Numerical Analysis of Mixed Convective Peristaltic Flow in a Vertical Channel in Presence of Heat Generation without using Lubrication Theory

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1 Journal of Applid Fluid Mchanics, Vol. 10, No. 6, pp , Availabl onlin at ISSN , EISSN DOI: /acadpub.jafm Numrical Analysis of Mixd Convctiv Pristaltic Flow in a Vrtical Channl in Prsnc of Hat Gnration without using Lubrication Thory B. Ahmd, T. Javd, A. H. Hamid and M. Sajid Dpartmnt of Mathmatics and Statistics, Intrnational Islamic Univrsity Islamabad Pakistan Corrsponding Author bilalmaths7@yahoo.com (Rcivd April 24, 2017; accptd August 8, 2017) ABSTRACT In this papr, hat transfr analysis of pristaltic mixd convction flow through a vrtical channl is prsntd in addition, ffcts of hat gnration ar also invstigatd. Th mathmatical modl is rprsntd by th systm of non-linar partial diffrntial quations. Th analysis is mad in th prsnc of non-zro wav and Rynolds numbrs. Th rsults of th long wavlngth assumption in a crping flow can b dducd. Ths rsults thus prdict nw faturs in th pristaltic transport in th absnc of th approximation of long wav lngth and low Rynolds numbr. Th modrat finit lmnts basd tchniqu has bn usd to comput th highly accurat solution of th govrning problm. To nsur th accuracy of th computd solution, th rsults obtaind ar validatd against th availabl rsults in th litratur and found good agrmnt. Th obtaind rsult ar prsntd through graphs and th influnc of involvd prtinnt paramtrs is analyzd. Kywords: Mixd convction; Pristaltic flow; Hat gnration; Numrical analysis; Non-zro Rynolds numbr. 1. INTRODUCTION Importanc of fluid mchanics in industrial and tchnological applications attractd th attntions of scintists, nginrs and mathmatician to study th fluid bhavior in diffrnt gomtris. Th manufacturing procsss lik glass fibr production, shts production, hot rolling, continuous casting, xtrusion procss, coating and papr production tc. involvs fluid mchanism. Furthrmor, advancmnts in fluid mchanics also assists in dvloping of hating and vntilating systms for industrial and domstic purposs. Th dsigning of piplin systms also rquirs th basic undrstanding of fluid flows. In th fild of mdical scinc, th importanc of fluid mchanics incrass significantly as it is usful in dsigning of blood substituts, artificial harts, hart-lung machins, brathing aids and othr such typ of dvics, which all obys th basic principls of fluid mchanics. Th flow gnratd by sinusoidal motion of th boundary of a vssl is known as pristaltic flow. Pristalsis is a common mchanism found in physiological and industrial procsss. Th most common xampls of physiological flows involving pristalsis includ flow of urin from kidny to th bladdr, flow of chym in small intstin, blood flow through capillaris, sprmatic fluid transport in fmal rproductiv tract tc. Th transport in corrosiv fluid in th nuclar industry, diabts pumps and pharmacological dlivry systms involv pristaltic mchanism. In th litratur diffrnt aspcts of th pristaltic flow ar discussd analytically and xprimntally. Th prsnc of non-linarity du to th finit Rynolds numbr maks thortical invstigations mor difficult. Many analytical studis involvs prturbation tchniqus through which an xplicit form of th solution is obtaind that hlps in undrstanding physical ffcts of th paramtrs. Pristaltic mchanism first studid by Latham (1966) both thortically and xprimntally. This work opn nw vnus of rsarch to undrstand pristaltic mchanism in th prsnc of nw aspcts. A numbr of articls (Winbrg t al. 1971; Lw t al. 1971; Ali t al. 2010; Manton 1975; Jaffrin 1973; Dharmrndra Tripathi 2013; Rao and Usha 1995; Narahari and Srnadh 2010; Shapiro 1967) addrssing th pristaltic phnomnon for diffrnt typ of gomtris undr th assumption of long wavlngth and low Rynolds numbr, small amplitud ratio and wav numbr ar availabl. Th lubrication thory was usd by Shapiro t al. (1969) to invstigat th pristaltic phnomnon in a two dimnsional channl using a wav fram of

