a Fr þ a Gð2 h þ 3Þ a Cð2a þ 1Þ k1 k1

2rudr, which, by virtue of

@a @p Re 8 @b 1 þ ka1 a sin A ¼ 4 1 þ kb2 b Q: @x Fr @t @t h

h Gkb2 t 2ab ; a k1 Cð2a b þ 1Þ

ð15Þ

It is clear from Eq. (15) that the stream function is independent of fractional parameters, material constants, Reynolds number, Froude number and inclination angle of tube.

kb t 4a t 4ab 4 h G 32a ; Cð4a þ 1Þ C ð4 a b þ 1Þ k1

ð21Þ

.. . 3. Solution of the problem by HAM Liao [29] developed a method HAM, which is a method to find series solutions of various types of linear and non-linear differential equations. HAM is based on homotopy and a fundamental concept of topology. It allows freedom in choosing initial approximations and auxiliary linear operators which often help to transfer the complicated non-linear problem to its simpler form. Eq. (14) can be simplified as

b @af 1 b @ G; þ f ¼ 1 þ k a 2 @ta k1 @t b where f ðx; tÞ ¼ @p Re sin A and G ¼ k8a Fr @x 1 HAM (see Appendix), we get

ð16Þ

Q þh2 1þ/ð3/2 =8Þ . Using h4

Re Z 0 ðx; tÞ ¼ p0 sin A ð17Þ Fr Re a a b p sin A t t G Z 1 ðx; tÞ ¼ h 0 Fra hGkb2 ; ð18Þ Cða þ 1Þ Cða b þ 1Þ k1

and so on. Proceeding in this manner the components Zn, n P 0 of the HAM can be completely obtained and the series solution is thus entirely determined. Finally, we approximate the analytical solution f(x, t) by the truncating the series

f ðx; tÞ ¼

@p Re sin A ¼ lim UN ðx; tÞ N!1 @x Fr

where UN ðx; tÞ ¼

ð22Þ

PN1

n¼0 Z n ðx; tÞ.

From Eq. (22), the pressure gradient is given as

@p Re ¼ sin A þ lim UN ðx; tÞ N!1 @x Fr

ð23Þ

The pressure difference (Dp) and frictional force (F) across one wavelength are given by

Author's Personal Copy D. Tripathi / Mathematical Biosciences 233 (2011) 90–97

Fig. 2. h-curves for the functions pressure at different values of fractional parameter and / ¼ 0:4; Q ¼ 0:1; t ¼ 1; b ¼ 4=5; k1 ¼ 1; k2 ¼ 1; Re ¼ 0:1; Fr ¼ 0:01 and A ¼ p=3.

Dp ¼ Z F¼

Z

0

0 1

93

Fig. 4. Pressure vs. averaged flow rate for various values of b at / = 0.4, t = 1, a = 1/5, k1 = 1, k2 = 1, Re = 0.1, Fr = 0.01 and A = p/3.

1

@p dx; @x @p 2 dx: h @x

ð24Þ ð25Þ

4. Convergence of the HAM solution The approximate analytic solution of the problem is obtained by HAM and it is given in Eq. (23). It is needed to check the convergence of the series (23), which is important to claim the numerical solution is accurate. Liao [29] pointed out that the convergence and rate of approximation for the HAM solution strongly depends on the values of auxiliary parameter h. One can check the range of the admissible values of h by drawing the so called h-curves. Fig. 2 shows that the admissible range for the values of h is 1.0 6 h 6 0.2. It is evident that the series (23) converge in the whole region of x when h = 1. So we can take the value of h is 1.

Fig. 5. Pressure vs. averaged flow rate for various values of k1 at / = 0.4, t = 1,a = 1/ 5, b = 4/5, k2 = 1, Re = 0.1, Fr = 0.01 and A = p/3.

5. Numerical results and discussion A thorough examination of the effects of emerging parameters like fractional parameters (a & b), relaxation time (k1), retardation time (k2), Reynolds number (Re), Froude number (Fr) and inclination angle (A) on pressure (Dp) and frictional force (F) which determine viscoelastic characteristic of the fluid, on the flow pattern, is carried out by computer simulation. Initial pressure gradient (t = 0) has been taken zero (i.e. p0 = 0) and auxiliary parameter are properly chosen as h = 1 in present analysis. The numerical results for the pressure and the frictional force are given by the expressions (24) and (25) and are determined by Mathematica software. Graphs (Figs. 3–9), are plotted between pressure across one wavelength (Dp) and averaged flow rate ðQ Þ, that exhibit a linear

Fig. 6. Pressure vs. averaged flow rate for various values of k2 at / = 0.4, t = 1, a = 1/ 5, b = 4/5, k1 = 1, Re = 0.1, Fr = 0.01 and A = p/3.

