Convective boundary conditions effect on peristaltic flow of a MHD Jeffery nanofluid

Transcription

1 Appl Nanosci (6) 6: DOI.7/s ORIGINAL ARTICLE Convective bondar conditions effect on peristaltic flow of a MHD Jeffer nanoflid M. Kothandapani J. Prakash Received: 6 Janar 5 / Accepted: 9 March 5 / Pblished online: 7 April 5 The Athor(s) 5. This article is pblished with open access at Springerlink.com Abstract This work is aimed at describing the inflences of MHD, chemical reaction, thermal radiation and heat sorce/sink parameter on peristaltic flow of Jeffer nanoflids in a tapered asmmetric channel along with slip and convective bondar conditions. The governing eqations of a nanoflid are first formlated and then simplified nder long-wavelength and low-renolds nmber approaches. The eqation of nanoparticles temperatre and concentration is copled hence, homotop pertrbation method has been sed to obtain the soltions of temperatre and concentration of nanoparticles. Analtical soltions for axial velocit, stream fnction and pressre gradient have also constrcted. Effects of varios inflential flow parameters have been pointed ot throgh with help of the graphs. Analsis indicates that the temperatre of nanoflids decreases for a given increase in heat transfer Biot nmber and chemical reaction parameter, bt it possesses converse behavior in respect of mass transfer Biot nmber and heat sorce/sink parameter. Kewords Peristalsis Chemical reactions Jeffer nanoflid Tapered asmmetric channel Convective bondar conditions & M. Kothandapani mkothandapani@gmail.com J. Prakash prakashjaavel@ahoo.co.in Department of Mathematics, Universit College of Engineering Arni, (A Constitent College of Anna Universit Chennai), Arni 63 36, Tamil Nad, India Department of Mathematics, Arlmig Meenakshi Amman College of Engineering, Vadamavandal 64 4, Tamil Nad, India Introdction Nowadas, the std of nanoflids flow has created significant interest becase of its wide ranging application in medical, biochemistr and indstrial engineering. The nanoflids are a new class of flids platform patterned b dispersing nanometer-sized materials in base flids. Choi (995) experimentall proved that the sspension of solid nanoparticles with tpical length scales of 5 nm with high thermal condctivit enhances the effective thermal condctivit and the convective heat transfer coefficient of the base flid. In his other work (Choi et al. ), it was also shown that the addition of a small amont (less than % b volme) of nanoparticles to conventional heat transfer liqids increases the thermal condctivit of the flid p to approximatel two times. An elaborated analsis of nanoflids examined b Bongiorno (6) was broght ot that this massive increase in the thermal condctivit occrs owing to the presence of two main effects sch as the Brownian diffsion and the thermophoretic diffsion of the nanoparticles. Masda et al. (993) described that the effective thermal condctivit of nanoflids is higher to enhance the heat transfer as compared to conventional heat transfer. This phenomenon sggests that the possibilit of sing nanoflids in advanced nclear sstems b Bongiorno and H (5). Mekheimer and Abd Elmabod (8) have pointed ot that the cancer tisses ma be destroed when the temperatre reaches 4 45 C. Some recent stdies of nanoflids are given in the references (Anoop et al. 9 Gorla and Hossain 3 Wang and Mjmdar 8 Kakaç and Pramanjaroenkij 9 Srinivasachara and Srender 4 Ellahi et al. 4). The field of peristaltic flow is another significant area, which has latel been paing attention of man researchers. Peristalsis is a mechanism, which is formed b sccessive