2 rfrnc. On th othr hand, Fung and Yih (1968) usd fixd fram of rfrnc for invstigating pristaltic flow without using lubrication approach. Th non-linar trms gt vanishd through th approach by Shapiro t al. (1969) and problm bcom simplr on comparison to Fung and Yih (1968) approach. In this approach ffcts of Rynolds numbr and wav numbr (rats of charactristic lngth of th channl/tub to th wav lngth of th pristaltic wav) on various flow charactristics cannot b invstigatd. Du to simplicity of th approach, this thory is usd widly to study th pristaltic flow of Nwtonian and non- Nwtonian fluid in numbr of scnarios. Jaffrin (1973) discussd th prturbation solution of two dimnsional pristaltic flow in a channl. Zin and Ostrach (1970) also discussd th pristaltic mchanism. Takabatak (1989) and Ayukawa (1990) mployd th finit diffrnc mthod to numrically simulat two-dimnsional pristaltic flow in a channl for modrat valus of th Rynolds numbrs and wav numbrs. Thy showd that th prturbation solution of Jaffrin (1973) but Zin and Ostrach (1970) prsntd th rsult in a narrow rang and in accordanc of Rynolds numbr th rflux phnomnon dos not caus th chang th whol situation in th flow. Th application of immrsd boundary tchniqu to simulat th transport solid particl in a two-dimnsional channl by pristalsis was initiatd by Fouci (1992). Mkhimr (2008) discussd th magntohydrodynamic ffct in th natural limitation of pristaltic phnomnon, which is not valid for modrat valus of both Rynolds numbr and wav numbr. Considrabl invstigation for pristaltic transport pumping problm ar formulatd (Ekstin 1970; Hung and Brown Sajid t al. (2015) invstigatd th mixd convction bhavior of an Oldroyd-B fluid. Vajravlua t al. (2007) studid hat transfr analysis in pristaltic flow through a vrtical porous annulus undr long wavlngth approximation. A not on pristaltic transport in an asymmtric channl with hat transfr is prsntd by Srinivas t al. (2008). In axisymmtric, channl mixd convctiv hat and mass transfr analysis wr prsntd by Srinivas t al. (2011). In both studis, thy apply th long wavlngth approximation. Mixd convction in pristalsis transport modl nowadays gain dp intrst by th rsarchrs, so th articls rlatd to pristaltic motion in diffrnt physical situation of mixd convction ar found in litratur (Sayd t. al 2015; Tanvr t al. 2017; Hayat t al. 2017; Sucharitha 2017). K. Ramsh (2016) rcntly studid th pristaltic flow in th prsnc of slip and convctiv conditions through porous mdium. H obtaind simplification using th lubrication thory. Rcntly, Hamid t al. (2017) studid th pristaltic flow numrically for two dimnsional non- Nwtonian fluid without mploying th lubrication thory. Th aim of this articl is to analyz th mixd convction in pristaltic flow in a vrtical channl in th prsnc of hat gnration and stramlin curvatur ffcts by dropping th assumption of long wavlngth and low Rynolds numbr. First, w dvlop th problm in wav fram by using formulation of stram function and vorticity. Th prssur is liminatd by cross diffrntiation and discussd by xplicit xprssion givn in lattr part. Th numrical solution of th problm is computd with th hlp of finit lmnt mthod which giv us th librty to st high valus of Rynolds numbr and othr suitabl valus of th involvd paramtrs. Th rsult ar discussd in th last part with diffrnt vrsions of prsntation. 2. PROBLEM DEVELOPMENT Considr th motion of Nwtonian fluid through a vrtical channl having innr width siz 2. Th flow is assumd in such a way that propagation of wavs is along th x-axis with vlocity c and y is along normal to th channl. Th pristaltic wall of channl ar assumd at som amount of tmpratur and obys th sinusoidal wav shap rprsntd by 2 ( X ct) HX, tab cos, (1) whr b is wav amplitud, rprsnts wav lngth and h is th symbol for man distanc of th wall from th cntral axis as shown in Fig. 1. Th coordinats of vlocity, prssur and tmpratur in fixd and moving fram of rfrnc ar rlatd by st of following xprssion * * * x X ct, y Y, u Uc, * * * v V, p p, T T, * * 1 (2) whr ( u, v ) and ( UV, ) rprsnt th componnts of vlocity vctors in moving fram of rfrnc and fixd fram of rfrnc rspctivly. Fig. 1. Th gomtry of considrd twodimnsional pristaltic channl. In fixd fram of rfrnc, th continuity, momntum and nrgy quations for th assumd problm ar 1814