Fig. 7. Pressure vs. averaged flow rate for various values of Re at / = 0.4, t = 1, a = 1/ 5, b = 4/5, k1 = 1, k2 = 1, Fr = 0.01 and A = p/3.

Fig. 3. Pressure vs. averaged flow rate for various values of a at / = 0.4, t = 1, b = 4/5, k1 = 1, k2 = 1, Re = 0.1, Fr = 0.01 and A = p/3.

relation between them, for varying fractional parameters, relaxation time, retardation time, Reynolds number, Froude number and inclination angle. In fact, it measures the magnitude of



Fig. 8. Pressure vs. averaged flow rate for various values of Fr at / = 0.4, t = 1, a = 1/ 5, b = 4/5, k1 = 1, k2 = 1, Re = 0.1 and A = p/3.

Fig. 12. Frictional force vs. averaged flow rate for various values of k1 at / = 0.4, t = 1, a = 1/5, b = 4/5, k2 = 1, Re = 0.01, Fr = 0.01 and A = p/3.

Fig. 9. Pressure vs. averaged flow rate for various values of A at / = 0.4, t = 1,a = 1/5, b = 4/5, k1 = 1, k2 = 1, Re = 0.1 and Fr = 0.01.

Fig. 13. Frictional force vs. averaged flow rate for various values of k2 at / = 0.4, t = 1, a = 1/5, b = 4/5, k1 = 1, Re = 0.01, Fr = 0.01 and A = p/3.

3 Re=0.01

F 2

Re=0.02 Re=0.03

1

Re=0.04

0.02

0.04

0.06

0.08

0.1

Q -1 -2 Fig. 10. Frictional force vs. averaged flow rate for various values of a at / = 0.4, t = 1, b = 4/5, k1 = 1, k2 = 1, Re = 0.01, Fr = 0.01 and A = p/3.

Fig. 14. Frictional force vs. averaged flow rate for various values of Re at / = 0.4, t = 1, a = 1/5, b = 4/5, k1 = 1, k2 = 1, Fr = 0.01 and A = p/3.

Fig. 11. Frictional force vs. averaged flow rate for various values of b at / = 0.4, t = 1, a = 1/5, k1 = 1, k2 = 1, Re = 0.01, Fr = 0.01 and A = p/3.

Fig. 15. Frictional force vs. averaged flow rate for various values of Fr at / = 0.4, t = 1, a = 1/5, b = 4/5, k1 = 1, k2 = 1, Re = 0.01 and A = p/3.

pressure that can stop transportation. Therefore, once pressure assumes zero value, maximum amount of flow can take place. From

the figures, it is observed that different regions on the basis of the values of pressure gradient are examined in the present study, the

Author's Personal Copy D. Tripathi / Mathematical Biosciences 233 (2011) 90–97

Fig. 16. Frictional force vs. averaged flow rate for various values of A at / = 0.4, t = 1, a = 1/5, b = 4/5, k1 = 1, k2 = 1, Re = 0.01 and Fr = 0.01.

region for Dp > 0 is entire pumping region, the region for Dp = 0 is free pumping zone and the region for Dp < 0 is co-pumping region. Fig. 3 shows the pressure vs. averaged flow rate for various values of first fractional parameter (a) at / = 0.4, t = 1, b = 4/5, k1 = 1, k2 = 1, Re = 0.1, Fr = 0.01, A = p/3. It is observed that the pressure decreases in entire pumping region, free pumping region and copumping region but after a certain value of averaged flow rate in co-pumping region it increases with increasing the magnitude of a. The effect of second fractional parameter (b) on pressure is

