Applied Mathematics and Computation

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1 Applied Mathematics and Computation 244 (214) Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: Effect of coupled radial and axial variability of viscosity on the peristaltic transport of Newtonian fluid Mehdi Lachiheb Faculty of Siences, Taibah University, Madinah Munawwarah, Saudi Arabia article Keywords: Peristaltic motion Newtonian fluid Trapping Reflux info abstract Many authors assume viscosity to be constant or a radius exponential function in Stokes equations in order to study the peristaltic motion of a Newtonian fluid through an axisymmetric conduit. In this paper, viscosity is assumed to be a function of both the radius and the axial coordinate. More precisely, it is dependent on the distance from the deformed membrane given the fact that the change in viscosity is caused by the secretions released from the membrane. The effect of this hypothesis on the peristaltic flow of a Newtonian fluid in axisymmetric conduit is investigated under the assumptions of long wavelength and low Reynolds number. The expressions for the pressure gradient and pressure rise per wavelength are obtained and the pumping characteristics and the phenomena of reflux and trapping are discussed. We present a detailed analysis of the effects of the variation of viscosity on the fluid motion, trapping and reflux limits. The study also shows that, in addition to the mean flow parameter and the wave amplitude, the viscosity parameter also affects the peristaltic flow. It has been noticed that the pressure rise, the friction force, the pumping region and the trapping limit are affected by the variation of the viscosity parameter but the reflux limit and free pumping are independent of it. Ó 214 Elsevier Inc. All rights reserved. 1. Introduction Several works have analyzed the mechanics of peristaltic pumping of a Newtonian fluid through an axisymmetric conduit. One of the remaining works is the treatment of the Stokes equation under an infinitely long wavelength approximation and a negligible Reynolds number in order to evaluate the velocities and pressures at each point of the geometrical model for a viscous fluid. Most works used the sine function for the geometry of the intestinal wall surface and a constant or an exponential function or its linear approximation for viscosity. For example, Latham [1] was the first to investigate the mechanism of peristalsis in relation to mechanical pumping. Shapiro et al. [2] studied the peristaltic motion of Newtonian fluid with constant viscosity, under long-wavelength and low-reynolds number assumptions, for plane and axisymmetric flow. They discussed the pumping characteristics and the phenomena of reflux and trapping. Shukla et al. [3] investigated the effects of peripheral-layer viscosity on peristaltic transport of a bio-fluid in uniform tube, the shape of interface is specified by them independently of the fluid viscosities. The invalidity of their analysis was proved by Brasseur et al. [4] with the limit of infinite peripheral layer viscosity, since the conservation of mass is not satisfied. They presented the effect of the peripheral layer on the fluid motion and discussed the pumping characteristics and the phenomena of reflux and trapping. They concluded that, for a peristaltic motion, a peripheral layer more viscous than the core fluid improves the pumping performance, Address: Faculté de siences de Gabes, Université de Sud, Gabes, Tunisie /Ó 214 Elsevier Inc. All rights reserved.