3 U V 0, X Y U U U p U V t X Y X U U g 0, T T T X Y V V V p U V t X Y Y X Y V V, U T T cp U V t X Y * T T Q 0. X Y (3) (4) (5) (6) in which is th viscosity is th dnsity, g is th acclration causd by gravity, T is thrmal xpansion cofficint, cp is spcific hat at constant * prssur, is th thrmal conductivity and Q 0 rprsnts constant hat gnration with in th flow domain. Th boundary conditions of th problm ar U T V 0, 0, 0 at Y 0, Y Y H U 0, V, T T1 at Y H. T (7) (8) Equations that govrns th flow in wav fram ar as follows * * u v 0, * * x y * * * * u * u p u v * * * x y x 2 * 2 * u u * g *2 *2 T T T0, x y (9) (10) * * 2 x ( x ) a bcos, (13) and th no slip and th symmtry condition on th plans =0 and = rspctivly can b xprssd as follows * * * u T * v 0, 0, 0 at y 0, * * y y * * * 2 b 2 x u c, v sin, * * * T T1 at y ( x ). (14) (15) As both plans =0 and =, prsnt stramlin and flow rat is constant in cross sction of th channl in wav fram, thrfor * 0 on * * q on * y 0, * y ( x ), (16) whr = h is th rlation btwn flow rat in th wav fram of rfrnc and th tim man flow in th laboratory fram. Dfining nw variabls * * * * x y u v a x, y, u, v,, a c c * 2 * hx ( ) a * * h, p p ( x ),, a c h * * * * q T T0,, q,, ca c / a ch T1 T0 ca ga T1 T0 R, Gr, c 2 c Qa Pr,. * * k ( T T ) 0 1 Eqs., (9) (12) taks th form 2, x y 0 (17) * * * * v * v p u v * * * x y y 2 * 2 * v v, *2 *2 x y * * 2 * 2 * * T * T * T T cp u v Q * * *2 *2 0 x y x y (12) (11) 2 R Gr, y x x y y 2 RPr, y x x y (18) (19) whr is Rynolds numbr, is wav numbr, is Grashof numbr, is hat gnration paramtr and is Prandtl numbr. Th boundary x, y ar as follow conditions in trms of Th configuration of th pristaltic wall can b writtn as 1815

