CHAPTER-I1 UNSTEADY PERISTALTIC FLOW OF A JEFFREY FLUID IN AN ASYMMETRIC CHANNEL

Transcription

1 CHAPTER-I1 UNSTEADY PERISTALTIC FLOW OF A JEFFREY FLUID IN AN ASYMMETRIC CHANNEL

2 1. INTRODUCTION Peristaltic pumping is a form of fluid transport that occurs when a progressive wave of area of contraction or expnnsion propagates along the length of a distensible tube containing the fluid. Physiologically it is inherent neuromuscular property of nny tubular smooth ~nuscle structure such as gastro- intestinal tract, bile duct, ureters and other glandular ducts wherein the irritation at thc epithelial lining of the duct can cause u contrnctile ring to $read along the tube, pushing the tluid contents iihcad (Gyton. 1986). Engineers developed peristaltic pumps having industrial and physiological applications adapting the principle of peristalsis. The first attempt to study the fluid dyniimics aspects of pcristaltic transport with an expcrimcntal is by Latham (1966). Various theoretical and experimental attempts have been made to understand peristaltic action in both mechanical and physiological situations under various approximtltions. The results of the experiments were found to be in good ugrccment with the theoretical results of Shapiro (1967). Based on the experimcrital work, Rum and Parks (1967) investigated thc peristaltic transport of a viscous fluid thmugh a pipe and a channel. Shapiro ct al. (1969) analyzed peristaltic pumping with long wavelength and low Reynolds numher, in a wave frame of rct'erence, whereas Fung and Yih (1968) and Yih and Fun# (1969) obtained analytical solutions of peristaltic flow by assuming small a~nplitude but arbitrary Reynolds number, in a fixed frame of reference. Many of the mathematical models on peristaltic transport considered in the literature deal with n prescribed train of waves moving with constant speed on the flexible boundaries and they are investigated in either a laboratory framc or a wave frame moving with constant velocity of the wave. Numerical techniques were used by Brown and Hung (1974) and Takabatake and Ayukawa (1982) for channel flow, and Takabatake el al. (1 988) for axisyrnmetric tube flow.

3 Many researchers considered the fluid to behave like a Newtonian fluid for physiological peristalsis including the flow of blood in arterioles. But such a model cannot be suitable for blood flow unless the non - Newtonian nature of the fluid is included in it. Peristaltic transport of non-newtonian fluids in a tube was first studied by Raju and Devanathan (1972), by considering the blood as a power-law fluid. They employed the perturbation technique used by Chow (1970) to solve the model of the flow in a cylindrical tube with a sinusoidal wave of small amplitude. Subsequently Raju and Devanathan (1974) investigated the peristaltic flow of a viscoelastic fluid in a tube considering the constitute equation of simple fluid with fading memory. Girija Devi and Devanathan (1975) extended the same problem of Raju and Devanathan (1972) to replace power-law nature of the fluid by micropolar nature. Peristaltic flow of a power-law fluid in a channel under long wavelength approximation was investigated by Radhakrishnarnacharya (1982). Rath and Reese (1984) studied the peristaltic transport of non - Newtonian fluids containing small spherical particles. Srivastava and Srivastava (1984) was investigated the peristaltic flow of blood in small blood vessels using the Casson fluid model. Peristaltic flow of non - Newtonian fluids with the application to the vas deferens and small intestine was studied by Srivastava and Srivastava (1985). Consequently, peristaltic transport of power-law fluid was investigated by Srivastava and Srivastava (1988) with the application to the ductus deferens of the reproductive tract. Thc effect of an endoscope on the peristaltic flow of a Jeffrey fluid was studied by Mayat et al. (2006). Hayat et nl. (2007) analyzed the peristaltic pumping of a Jeffrey fluid in an misymmetric tube. The effects of an endoscope and magnetic field on thc peristaltic flow of a Jefiey fluid was investigated Hayat et al. (2008). The non-steady peristaltic flow of a Newtonian fluid has been studied by Li and Brasseur (1993) in finite length tubes with lubrication approach. Peristaltic flow of compressible viscous fluid has been investigated by Antonovskii and Ramkissoon (1997) with a time-dependent pressure drop. The peristaltic flow in a tapered channel and tube was first investigated by Gupta

