Peristaltic transport of a conducting fluid through a porous medium in an asymmetric vertical channel

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1 Available online at Advances in Applied Science Research,, (5):4-48 ISSN: CODEN (USA): AASRFC Peristaltic transport of a condcting flid throgh a poros medim in an asmmetric vertical channel G. Rami Redd and S. Venkataramana Dept. of Mathematics, S.V.Universit,Tirpati, A.P. India. _ ABSTRACT The aim of this problem is to std the peristaltic motion of a viscos condcting flid throgh a poros medim in an asmmetric vertical channel b sing Lbrication approach. The expressions for velocit and pressre rise are obtained. The effects of Darc nmber, phase shipt and Hartmann nmber on flow characteristics are stdied in detail. Ke words: Peristaltic transport, MHD, poros medim, Lbrication approach. _ INTRODUCTION The std of the mechanism of peristalsis in both mechanical and phsiological sitations has recentl become the object of scientific research. Several theoretical and experimental attempts have been made to nderstand peristaltic action in different sitations. A review of mch of the earl literatre is presented in an article b Jaffrin and Shapiro (5). A smmar of most of the experimental and theoretical investigations reported with details of the geometr, flid Renolds nmber, wavelength parameter, wave amplitde parameter and wave shape has been stdied b Srivastava and Srivastava (6). Flow throgh a poros medim has been stdied b a nmber of workers emploing Darc s law is given b A.E. Scheidegger (4). Some stdies abot this point have been made b Varshne (8) and EL-Dave and EL-Mohendis (). Elshehawe et al., (4) stdied peristaltic motion of a generalized Newtonian flid throgh a poros medim. Ramiredd et.al. () stdied peristaltic transport of a condcting flid in an inclined asmmetric channel. Peristaltic motion of a generalized Newtonian flid nder the effect of a transverse magnetic field is stdied b Elshehawe et al.,(3). Satanaraana et a l(3) stdied Hall crrent effect on magnetohdro dnamics Free-convection flow past a semi-infinite vertical poros plate with mass transfer. Flow throgh a poros medim has been of considerable interest in recent ears particlarl among geophsical flid dnamicists. Examples of natral poros media are beach sand, standstone, limestone, re bread, wood, the hman lng, biledct, gall bladder with stones and in small blood vessels. 4

2 G. Rami Redd et al Adv. Appl. Sci. Res.,, (5):4-48 The first std of peristaltic flow throgh a poros medim is presented b Elshehawe et al., (3). The interaction of peristaltic flow with plsatile flid nder the effect of a transverse magnetic field throgh a poros medim bonded b a two-dimensional channel is stdied b Afifi and Gad (). Mekheimer and Arabi (8) stdied the non-linear peristaltic transport of MHD flow throgh a poros medim. Mekheimer (9) stdied peristaltic flow of blood nder effect of a magnetic field in non-niform channels. He observed that the pressre rise for a cople stress flids (as a blood model) is greater than for a Newtonian flid and is smaller for a magnetohdrodnamic flid than for a flid withot an effect of a magnetic field. Non-linear peristaltic transport throgh a poros medim in an inclined planar channel has been stdied b Mekheimer () taking the gravit effect on pmping characteristics. Peristaltic transport of a viscos flid in an inclined asmmetric channel has been stdied b Sbba Redd (7). Recentl the effects of heat transfer on MHD nstead free convection flow past an infinite/semi infinite vertical plate was analzed b [6,7,, 5]. This paper deals with the effects of Darc nmber, phase shipt and Hartmann nmber in the peristaltic motion of a viscos condcting flid throgh a poros medim in an asmmetric vertical channel in the presence of magnetic field.. Mathematical formlation and Soltion We consider the peristaltic transport of a viscos condcting flid throgh an asmmetric channel with flexible walls and asmmetr being generated b the propagation of waves on the channel walls travelling with same speed c bt with different amplitdes and phases. We assme that a niform magnetic field strength B is applied in the transverse direction to the direction of the flow (i. e., along the direction of the -axis) and the indced magnetic field is assmed to be negligible. Fig. shows the phsical model of the asmmetric channel. The channel walls are given b π Y H ( X, t) a + b cos ( X ct) (a) λ π Y H ( X, t) a b cos ( X ct) + θ λ (b) O Y a b a.5 H (X, t) c b λ B H (X, t) θ g X Fig. Phsical Model 4