4 2 2 0, 0, y 0, =0 at y 0, x x q, 1, 1, y =2 sin2 x at y ( x), x whr u,, y x y v u v and =. x x y 3. NUMERICAL ANALYSIS (20) (21) In th prvious sctions th obtaind quations ar valid for non-zro Rynolds and wav numbrs and cannot b transfrrd to ordinary diffrntial quations. Finit lmnt mthod has bn prfrrd in th currnt invstigation du to its svral advantags ovr othr mthods gnrally usd to solv numrical problms. Th formost advantag of finit lmnt mthod is that it works with nonuniform msh which rsults in mor accurat numrical approximation particular whn you ar daling with complx gomtris and irrgular boundary conditions. Providing th usr librty to choos th shap functions and typs of lmnts according to th problm. This approach rturns mor accurat solutions with lss computational cost. To dal with th partial diffrntial quations finit lmnt mthod of Glrikin s approach is applid to solv Eqs. (17)-(19) subjct to th boundary conditions (20) in a finit rgion of numbr of wavs in a wav fram having two nd sctions with on fixd and othr moving boundary. Bcaus of th continuity of th flow w considr singl wav at a tim and thn follow th nxt and so on. In all th cass, th computd rsults ar highly convrgnt and satisfy th tolranc of , 10 and T 10. Th numrical computations ar mad by using nonuniform mshing of quadratic lmnts with th hlp of pdtool availabl in MATLAB. Th stram function, vorticity as wll as tmpratur functions ar approximatd by n n n Nk k Nk k Nk k k1 k1 k1 (22),,, in which k, k and k ar lmnt nodal approximation of, and. Thn Galrkin finit lmnt is ndorsd to govrning Eqs. (17)- (19) as follows 2 w1 d 0, x y (23) R y x x y w2 0, d (24) 2 Gr x y y RPr y x x y w3 0, d (25) 2 x y whr ww 1, 2 and w3 ar wight functions. Aftr simplifying Eqs. (23)-(25), w obtain 2 w1 w1 d x x y y w d w d 1 1, n Rw2 d y x x y 2 w2 w2 d x x y y Grw2 d w2 d, y n RPr w3 d y x x y 2 w3 w3 d x x y y w3 dw3 d. n (26) (27) (28) By considring th discrtizd domain, w plugin Eq. (22) in (26)-(28) to gt i k kii kii n, i (29) B A S k Aki i R Ckij i i Gr B1 ki i S n, (30) i i i k k Aki i RPr Ckij i i Sn S, (31) i whr i 2 Nk Ni Nk N A i ki d, x x y y (32) N B, 1 i ki NNd k i B ki Nk d, y N N i j Nj N C i kij Nk d, y x x y 1816

5 k k S Nkd and Sn NkSkd. Th global systm in form of matrix is dfind as KA = F, (33) in clos agrmnt with corrsponding rsult of Srinivas t al. (2011). Th solution of Srinivas t al. (2011) is analytical approximation computd by prturbation mthod. whr Bki A 0 ki Kij Aki RCkiji GrB1 ki, 0 RPrCkiji A ki k k S n Ak k, F 0. k k k Sn S (34) Th obtaind non-linar quations ar thn solvd by using famous Nwton-Raphson mthod. Th solution procss is itratd unlss th rror btwn two conscutiv itrations is bcom not lss than Fig. 2. Comparison of vlocity profil for prsnt rsults with Srinivas t al. (2011). 4. ANALYSIS OF THE PRESSURE Sinc th flow is causd by th infinit train of sinusoidal wavs of sam priod, thrfor valuat th prssur at th cntral part of considrd domain which covrs th rgion of on wav lngth, Th dimnsionlss form of th xprssions for prssur gradint is obtaind from two dimnsional form of Naivr-Stoks quations for stady flow as (a) P R x 2 y x x y y y Gr. (35) P 2 R y 2 x y x y x. x (36) Th prssur-ris pr wav lngth in th wav fram ar dfind as dp p dx. (37) dx 0 5. RESULTS AND DISCUSSION In this sction, w analyzd th rsults of th modlld problm in trms of vlocity distribution, hat charactristics, trapping phnomnon and ris in prssur against diffrnt involvd flow paramtrs. 5.1 Vlocity Fild and Tmpratur Charactristics A comparison of th obtaind numrical rsults with th analytical rsults of Srinivas t al. (2011) in th spcial cas ar prsntd in Fig. 2. It is obsrvd that our rsults in th limiting cas ( = 0, = 0 ar (b) Fig. 3. (a) Longitudinal vlocity distribution and (b) tmpratur profil for diffrnt valus of whn othr paramtrs ar fixd at =., =., =., =., =., =.. Th longitudinal vlocity and th tmpratur profil ar prsntd in th Figs. 3 6 to s th ffct of Rynolds numbr, tim man flow rat, Grashof numbr and hat gnration paramtr. From ths figurs, it is obsrvd that both vlocity and tmpratur achiv maximum in th vicinity of th cntr of th channl. Morovr, parabolic bhavior by vlocity and tmpratur graphs is obsrvd in al cass. In Fig. 3(a), w obsrv th ffct of Rynolds numbr on vlocity distribution. W s that nar th cntr of th channl, incras in causs a dcras in vlocity whil an opposit bhavior is obsrvd at th wall. So, it prdicts that dominant inrtial ffcts 1817