2 762 M. Lachiheb / Applied Mathematics and Computation 244 (214) while a less-viscous peripheral layer degrades performance. Elshehawey et al. [5] studied the peristaltic motion of an incompressible Newtonian fluid through a channel in the presence of three layers flow with different viscosities. Srivastava et al. [6,7], discuss the pressure rise, peristaltic pumping, augmented pumping and friction force for Newtonian fluid on peristaltic motion, with variable viscosity in uniform and non-uniform tubes and channel under zero Reynolds number and long wavelength approximations and comparison with other theories are given. Their computational results indicate that the pressure rise decreases as the fluid viscosity decreases. The difference between two corresponding values (for constant and variable viscosity) of the pumping region increases with increasing amplitude ratio. Further, the free pumping increases as viscosity of fluid decreases. The pressure rise, in the case of non-uniform geometry is found to be much smaller than the corresponding value in the case of uniform geometry. Abd El Hakim et al. [8,9] have investigated the effect of endoscope and fluid with variable viscosity on peristaltic motion. Abd El Hakim et al. [1] choose the viscosity parameter in the exponential function as a perturbation parameter in order to study the hydromagnetics flow of fluid with variable viscosity in a uniform tube with peristalsis under a negligible Reynolds number, a small magnetic Reynolds number, sine harmonic traveling wave and infinitely long wavelength approximation. The effect of variable viscosity on the peristaltic transport of a Newtonian fluid in an asymmetric channel has been studied by Hayat and Ali [11]. They showed that, in addition to the effect of mean flow parameter, the wave amplitude also affects the peristaltic flow. Recently a number of analytical, numerical and experimental studies of peristaltic flows of different fluids have been reported. Rathod and Asha [12] studied the effects of magnetic field and an endoscope on peristaltic motion and Rathod and Devindrappa [13] studied the slip effect on peristaltic transport of a porous medium in an asymmetric vertical channel by Adomian decomposition method. Hina et al. [14] studied the Slip Effects on magnetohydrodynamic Peristaltic Motion with Heat and Mass Transfer. A mathematical model has been developed by Misra and Maiti [15] with an aim to study the peristaltic transport of a rheological fluid for arbitrary wave shapes and tube lengths. The previously mentioned works are based on Newtonian fluids with a viscosity that is either constant or function of the radius only. This is inconsistent with the situation in natural peristaltic motions. For instance, chyme motion within the small intestine is subjected to water intake at a first stage, then to water absorption accompanied by some acids released from the intestinal membrane to facilitate digestion and enable the absorption of necessary nutrients such as proteins, fat and vitamins [16 18]. The movement of blood in the blood vessels is characterized by a concentration of red blood cells near the axis and of white blood cells and plasma near the membrane, which makes the viscosity decrease closer to the wall [19]. This shows that viscosity depends on the distance from the membrane in its deformed state, which again contradicts the assumptions of constant or radially varying viscosity since these imply a constant viscosity along lines parallel to the axis. This led us in this work to consider the viscosity to be a function of two variables. We assume that the viscosity remains constant on any surface such that the radius is proportional to that of the membrane with a factor less than unity. This choice is consistent with the observation mentioned above concerning the distribution of viscosity in the blood vessels and the small intestine. It is expected to provide more accurate and realistic explanations of the mechanisms of peristaltic movement and blood circulation and the associated reflux and trapping phenomena. Moreover, in order to make this work even more general an additional parameter has been introduced into the viscosity function. This parameter indicates the gradient of the viscosity. We further studied the effect of this parameter on pumping, reflux and trapping and compared it with published results based on the assumptions of viscosity being either constant or function of a single variable. 2. Formulation and analysis We consider the creeping flow of an incompressible Newtonian fluid with variable viscosity through an axisymmetric form in a uniform tube wall thickness with a wave traveling down its wall. We use the cylindrical coordinate system Fig. 1. Peristaltic transport through a tube.

3 M. Lachiheb / Applied Mathematics and Computation 244 (214) ðz; RÞ, where the z axis lies along the centerline of the tube, and r transverse to it. The geometry of the wall surface is described in Fig. 1 and it is given by ^HðZÞ ¼a þ b sin 2p k ðz c tþ ð1þ where a is the radius of the tube at inlet, k is the wavelength, c is the wave speed and t is the time. Let U and W be the velocity components in the radial and axial directions and W the stream function. In these coordinates the flow is unsteady but we can eliminate the time variable if we choose moving coordinates ðz; rþ which travel in the Z direction with the same speed as the wave. Then the flow can be treated as steady. The coordinate frames are related through: z ¼ Z ct r ¼ R ð2þ w ¼ W c u ¼ U w ¼ W r 2 =2 ð3þ HðzÞ ¼a þ b sin 2p k z where u and w are the velocity components in the moving coordinates. The following equations are the continuity and Navier Stokes equations and the boundary conditions in the moving coordinates: 1 ¼ þ @u 2lðrÞ q w þ þ 2lðrÞ r þ u @z r w ¼ u ¼ if r ¼ ð8þ ð4þ ð5þ ð6þ ð7þ w ¼ c u ¼ if r ¼ H where P is the pressure and l is the viscosity function. We define the Reynolds number Re ¼ caq l where l is the viscosity on the axis, and the wave number d ¼ a k ð9þ ð1þ ð11þ and the non-dimensional variables: r ¼ r a z ¼ z k u ¼ ku ac w ¼ w c P ¼ a2 P l l ¼ t ¼ c t ckl l k H ¼ H a ¼ 1 þ / sinð2pzþ where / ¼ b is the amplitude ratio. a Replacing Eq. (12) in (5) (9) and using the long wavelength approximation and the negligible wave number we obtain: 1 þ ¼ ð12þ ¼ ¼ 1 ð14þ ¼ u ¼ at r ¼ ð16þ