3 G. Rami Redd et al Adv. Appl. Sci. Res.,, (5):4-48 Where b, b are amplitdes of the waves, λ is the wavelength, a + a is the width of the channel, θ is the phase difference ( θ π) and t is the time. We introdce a wave frame of reference (, ) x moving with velocit c in which the motion becomes independent of time when the channel length is an integral mltiple of the wavelength and the pressre difference at the ends of the channel is a constant (Shapiro et al., (969)). The, x, transformation from the fixed frame of reference ( X Y ) to the wave frame of reference ( ) is given b x X - c t, Y, U - c, v V and p( x) P( X, t),, U, V are the velocit components, p and P are pressres in the wave where ( v) and ( ) and fixed frames of reference, respectivel. The eqations governing the flow in wave frame of reference are given b v +, () x ρ v B x x ε x k v v p µ v v µ ρ + v + + v. x ε x k p + + µ + σ µ e, Where σ e is the electrical condctivit of the flid, ε and k are the porosit and permeabilit of the poros medim, ρ is the densit and µ is the viscosit of the flid. Introdcing the following non-dimensional variables x v a a x,,, v, δ, d λ a c cδ λ a pa H H b b p, h, h,,. µ cλ a a a a φ φ in the governing eqations (-4), and dropping the bars, we get ( ) h + φ cos π x, h d φ cos π x + θ (5) v +, (6) x p Re δ + v + δ + M, x x ε x Da 3 v v p δ v v δ Re δ + v + δ + v. x ε x Da (3) (4) (7) (8) 4

4 G. Rami Redd et al Adv. Appl. Sci. Res.,, (5):4-48 ρac where Re is the Renolds nmber, µ the Hartmann nmber. k Da is the Darc nmber and M B a σ is a µe Using long wavelength (i.e., δ << ) and negligible inertia (i.e., Re ) approximations, we have p, N. (9) where N ε + M Da. The corresponding non-dimensional bondar conditions are given as - at h and h () Solving eqation (9) sing the bondar conditions (), we get c N c N N + / N [ sinh Nh sinh Nh ] where c cosh Nh sinh Nh cosh Nh sinh Nh cosh + sinh / () c [ ] + N Nh Nh [ ] / cosh cosh [ cosh Nh sinh Nh cosh Nh sinh Nh ] and. The volme flow rate in the wave frame is given as q h h d c c + M M ( sinh Nh sinh Nh ) ( cosh Nh cosh Nh ) From (), we have 3 qn D + D N D h h ND where ( ) ( h h ) N. () (3) D cosh Nh sinh Nh cosh Nh sinh Nh and ( cosh cosh ) ( sinh sinh ) D Nh Nh Nh Nh. 43

5 G. Rami Redd et al Adv. Appl. Sci. Res.,, (5):4-48 The instantaneos flx at an axial station is given b h Q( x, t) ( + ) d q + h h. (4) h The average volme flow rate over one wave period (T λ / c ) of the peristaltic wave is defined as T T Q Qdt ( q h h ) dt q d. T T (5) The pressre rise over one wave length of the peristaltic wave is given b 3 qn D + DN p D ( h h ) ND. (6) 3 ( Q d ) N D + DN QI + I D h h ND ( ) where I 3 N D and D ( h h ) ND I 3 ( + d) N D + DN. D h h ND ( ) The eqation (4.6) can be rewritten as p I Q. (7) I RESULTS AND DISCUSSION The variation of velocit with for different vales of M with φ.7, φ., d, δ.5, x.5, ε.5, Da.8 and θ for (i).5; ( ii) ; ( iii) as depicted in Fig. It is observed that the maximm velocit U increases with the increase in Hartmann nmber M for all the three cases -.5, and. The similar behavior observed for phase shipt θ π / 4 as shown in Fig 3. The Fig 4 shows the variation of velocit µ with λ for different vales of Darc nmber Da with φ.7, φ-, d, n.5,x.5 ε.5, M.5 and θπ/4 for (i) and (ii). It is observed that as the Darc nmber Da decreases the maximm velocit increased. Bt when Da <., the change in maximm velocit is negligible. Using eqation (7), we have plotted the variation of time-averaged volme flow rate Q with pressre rise p for different vales of phase shipt θ with φ.7, φ., d, δ, ε. M.5 and Da.. The pmping increases as the phase shipt θ decreases, where as phase shipt θ increases. The free pmping as well as co-pmping both increases is show in Fig 5. 44