6 to viscous forcs in th cntr of th channl causs dcras in vlocity of th fluid whil in th rgion of th wall dominanc of inrtial forcs nhancs th vlocity. Fig. 3(b) illustrats that an incras in riss th tmpratur ovr th whol cross-sction. This obsrvation is not rportd in arlir studis and it attributd to strong inrtial ffcts inducd for larg valus of Rynolds numbr. Morovr, long wavlngth and low Rynolds numbr thory is not abl to prdict such non-linar ffct. natural. Hnc Grashof numbr also hlps to control th hat in th fluid flow with hat gnration paramtr and its ffcts ar opposit to th ffct of hat gnration paramtr. (a) (a) (b) Fig. 5. (a) Longitudinal vlocity distribution and (b) tmpratur profil for diffrnt valus of whn othr paramtrs ar fixd at =, =., =., =., =.,=.. (b) Fig. 4. (a) Longitudinal vlocity distribution and (b) tmpratur profil for diffrnt valus of whn othr paramtrs ar fixd at =, =., =., =., =.,=.. Figurs 4(a) and 4(b) show th ffct of tim man flow rat on vlocity distribution and tmpratur profil, rspctivly. Both figurs show that th ris in volum flow rat nhancs longitudinal vlocity and tmpratur profil. Fig. 5(a) prsnts th ffct of Grashof numbr on longnitudanl vlocity profil. W s that th bhavior of vlocity du to Grashof numbr at th wall and nar th cntr is diffrnt. In th rgion 00.2, w obsrv that dominanc of buoyancy forcs rducs th vlocity but aftr this rgion 0.2, incras in vlocity is obsrvd by sam bhavior of. Fig. 5(b) xhibits th dcrasing bhavior in tmpratur distribution by nhancing in th whol domain. As incras in Grashof numbr corrsponds to nhanc th buoyancy forcs causd by tmpratur diffrnc, which suffrrs its ffct in th cntr of th channl and so causs th dcras in ris in tmpratur. In Fig. 6(a), w obsrv that for th vlocity distribution, hat gnration paramtr xhibits sam bhavior as in cas of Grashof numbr but in Fig. 6(b) tmpratur incrass du to incras in th hat gnration paramtr, which is quit (a) (b) Fig. 6. (a) Longitudinal vlocity distribution and (b) tmpratur profil for diffrnt valus of whn othr paramtrs ar fixd at =, =., =., =., =., =

7 variation of Rynolds numbr. It is obsrvd that siz of bolus magnifis with th incras in Rynolds numbr and as far as th numbr of bolus ar concrn, w obsrv that numbr of boluss also incrass with incras in. It is du to th fact that incras in inrtial ffcts causs ris in vlocity profil of th fluid so thsolution surfac attains mor hight. Fig. 8 shows th variation of stramlins for diffrnt valus of Grashof numbr. It is noticd that th trappd bolus xits nar th cntral rgion of th channl for smallr valus of Grashof numbr. Morovr, it movs in lowr rgion of th boundary wall with incras in Grashof numbr as shown in Fig. 8. Fig. 7. Variation of stramlins in wav fram of rfrnc for diffrnt valus of with fixd valus of =.,=.,=., =., =., = Trapping and Vorticity Th bhavior of stramlins in wav fram for stationary wall is mostly similar as that of wall but somtim situation ariss that pattrn of stramlins can split and ncloss a bolus of fluid particls in closd stramlins form so that circular rgion is cratd. On th othr hand, in th fixd fram, th wavs trappd th fluid bolus and traps it with spd of wav. To xamin th variation of stramlins, w plottd contours of stramlins for diffrnt valus of th paramtr involvd as shown in Figs Figur 7 shows th bhavior of stramlins with Fig. 8. Variation of stramlins in wav fram of rfrnc for diffrnt valus of with fixd valus of =, =., =., =., =., =