4 and Seshadri (1976). Following their analysis many other peristaltic flow problems are reported by Srivastava et al. ( 1983), Mekheimer (2002). Eytan et al. (2001) investigated the peristaltic flow in a tapered channel and discussed the application to embryo transport within the uterine cavity. However, the problem of peristaltic flow of a Jeffiy fluid in a finite length asymmetric channel has received a little attention. Hence, an attempt is made to analyze the problem of unsteady two-dimensional peristaltic flow of a Jgffrey fluid in a finite length i~symn~etric channel under the assumptions of long wavelength and low Reynolds number. l'hc fluid flow is investigated in a fixed frame of reference. Expressions ['or the axial velocity, axial prcssure rise and llux are obtained analytically. 'l'he effect of material constant &on thc flow characteristics are studied in detail. 2. MATHEMATICAL FORMULATION AND SOLUTION An unstcady two-dimensional flow of a Jefiey fluid in an asymmetric channel with flexible walls is considered. It is ilssumcd that the progrcssivc sinusoidal waves propagate along the walls oi'the channel. Fluid noti inn within the channel is induced by two infinite trains of sinusoidal waves with same speed, amplitude but with diffcrcnt phases. that arc propogntcd dong thc channel walls. A rectangular co-ordinate system (x, y) is chosen such that x- axis lies along the centre line of the channel in the dircclion of wavc propagation and y-axis transverse to it, as shown in Fig. 1. The channel walls arc characterized by y = ti,l,fx,r) = -u + hcos (2, la) (upper wall), where 20 is the undisturbed width of the channel, b - the wavc amplitude, I - the time, h - the wavelength and f) - the phase difference.?he phase difference may be in the range0 s 6 s x, where 6' = x defines symmetric contractions and B = 0 defines the waves with in phase.

5 Fig. 1 The Physical Model The constitute equation for the Jeffrey fluid is 7 = --,;A (Y+~Y) (2.2) where,u is the viscosity function, A, - the ratio of the relaxation time to retardation time, kz - the retardation time, ); - the shear rate and dots over the quantities denotc differentiation with respect to time. The equations governing the flow field in a fixed frame, are where u, v are the velocity components, p is the pressure, r is the stress tensor,

6 =~[~+4(u~+17~)]au a S, Z' rm 1 +A The boundary conditions for ~ hc velocity ore 111,,,, =li(,,,, =O (2.6) (2.7) In order to write the governing equntions and thc boundary conditions in dimensionless form, the following non-dimcnsiilnal quantities arc introduced. - X - Cl v - LIT /xi1 x=- y=y, d,!!, ;=! I=- v=-- r=- p=.-.- R' a 1 c' A' d' /rc' pca' b H H +=-, h, =--1.,/h =--2. (2.8) a a a where 6 is a wave number and 4 - the amplitude ratio. In view of (2.8), thc Equations (2.3) - (2.5), alter dropping bars, rcducc to du + - AJ = 0 ax ity (2.9) (2.10) (2.1 1) where Rc = P is the Reynolds number,

7 and ~~=--$-[~+~(~d+~d)]i. ax + + Under the long wavelength approximation (6 << 1 ) and low Reynolds number (Re -+ 0) assumption the Equations (2.10) and (2.11) reduce to The dimensionless boundary conditions are ulv 4 =MI" h, =o' The derivation for the present two-dimensional asymmetric channel of Fig.1 is similar to the axisymmetric model of Li and Brasseur (1993). 3. SOLUTION OF THE PROBLEM Integrating Equation (2.12) twice with respect to y and using the Equation (2.13) results in u(x,y,r)= -- ( ' + ~ ~ ) ~ ~ ( ~ ~ l ) ( ~ - ~ ~ ~ - ~ ). (3.1) 2 ax where h,, h, are the functions of xand t Using Equation (3.1) in the Equation (2.91, integrating with respect to y and substituting of Equation (2.14) for y = hl, yields

8 Substituting the Equation (2.14) for the upper wall, in Equation (3.2) and integrating with respect to x yields where f,(r)can be obtained by integrating the Equation (3.3) over the finite length of the channel (x = 0 tox = 1. ). Thus where&, (t)= p(l,t)-p(o,~). Knowing thc velocity field, wc can obtain the volume flow rote, which in dimensionless form is givcn by q(x,t) = Jh' t,(.x,y,/)dy. (3.5) 1) 'l'hc net llow rate is obtained by integrating thc Equation (3.5) over onc cycle, yielding - Q(x) = 1'' q(.r, [)dl.. O Elad (1999). (3,O) For the case of4 -+ 0, our results coincide with the results ofi<ytan and 4. RESULTS AND DISCUSSION In order to get a the physical insight of the problem, the axial pressure, volume flow rate and time averaged volume llow rate arc computed numerically for different valucs of the emerging parameters, viz., pressure difference at the channel ends Ap,, the phave shia 0, amplitude ratio 9 and Jeffrey fluid parameter A, and are presented in figures