95

shown in Fig. 4 and it is observed that pressure increases in pumping region, it is same in free pumping region and it diminishes in co-pumping region with increasing the magnitude of b. The results are similar to the results obtained by the Tripathi [25]. From this observation, it is difficult to conclude that the combined effect of both fractional parameters on flow pattern of viscoelastic fluid with fractional Oldroyd-B model, as the effects of both parameters on pressure are different in all regions to each other. Therefore it is not possible to compare the behavior of flow pattern for fractional models with their ordinary models of viscoelastic fluids. The variation of pressure with the averaged volume flow rate for various values of relaxation time and retardation time (k1 and k2) at / = 0.4, t = 1, a = 1/5, b = 4/5, Re = 0.1, Fr = 0.01, A = p/3 is presented in Figs. 5 and 6. It is found that the pressure diminishes in all regions with increasing k1, whereas, the effect of k2 on pressure is that the pressure increases in pumping region, it is same in free pumping region and it diminishes in co-pumping region with increasing the magnitude of retardation time. It is physically interpreted that when relaxation time for viscoelastic fluids is large, then, the flow for this type of fluid requires less pressure, whereas, for large retardation time, more pressure is required for flow in pumping region. When k1 ? 0 and k2 ? 0, this model reduces for Newtonian fluids. The study of flow of viscoelastic fluids with this model with Newtonian fluids is not comparable, since effects of both parameters are quite opposite in pumping region.

(a)

(b)

(c)

(d)

Fig. 17. Streamlines in the wave frame at Q ¼ 0:8 for (a) / = 0.3, (b) / = 0.4, (c) / = 0.5 and (d) / = 0.6.



The effect of two non-dimensional parameters such as Reynolds number (Re), Froude number (Fr) on the pressure-averaged flow rate curve keeping other parameters fixed / = 0.4, t = 1, a = 1/5, b = 4/5, k1 = 1, k2 = 1, A = p/3 are depicted from Figs. 7 and 8. It is evident that the pressure increases with increasing the magnitude of Reynolds number while it decreases with increasing the magnitude of Froude number in all regions. A curve (Fig. 9) between Dp and Q is plotted under influence of varying inclination (0 6 a 6 p/2) of tube at fixed value of other parameters / = 0.4, t = 1, a = 1/5, b = 4/5, k1 = 1, k2 = 1, Re = 0.1, Fr = 0.01. It is observed that the pressure increases in interval 0 6 a 6 p/2 in all three regions. Since the value of sine function decreases in interval p /2 6 a 6 p, therefore, we can conclude that effect of inclination of tube on pressure in this interval is opposite to said result. Figs. 10–16 are prepared for the variations of frictional force with the averaged flow rate under the influence of all pertinent parameters (a,b, k1, k2, Re, Fr and A). The relation between frictional force and averaged flow rate is found to be linear. Frictional force increases in magnitude with decreasing the value of averaged flow rate. From Figs. 10 and 11, it is evident that the frictional force decreases up to a certain value of averaged flow rate then after it increases with increasing the magnitude of a while the effect of b on frictional force is opposite to that of a. Fig. 12 reveals that frictional force increases with increasing the magnitude of k1. The effect of k2 on frictional force is observed similar to that of b from Figs. 11 and 13. Figs. 14–16 show the effect of Re, Fr and A on frictional force and it is found that the behaviors of these parameters on frictional forces are opposite to that for pressure. The streamlines on the center line in the wave frame of reference are found to split in order to enclose a bolus of fluid particles circulating along closed streamlines under certain conditions. This phenomenon is referred to as trapping, which is a characteristic of pumping motion. Since this bolus appear to be trapped by the wave, the bolus moves with the same speed as that of the wave. Fig. 17(a-d) are drawn for stream lines for different values of the amplitude ratio (/ = 0.3–0.6) at Q ¼ 0:8. Figures reveals that the size of trapped bolus reduces when the magnitude of / increases. It is observed from Eq. (15) that the stream function is independent of (a, b, k1, k2, Re, Fr and A). This indicates the size of trapped bolus is unaltered with these parameters.

pressure rises by increasing the magnitude of Reynolds number, whereas, it falls by increasing the magnitude of Froude number in all the regions. Inclination of tube affects the flow pattern, in different intervals of angles, the conclusions are different. It is observed that the frictional force diminishes up to a certain value of averaged flow rate then after it increases with increasing the magnitude of a, whereas, the effects of b and k2 on frictional force are opposite to that of a. It is further revealed frictional force is an increasing function of k1 and Fr, while, it is decreasing function of Re and A (0 6 a 6 p/2). Finally, it is concluded that the trapping reduces with increasing the amplitude ratio, while, it is unaltered with other parameters. Appendix To solve Eq. (16) by means of HAM, we choose the initial approximation

f0 ðx; tÞ ¼ p0

Re sin A; Fr

and the linear operator

L½uðx; t; hÞ ¼

@ a uðx; t; hÞ ; @t a

0