8 incrasing in Rynolds numbr, isothrmal lins shows nhancd curvatur ffcts and formation of bolus appars whr complt bolus is dvlopd in th cntral rgion for = 12. Fig. 13 shows that incras in Grashof numbr incrass curvatur ffcts nar cntral wall and shows sam smoothnss of curvatur nar th pristaltic wall in all cass. Fig. 14 xhibits th ffct of incrasing hat gnration paramtr on isothrmal lins. It is obsrvd that isothrmal lins congrgatd at cntral wall and movs towards th wall channl. Variation in isothrmal lins for diffrnt valus of Prandtl numbr may b sn in Fig. 15. Fig. 9. Variation of stramlins in wav fram of rfrnc for diffrnt valus of with fixd valus of =, =., =., =., =., =.. Figur 9 provids th bhavior of hat gnration paramtr. From this figur, w obsrv th incras in th siz of bolus with incras in hat gnration paramtr. Fig. 10 shows incras in th siz of trapd bolus with incras in wav numbr. It is also obsrvd that for small, th bolus ar trappd nar th wall but whn w incras th wav numbr, th bolus movs to th cntral part of th rgion. Fig. 11 illustrats th ffct of tim man flow rat on th stramlins. W obsrvd a rapid incras in th numbr and siz of bolus with small incras in th flow rat. In Figs , w plottd th isothrmal lins for diffrnt flow paramtrs. In Fig. 12, w hav obsrvd that by Fig. 10. Variation of stramlins in wav fram of rfrnc for diffrnt valus of with fixd valus of =, =., =., =., =., =.. W can s that mor curvatur ffct in cntral rgion with mor numbr of bolus by incrass th valu of. Fig. 16 shows dcras in curvatur of isothrmal lins with incras in th wav numbr. 1820

9 Fig. 11. Variation of stramlins in wav fram for diffrnt valus of with fixd valus of =, =., =., =., =.,=.. Fig. 12. Variation of Isothrmal lins in wav fram of rfrnc for diffrnt valus of with fixd valus of =., =., =., =., =., =

10 B. Ahmd t al. / JAFM, Vol. 10, No.6, pp , Fig. 13. Variation of Isothrmal lins in wav fram of rfrnc for diffrnt valus of valus of =, =., =., =., =., =.. with fixd Fig. 14. Variation of Isothrmal lins in wav fram of rfrnc for diffrnt valus of valus of =, =., =., =., =., =.. with fixd 1822

11 B. Ahmd t al. / JAFM, Vol. 10, No.6, pp , Fig. 15. Variation of Isothrmal lins in wav fram of rfrnc for diffrnt valus of valus of =, =., =., =., =., =.. with fixd Fig. 16. Variation of Isothrmal lins in wav fram of rfrnc for diffrnt valus of valus of =, =., =., =., =., =.. with fixd 1823

12 B. Ahmd t al. / JAFM, Vol. 10, No.6, pp , Fig. 17. Variation of Isothrmal lins in wav fram of rfrnc for diffrnt valus of with fixd valus of =, =., =., =., =., =.. Fig. 18. Vorticity in wav fram of rfrnc for diffrnt valus of with fixd valus of =., =., =., =., =., =.. In Fig. 17, w s th ffct of tim man flow rat on isothrmal lins which xhibits that incras in this paramtr causs th bnding of isothrmal lins in th cntral part of th channl towards uppr on rgion. Th ffct of Rynolds numbr vorticity ar prsntd in Fig. 18. It can b obsrvd that vorticity is gnratd at th pristaltic wall and diffuss to th cntral rgion of th channl with incras in th valus of Rynolds numbr. 5.3 Prssur Fild Analysis Th prssur ris pr wav lngth has bn plottd in Figs against tim man flow rat for diffrnt rang of valus of involvd paramtrs. Thr ar following four typs of flow rgion. Whn > 0 which th flow is in rgions Δ > 0 and mans, in quadrant I thn it is Pristaltic/Positiv pumping. Fig. 19. Prssur ris pr wav lngth for varis valu of R. 1824