4 764 M. Lachiheb / Applied Mathematics and Computation 244 (214) w ¼ 1 at r ¼ H The instantaneous volume flow rate in the fixed and moving coordinate system are defined by: ^Q ¼ 2p q ¼ 2p Z ^H Z H RWdR r wdr From Eqs. (2) (4) and (18) it follows that: ^Q ¼ q þ pc ^H Using the dimensionless variables, we find Q ¼ q 2pa 2 c ¼ Z H rwdr The time-mean flow over a period T ¼ k at a fixed Z-position is defined as c Q ¼ 1 Z! T ^Qdt ¼ q þ pc a 2 þ b2 T 2 The dimensionless time-mean flow is! H ¼ Q 2pca ¼ Q þ þ /2 2 ð17þ ð18þ ð19þ ð2þ ð21þ ð22þ ð23þ From Eq. (15) @P ðrlðr; zþ Þ¼r. Since the pressure is independent to the variable r, lðr; ¼ r2 þ c 1ðzÞ, where c 1 ðzþ is a function dependent on z. Using the first condition of (16) we obtain c 1 ðzþ ¼. Then wðrþ ¼Mðr; þ c 2ðzÞ, where Mðr; zþ ¼ 1 r t dt from the first condition of (17), c 2 HðzÞ lðt;zþ 2ðzÞ ¼ 1, then wðrþ 1 ¼ rmðr; r and the shear stress at wall is given by sðzþ ¼lðt; ¼ 1 r¼h The dimensionless instantaneous volume flow rate in moving coordinate system is Q ¼ q ¼ R H rwdr. Replacing the velocity component w by the right hand side of (24), we obtain Q ¼ R 2pa 2 c tmðt; zþdt 2 ¼ R H Q þ 1 2 H2 ð27þ tmðt; zþdt ð25þ ð26þ and since w ¼ atr¼, it follows that w ¼ Q þ 1 R r tmðt; zþdt 2 H2 R H tmðt; zþdt 1 2 r2 ð28þ Since large quantities of water are secreted into the lumen of the small intestine during the digestive process, almost all of this water is also reabsorbed in the small intestine [16 18] and the movement of blood in the blood vessels is characterized by a concentration of red blood cells near the axis and of white blood cells and plasma near the membrane [19], which makes the viscosity decrease closer to the vessels membrane. This shows that viscosity depends on the distance from the membrane in its deformed state, which again contradicts the assumptions of constant or radially varying viscosity since these imply a constant viscosity along lines parallel to the axis. This led us in this work to consider the viscosity to be a function of two variables in the following form: lðr; zþ ¼f where f ðrþ ¼e ar. r HðzÞ ð29þ