6 G. Rami Redd et al Adv. Appl. Sci. Res.,, (5): M -.5 M M -.7 M M M Fig (i).the variation of velocit with for different vales of M with Fig 3(i).The variation of velocit with for different vales of M with, M M -.7 M M M M Fig (ii).the variation of velocit with for different vales of M with Fig 3(ii).The variation of velocit with for different vales of M with M -.7 M M M M M Fig (iii).the variation of velocit with for different vales of M with Fig 3(iii).The variation of velocit with for different vales of M with,. 45

7 Kalada Hart et al Adv. Appl. Sci. Res.,, (5): Fig 6 shows the variation of time-averaged volme flow rate Q with pressre rise p for different vales of Hartmann nmber M with φ.7, φ., d, δ, ε. θπ/4 and Da-.. As M increases the pmping is increases.. Da. Da Da Fig 4(i). The variation of velocit with for different vales of Da with.. Da. Da Da Fig 4(ii). The variation of velocit with for different vales of Da with. 46

8 Kalada Hart et al Adv. Appl. Sci. Res.,, (5): 5 M θ 5 M 5 θ π / 4 θ π / M θ π Fig 5. The variation of time-averaged volme flow rate Q with pressre rise p for different vales of phase shift θ with Da.. p Fig 6.The variation of time-averaged volme flow rate Q with pressre rise p for different vales of Hartmann nmber M with Da.. Acknowledgements: The athors sincerel thank Prof. S.Sreenadh (Dept. of Mathematics, S.V.Universit, Tirpati- 575) for sefl discssions. REFERENCES [] Afifi, N.A.S. and Gad, N.S. Acta Mechanica, 49 (), [] El-Dave, N.T. and El-Mohendis, S.M. Arabian J. Sci. Engrg., (995), 57. [3] Elshehawe, E.F., Mekheimer, KH.S., Kalads, S.F., Afifi, N.A.S. J. Biomath., 4 (), 999. [4] Elshehawe, El Saed, F. and El Sebaei, W.A.F. Int. J. Math. Math. Sci., 6, no.4 (), 7-3. [5] Jaffrin, M.Y. and Shapiro, A.H. Ann. Rev. Flid Mech. 3 (97), [6] Kango SK and Rana GC, Advances in Applied Science Research, () [7] Kavitha A, Hemadri Redd R, Sreenadh S, Saravana R and Srinivas ANS, Advances in Applied Science Research, Volme () [8] Mekheimer, K.H.S. and Al-Arabi, T.H. Int. J. Math. Sci., 6 (3), [9] Mekheimer, K.H.S. J. Poros Media, 6 (3), 9-. [] Mekheimer, K.H.S. Appl. Math. Compt., 53 (4), no.3, [] Ramiredd G, Sata Naraana P V, Venkataramana S, Applied Mathematical Sciences 4, (), [] Rathod VP and Asha SK, Advances in Applied Science Research, () -9 [3] Sata Naraana P V, Rami Redd G and Venkataramana S, Hall International Jornal of Atomotive and Mechanical Engineering 3, (), [4] Scheidegger, A.E. The Phsics of Flow Throgh Poros Media, 3 rd ed., Universit of Toronto Press, Toronto, Canada, 974. [5] Sri Hari Bab V and Ramana Redd GV, Advances in Applied Science Research, () [6] Srivastava, L.M. and Srivastava, V.P. J. Biomech., 7 (984), [7] Sbba Redd, M.V. Some stdies on peristaltic pmping of bioflids, Ph.D. thesis, Sri Venkateswara Universit, Tirpati, December 4. 47

9 Kalada Hart et al Adv. Appl. Sci. Res.,, (5): [8] Varshne, C.L. Ind. J. Pre and Appl. Maths, Vol., p. 558,