13 Fig. 20: Prssur ris pr wav lngth for varis valu of Pr. which mans, in quadrant IV thn it is fr pumping rgion. Prssur gradint is known as advrs prssur if Δ >0 and favorabl prssur gradint if Δ <0. In Fig.(19), th ffct of Rynolds numbr on prssur ris pr wav lngth has bn shown. W noticd that incras in causs incras in th prssur du to th dominanc of inrtial forcs ovr viscous forcs. It is du to th fact that whn w incras, it nhancs th inrtial ffct as compard to viscous ffcts, hnc mor prssur is rquird to maintain th flow in th channl. W furthr obsrvd that whn tim man flow rat approachs to 1, all th lins coincids which rflcts th sam amount of prssur at that point. It is obsrvd from Figs. (20) (22) that incras in Prandtl numbr Pr, Grashof numbr and hat gnration paramtr drops prssur ris pr wav lngth but in th cas of amplitud ratio and wav numbr, opposit bhavior is sn in Figs. (23) and (24) rspctivly. Fig. 21. Prssur ris pr wav lngth for varis valu of Gr. Fig. 23. Prssur ris pr wav lngth for varis valu of. Fig. 22. Prssur ris pr wav lngth for varis valu of. Whn th flow is in rgions Δ >0 and <0 which mans, in quadrant II thn it is Rtrograd/Backward pumping. Whn th flow is in rgions Δ <0 and <0 which mans, in quadrant III thn it is Co-pumping. Whn th flow is in rgions Δ <0 and <0 which mans, in quadrant IV thn it is augmntd pumping. Also whn th flow is in rgions Δ =0 and >0 Fig. 24. Prssur ris pr wav lngth for varis valu of. 6. CONCLUSION Th hat and mass transfr analysis for pristaltic flow in a vrtical channl has bn prsntd. Th govrning quations ar modlld in th absnc of long wav lngth approximation which allows 1825