5 M. Lachiheb / Applied Mathematics and Computation 244 (214) We assume that the viscosity remains constant on any surface such that the radius is proportional to that of the membrane with a factor less than unity. This choice is consistent with the observation mentioned above concerning the distribution of viscosity in the blood vessels and small intestine. Moreover, in order to make this work even more general the parameter a has been chosen positive for decreasing viscosity and negative for increasing viscosity. Replacing Eq. (29) in Mðt; zþ we obtain and Mðt; zþ ¼ 1 Z 2 H2 ðzþ 1 t HðzÞ s f ðsþ ds ¼ H2 ðzþ ð1 aþe a þ e at 1 at HðzÞ 2a 2 HðzÞ Z H tmðt; zþdt ¼ H 4 ð 6 þð6 6a þ 3a 2 a 3 Þe a Þ 4a 4 then and 4a 4 1 Q ¼ bh 4 2 H2 2a 4 Q þ 1 2 H2 sðzþ ¼ bh 3 w ¼ 1 Q þ 1 2 H2 6H 2 1 þ e ar ar H þ 6aHre H þ a 2 ð 1 þ aþe a 2e ar H r 2 2 r2 bh 2 where b ¼ð 6þð6 6aþ3a 2 a 3 Þe a Þ. The pressure rise DP and friction force F in their non-dimensional forms are given by DP ¼ F ¼ Z 1 Z 1 @z Replacing Eq. (3) in (33) and (34) we obtain dz DP ¼ a4 ðð4 þ 6/ 2 ÞH ð8/ /4 ÞÞ bð1 / 2 Þ 7 2 F ¼ a4 b / 2 þ 2 4H 2A ð1 / 2 Þ 3 2 If a approaches zero, we obtain the results given by [11] " w ¼ r 2 Q þ 1 # ð2h 2 r 2 Þ 2 H2 1 H 4 2 DP ¼ 8 ð2 þ 3/2 ÞH 4/ /4 ð1 / 2 Þ F þ /2 4H A ð1 / 2 Þ 3 2 ð3þ ð31þ ð32þ ð33þ ð34þ ð35þ ð36þ ð37þ ð38þ ð39þ

6 766 M. Lachiheb / Applied Mathematics and Computation 244 (214) Discussion of the results 3.1. Pumping characteristics The maximum pressure rise DP max against which the peristalsis works as a pump (i.e. DP corresponding to H ¼ ). In the same way the maximum volume flow rate H max is the value of H when the pressure rise DP ¼ (free pumping). These are given by DP max ¼ a4 1 2 /4 8/ 2 ð4þ bð1 / 2 Þ 7 2 H max ¼ 8/2 1 2 /4 4 þ 6/ 2 ð41þ When DP > DP max one obtains negative flux and when DP <, one gets H > H max as the pressure assists the flow which is known as copumping. The relation between the pressure rise and the flow rate and the friction force and the flow rate given by (33) and (34), respectively, are plotted in Figs Since the viscosity function, as reported in Srivastava and al.[7],ise ar with a ¼ (Shapiro et al. [2]) and a ¼ :1, Figs. 6 and 7 show the relation between the pressure rise and the friction force versus flow rate for different viscosity functions at / ¼ :2; :4, and :6. It is clear that the pressure rise and the friction force are less under radial and axial variability of viscosity than that for constant or under radial variability of viscosity as reported in [2]. The variation of DP and F given by (33) and (34), respectively, with H for / ¼ :4 for different values of viscosity parameter a is shown in Figs It is observed that the pressure rise and the friction force decrease with increasing a and the pumping region ð 6 DP 6 DP max Þ is less for radial and axial variability of viscosity than that for constant or radial variability of viscosity (Fig. 8) and it decreases as the viscosity parameter a increases (Fig. 9). But the free pumping is independent of alpha. Fig. 1 shows the shear stress distribution along the wall for different values of the viscosity parameter for the three viscosity functions e ar and f ðr=hþ ¼e ar=h and constant. Unlike in the viscosity function e ar, in our choice the viscosity is constant along the wall. Consequently, the shear stress function is independent of the gradient of the viscosity function f but it does depend on the viscosity at the wall which is itself dependant on the viscosity parameter alpha Trapping limit For certain combinations of H and / there is a region of closed streamlines. This region experiences a recirculation and moves with a mean speed equal to that of the wave. This is termed as trapping. Peristalsis exhibits this phenomenon when the tube is sufficiently occluded. The trapping limit is given by the value of H where w ¼ at some r just greater then zero. Having expressed w in a power series in terms of r, the flow-rate H for which trapping may occur lies between two extremes as given below "! # H ¼ 1 1 þ /2 6 þ a2 þð6 6aþ2a 2 Þe a 2 2 a 2 ð1 þð 1þaÞe a ð1 þ /Þ 2 ð42þ Þ and "! # H þ ¼ 1 1 þ /2 6 þ a2 þð6 6aþ2a 2 Þe a 2 2 a 2 ð1 þð 1þaÞe a ð1 /Þ 2 Þ ð43þ Fig. 2. The pressure rise DP versus the volume flow rate h with / ¼ :4 with decreasing viscosity.