9 Figs. 2-4 show the effects of 0, 4, A, on the pressure distribution along the channel axis. From Fig.2, it is noted that the maximum - to - minimum pressure difference increases with increasing phase shifle. From Fig. 3, it is observed that the maximum - to - minimum pressure difference decreases with increasingx,. Fig. 4 shows that the maximum - to - minimum pressure difference increases with increasing amplitude ratio4. Figs. 5-8, depict the effects of phase shift and overall pressure diffrencear~, on instantaneous flow rate q, during a singe cycle of contraction. It is observed that, the q oscillates with time in each case. The maximum flow variation is obtained for symmetric contractions (0 = n ) and decreases to zero as Furthcr, increasing thc overall pressure difference at the channel ends against the peristaltic direction decreases the instantaneous flow rate. More over when two waves are in the phase (0 = O), the flow rate curve for Ap, = 0 is straight line. Fig. 9 presents the instantaneous flow rate for various A, with 4 = 0.3, 0 = n/ 6, L = 2 and Ap,, = 0.6. It is found that the instantaneous flow rate decreases as A, increases. The total flow rate over one period as a function of phase shift 0, for different values of Ap,, with 4 = 0.3, L = 2, X = 0.4 is presented in Fig. 10. It is observed that, the net flow rate increases as the contraction becomes more symmetric (0-3 R). AS Ap, increases, the total flow rate 3 decreases, but the general shape of the graph remains similar. When Q<O, the peristaltic pumping is completely eliminated. Fig. 1 1 depicts the variation of with phase shift 0 for different values oj 4 with 4 = 0.3, L = 2 and Ap, = 0.6. It is noted that the a decreases with an

10 increase in A,. Further, it is observed that. the is' mcm for Newtonian fluid (A, - 0) than that of Jeffrey fluid 4 > CONCLUSIONS We developed a wall induced hid flow in a unifonn two dimensional asymmetric channel, under the assulnptions of long wave length and low Reynolds number. Fluid motion within the channel is induced by two - infinite trains of sinusoidal waves that are propagated along the channel walls with same speed, amplitude but with different phases. It is found that the maxi~num - to - minimum pressure diltercnce increilses with an illcrease in the phasc shift8. The maximum - to - minimum pressure dilr'crcncc is mow for Newtonian fluid than that of Jethey fluid. Also, the instclntancous tlow rule is more for Newtonian fluid th;m that of Jeltirey fluid. 'l'hc sune trcnd is obscrvcd for total flow rate$. Further. the present results arc in vcry gc~d agreement with the results of Eytan and Elad ( 1999) when A, - b 0. Communicated to "Journal of Interdisciplinary Malhematics"

11 Fig. 2 Pressure distributiot~ along the channel axis when Ap, = 0 for different phase 4 = 0.3and il, = 0.4atf = 0.5

12 Fig. 3 Pressure distribution along the channel axis when Ap,. = 0 For different A, with 4 = 0.3 and 0 --; rr lh atl= 0.5.

13 Fig. 4 Pressure distribution along the channel axis when Ap, = 0 for different values of amplitude ratio@ with A, =0.4 and B= r/6att =0.5.

14 Fig. 5 Instantaneous flow rate for dittcrcnt vtllucs of Ap, during a single period with 16 = 0.3, 1, = 2and A, = 0.4 a1 0 = R.

15 Fig. 6 Instantaneous flow rate for different values of &,during a single period with4 = 0.3, L = 2andA, = 0.4 at8 = ni2.

16 t Fig. 7 Instantaneous flow ralc for difrcrcnt values of Ap,. during a single period with4 = 0.3, L = 2and A, = 0.4 at0 = n/3.

17 I. I - 0 \ \ 8 I I '. \ /* \ /' / # -0.2 \ -, / /' / 8 * / \,, \ 8 Ap, = 0.0 \ ---- Ap,=O.6 \ / -.- Ap, =1.2 \ 0 % \ -0.5 Fig. 8 Instantaneous flow rate for different values of Ap, during 'a single period with 4 = 0.3, L = 2 and A, = 0.4 at B = 0. t

18 Fig. 9 Instantaneous tlow rate for different valucs of A, during a single period with (=0.3, B= ~16, L = 2and Ap,, ~ 0.6.

19 -. /- I I0 ' 0 -/ 0' / 0- /* _,- - - A&=O.O Ap, = 0.6 / -.- ApL=1.2 Fig. 10 The total flow rate over one period versus the phase shift B for different values of Ap, with q5 = 0.3, L = 2 and 4 =0.4.

20 -0.2 Bin Fig. I I, The total flow rate over one period versus the phase shift 6 for different values of 4 with 4 = 0.3. l, = 2 and 4, = 0.6.