14 us to obsrv th ffct of all th paramtr with modrat valus. Th numrical solutions for stram function, prssur ris pr lngth as wll as tmpratur profil ar obtaind. Th ffct of paramtrs on th vlocity, hat transfr, and th trapping du to th pristaltic wall ar discussd in dtail. From th analysis th main outcoms for diffrnt flow charactristics ar summarizd. Incras in hat gnration rducs th vlocity nar th cntral rgion and improvs th vlocity in nighbor of pristaltic wall. It also nhancs th siz of bolus and curvatur ffct on isothrmal lins and drops th prssur. Incras in th Grashof numbr causs fall in vlocity nar th cntral rgion and riss th vlocity nighbor of pristaltic wall and drops th prssur. Incras in tim man flow rat supports th nhancmnt of vlocity, tmpratur and siz of bolus. Incras in wav numbr incrass siz of bolus, rducs curvatur ffct of isothrms and riss th prssur. REFERENCES Ali N., M. Sajid, T. Javd and Z. Abbas (2010). Hat transfr analysis of pristaltic flow in a curvd channl. Int. J. Hat and Mass Transfr 53, Dharmndra, T. and O. Anwr Bg (2013). Study of transint pristaltic hat flow through a finit porous channl. Math. and Computr Modling 57, Ekstin, E. C. (1970). Exprimntal and thortical prssur studis of pristaltic pumping SM. Thsis, Dpt. Mch. Eng., Massachustts Institut of Tchnology, Cambridg, MA. Fauci Lisa, J. (1992).Pristaltic pumping of solid particls. Comp. Fluid 21, Fung, Y. C. and C. S. Yih (1968). Pristaltic Transport. J. of Appl. Mch Hamid, A. H., Tariq Javd, B. Ahmd and N. Ali. (2017). Numrical study of two-dimnsional non-nwtonian pristaltic flow for long wavlngth and modrat Rynolds numbr. Brazilian Socity of Mchanical Scincs and Enginring. Hayat, T., S. Farooq, A. Alsadi and B. Ahmad (2017). Numrical study for Sort and Dufour ffcts on mixd convctiv pristalsis of Oldroyd 8-constants fluid. Intrnational Journal of Thrmal Scincs 112, Hung, T. K. and T. D. Brown. (1975). Solid-Particl motion in two-dimnsional pristaltic flows. Journal of Fluid Mchanics 73, Jaffrin, M. Y. (1973). Inrtia and stramlin Curvatur Effcts on Pristalsis. Intrnational Journal of Enginring. Scinc Latham T. W. (1966). Fluid motion in a pristaltic pump. MIT, Massachustts. Lw, H. S., Y. C. Fung and C. B. Lownstin (1971). Pristaltic carrying and mixing of chym in th small intstin. Journal of Biomchanics 4, Manton, M. J. (1975). Long-Wavlngth Pristaltic pumping at low Rynolds numbr. J.of Fluid Mch. 68(3), Mkhimr, Kh. S. (2008). Pristaltic flow of a magnto-micropolar fluid, Effct of inducd magntic fild. Intrnational J. of Appl. Mathmatics. Narahari, M. and S. Srnadh (2010). Pristalatic transport of a Bingham fluid in contact with a Nwtonian fluid. Intrnational J. of Applid Math. Mchanics 6(11), Ramsh, K. (2016). Effcts of slip and convctiv conditions on th pristaltic flow of coupl strss fluid in an asymmtric channl through porous mdium. Computr Mthods and Programming in Biomdicn 135, Rao, A. R. and S. Usha (1995). Pristaltic transport of two immiscibl viscous fluid in a circular tub. J. of Fluid Mchanics 85, Sajid, M., B. Ahmd and Z. Abbas (2015). Stady mixd convction stagnation point flow of MHD Oldroyd-B fluid ovr strtching sht. Journal of th Egyptian Mathmatical Socity 23(2), Sayd, H. M., E. H. Aly and K. Vajravlu (2016). Influnc of slip and convctiv boundary conditions on pristaltic transport of non- Nwtonian Nano-fluids in an inclind asymmtric channl. Alxandria Eng. J. 55 (3) Shapiro, A. H. (1967). Pumping and rtrograd diffusion in pristaltic wavs. In Proc. Workshop Urtral Rftm Childrn, Nat. Acad. Sci., Washington, DC, 1, Shapiro, A. H. M. Y. Jaffrin and S. L. Winbrg (1969). Pristaltic Pumping with Long Wavlngth at Low Rynolds Numbrs. Journal of Fluid Mchanics 37, Shapiro, A. H., M. Y. Jaffrin and S. L. Winbrg (1969). Pristaltic Pumping with Long Wavlngth at Low Rynolds Numbrs. Journal of Fluid Mchanics 37, Srinivas, S. and M. Kothandapani (2008). Pristaltic transport in an asymmtric channl with hat transfr: A not, Intrnational Communications in Hat and Mass Transfr 35 (4), Srinivas, S., R. Gayathri and M. Kothandapani. (2011). Mixd convctiv hat and mass transfr in an asymmtric channl with pristalsis. Comm. in Nonlinar Scinc and numrical simulation 16, Sucharitha, G., P. Lakshminarayana and N. Sandp. (2017). Joul hating and wall flxibility ffcts 1826

15 on th pristaltic flow of magntohydrodynamic nanofluid. Intrnational Journal of Mchanical Scincs 131, Takabatak, S., K. Ayukawa and M. Sawa (1989). Finit lmnt Analysis of Two dimnsional Pristaltic flow (1st Rport, Finit lmnt solution). Japan Socity of Mch. Eng. 53, Tanvr, A., T. Hayat, A. Alsadi and B. Ahmad. (2017). Mixd convctiv pristaltic flow of Sisko fluid in curvd channl with homognous-htrognous raction ffcts. Journal of Molcular Liquids Vajravlu, K., G. Radhakrishnamacharya and V. Radhakrishnamurty (2007). Pristaltic flow and hat transfr in a vrtical porous annulus with longwav approx. Intr. J. of Non-Linar Mch. 42, Winbrg, S. L., E. C. Eckstin and A. H. Shapiro (1971). An xprimntal study of pristaltic pumping. Journal of Fluid Mchanics 49, Zin, T. F. and S. A. Ostrach (1970). A Long wavlngth Approximation to Pristaltic Motion. Journal of Biomch