7 M. Lachiheb / Applied Mathematics and Computation 244 (214) Fig. 3. The friction force F versus the volume flow rate h with / ¼ :4 with decreasing viscosity. Fig. 4. The pressure rise DP versus the volume flow rate h with / ¼ :4 with increasing viscosity. Fig. 5. The friction force F versus the volume flow rate h with / ¼ :4 with increasing viscosity. If a approaches zero, we have the range for Newtonian fluid with constant viscosity reported in [2]. Like in the case of H constant viscosity, þ Hþ is greater than 1 for all amplitude ratios h. Fig. 11 shows the lower limit of trapping region versus Hmax Hmax the amplitude ratio with various values of viscosity parameter including zero (constant viscosity). The trapping region is smaller for radial and axial variability of viscosity than that for constant viscosity and it decreases as the viscosity parameter a.infig. 12, the trapping contours are plotted with various values of the viscosity parameter a and it can be seen that the size of trapping increases as a decreases Reflux limit Reflux refers to the presence of fluid particles that move, on the average, in a direction opposite to the net flow. Since the laboratory frame has been found suitable for the study of reflux, W the flow-rate corresponding to Z and t, is transformed as

8 768 M. Lachiheb / Applied Mathematics and Computation 244 (214) Fig. 6. The pressure rise DP versus the volume flow rate h. Fig. 7. The friction force F versus the volume flow rate h. Fig. 8. The maximum pressure rise DP max versus / with a ¼ :1. W ¼ w þ 1 2 r2 ðw; zþ ð44þ Averaged over one cycle, this becomes H W ¼ w þ 1 2 Z 1 r 2 ðw; zþdz ð45þ

9 M. Lachiheb / Applied Mathematics and Computation 244 (214) Fig. 9. The maximum pressure rise DP max versus the parameter viscosity a with various values of amplitude ratio /. Fig. 1. The shear stress at wall versus the variable z with various values of viscosity parameter a and with / ¼ :4 and H ¼ :1. Fig. 11. The lower limit of trapping region with various values of viscosity parameter a.

10 77 M. Lachiheb / Applied Mathematics and Computation 244 (214) Fig. 12. Streamlines in the wave frame showing the appearance of the trapping with various values of viscosity parameter a and with / ¼ :6 and H ¼ :1. In order to evaluate the reflux limit, H W is expanded in power series in terms of a small parameter about the wall, where ¼ w F and it is subjected to the reflux condition H W H > 1 as! ð46þ The coefficients of the first two terms in the expansion of r, i.e., r ¼ h þ a þ a 1 2 þ are found by using (3), as given below a ¼ 1 h ; a 1 ¼ ð2f þ H2 Þa 4 e a bh 2 2H 5 b Using (45) (48) it follows that the reflux occurs whenever H < 1 and it is independent to the parameter a. That is, like in Hmax the case of Newtonian fluids with constant viscosity, reflux does take place in the entire domain. 4. Conclusion A coupled radial and axial variability of viscosity is considered to study the peristaltic transport of Newtonian fluid with variable viscosity in axisymmetric conduit. The effects of the viscosity function s slope and the amplitude ratio on the pressure rise, the friction force, trapping and reflux limits and pumping region are studied. It is observed that an increase in the viscosity parameter decreases the pressure rise and the frictional forces. Moreover, the pressure rise, the friction force, the pumping region and the trapping limit are affected by the variation of the viscosity parameter but the reflux limit and free pumping are independent of it. It is found that the pressure rise, the friction force and the pumping region are lower under radial and axial variability of viscosity than that for constant or under radial variability of viscosity. Moreover, the trapping region and the size of trapping is smaller for radial and axial variability of viscosity than that for constant viscosity. ð47þ ð48þ

11 M. Lachiheb / Applied Mathematics and Computation 244 (214) References [1] T.W. Latham, Fluid motions in a peristaltic pump (M.S. Thesis), M.I.T., [2] A.H. Shapiro, M.Y. Jaffrin, S.L. Weinberg, Peristaltic pumping with long wavelengths at low Reynolds number, J. Fluid Mech. 37 (1969) [3] J.B. Shukla, R.S. Parihar, B.R.P. Rao, S.P. Gupta, Effects of peripheral layer viscosity on peristaltic transport of a bio-fluid, J. Fluid Mech. 97 (198) [4] J.G. Brasseur, S. Corrsin, N.Q. Lu, The influence of peripheral layer of different viscosity on peristaltic pumping with Newtonian fluids, J. Fluid Mech. 174 (1987) [5] E.F. Elshehawey, Z.M. Gharsseldien, Peristaltic transport of three-layered flow with variable viscosity, Appl. Math. Comput. 153 (24) [6] L.M. Srivastava, V.P. Srivastava, S.N. Sinha, Peristaltic transport of a physiological fluid. Part-I. Flow in non-uniform geometry, Biorheology 2 (1983) [7] L.M. Srivastava, V.P. Srivastava, S.N. Sinha, Peristaltic transport of a physiological fluid. Part-II. Flow in uniform geometry, Biorheology 2 (1983) [8] A.E.M. El Misery, A.H. El Naby, A.H. El Nagar, Effects of a fluid with variable viscosity and an endoscope on peristaltic motion, J. Phys. Soc. Jpn. 72 (23) [9] A.H. El Naby, A.E.M. El Misery, I.I. El Shamy, Effects of an endoscope and fluid with variable viscosity on peristaltic motion, Appl. Math. Comput. 158 (24) [1] A.H. El Naby, A.E.M. El Misiery, I.I. El Shamy, Hydromagnetic flow of fluid with variable viscosity in a uniform tube with peristalsis, J. Phys. A: Math. Gen. 36 (23) [11] T. Hayat, N. Ali, Effect of variable viscosity on the peristaltic transport of a Newtonian fluid in an asymmetric channel, Appl. Math. Modell. 32 (28) [12] S.K. Asha, V.P. Rathod, Peristaltic transport of a magnetic fluid in a uniform and non-uniform annulus, Int. J. Math. Arch. 12 (211) [13] V.P. Rathod, L. Devindrappa, Slip effect on peristaltic transport of a porous medium in an asymmetric vertical channel by Adomian decomposition method, Int. J. Math. Arch. 4 (213) [14] S. Hina, T. Hayat, A. Alsaedi, Slip effects on MHD peristaltic motion with heat and mass transfer, Arabian J. Sci. Eng. 39 (214) [15] J.C. Misra, S. Maiti, Peristaltic transport of a rheological fluid: model for movement of food bolus through esophagus, Appl. Math. Mech. (AMM) 33 (212) [16] J.L. Turvill, M.J. Farthing, Water and electrolyte absorption and secretion in the small intestine, Curr. Opin. Gastroenterol. 15 (1999) [17] L.S. Costanzo, Physiology, 4th ed., Elsevier, 29. [18] M.L. Lucas, Enterocyte chloride and water secretion into the small intestine after enterotoxin challenge: unifying hypothesis or intellectual dead end?, J Physiol. Biochem. 64 (28) [19] B. Grobelnik, Blood Flow, Postgradute seminar, University in Ljubljana, Faculty of Mathematics and Physics, 28.