2 34 Appl Nanosci (6) 6: waves of contraction/expansion that pshes flid (or flidlike contents) forward. The first investigation as peristaltic motion was done b Latham (966). The phenomenon of peristalsis mechanism has been analzed in detail b varios researchers for different flids nder different conditions with references to phsiological and mechanical sitation (Fng and Yih 968 Shapiro et al. 969 Akbar and Nadeem Mekheimer Kothandapani and Srinivas 8a Vajravel et al. Haat et al. a, ). Recentl, attention (Akbar et al. a Srinivas and Kothandapani 9 Kothandapani and Srinivas 8b Srinivas et al. 9 Haat et al. b) has been extended on the std of peristaltic flow in the presence of heat transfer. At present, onl a few nmbers of stdies of peristaltic transport of nanoflids are available, however in the literatre, despite important applications in medical and indstrial engineering sstems. In view of the application of endoscope, peristaltic flow of a nanoflid between two concentric tbes has been investigated b Akbar and Nadeem (). The inflences of wall properties on the peristaltic flow of a nanoflid have been stdied b Mstafa et al. (). Akbar et al. b, c investigated peristaltic flow of a nanoflid with slip effects sing a homotop pertrbation method. The indicated that the pressre rise decreases with the increase in thermophoresis parameter, whereas increasing in the Brownian motion parameter and the thermophoresis parameter indces a rise in temperatre of the nanoflids. Mixed convection peristaltic flow of magnetohdrodnamic (MHD) nanoflid was analzed b Haat et al. (4a). Conseqence of their investigation (Haat et al. 4b), the peristaltic transport of viscos nanoflid in an asmmetric channel has been presented in accont of the convective conditions. Akbar et al. (4) stdied the wall-generated flid motion of a Newtonian nanoflid in an asmmetric channel in the presence of thermal and velocit slip effects. Nadeem et al. (4) examined the peristaltic flow of Williamson nanoflid in a crved compliant wall. More recentl, to simlate the intraterine flid motion in a sagittal cross section of the ters, the indced flid flow in a finite tapered asmmetric channel has been modeled (Etan et al. ). In spport of extension of or earlier works (Kothandapani and Prakash 5a, b, c the effect of chemical reaction and convective bondar conditions on the peristaltic flow of Jeffer nanoflids is taken into the accont. Even the std of peristaltic flow of an electricall condcting Jeffer nanoflid inclding the slip effect in the tapered asmmetric channel is also not available. In this paper, we have discssed the inflence of nanoflids on peristaltic transport of a Jeffer flid model nder the effects of slip, magnetic field, chemical reactions, heat sorce/sink parameter and thermal radiation parameter. The paper is arranged as: the mathematical formlation of the present problem is given in Mathematical formlation of the problem. In Soltions procedre, analtical soltions are evalated for the velocit, nanoparticles temperatre and concentration with the help of homotop pertrbation method. The graphical reslts of the problem are represented in Reslts and discssion. Final section contains Conclding remarks. Mathematical formlation of the problem We consider the peristaltic transport of incompressible non-newtonian nanoflids (Jeffer model) in an infinite two-dimensional tapered asmmetric channel. The tapered channel asmmetr is prodced de to non-niform channel having different amplitdes and phase difference with the same speed of peristaltic waves. We denote axial and transverse directions b X and Y, respectivel. Here, U and V are components of velocit in the axial and transverse directions, respectivel. Let Y ¼ H and Y ¼ H be the right and left bondaries of walls (see Fig. ). the ambient vales of T and C as tend to H are denoted b T and C and tend to H are denoted b T and C : These are defined b H ð X t Þ¼dm X a sin p ðx k ct Þþ/ ðaþ H ð X t Þ¼dþm X þ a sin p ðx k ct Þ ðbþ where d is the half-width of the channel at the inlet of flow, a and a are the amplitdes of right and left walls, respectivel, c is the phase speed of the wave, m ð\\þ is the non-niform parameter, k is the wavelength, the phase difference / is the phase difference which varies in the range / p when / ¼ then channel is ot of phase. Frther a a d and / satisf the condition k' U U = C = C α' Y = T T kh = hh ( T T ) Y B X H H c C km = hm ( C C ) Y a Fig. Geometr of the problem d λ φ d a C = Ck' U U = α ' Y T = T T kh = hh ( T T ) Y C km = hm ( C C ) Y Y

3 Appl Nanosci (6) 6: a þ a þ a a cosð/þðdþ : ðþ The expression for Jeffer nanoflids is T ¼pIþS ð3þ S ¼ l ð_c þ k c Þ : þ k where T and S are Cach stress tensor and extra stress tensor, respectivel, p is the pressre, I is the identit tensor, k is the ratio of relaxation to retardation times, k is the retardation time, l is the coefficient of viscosit of the flid, _c is the shear rate and dots over the qantities indicate differentiation with respect to time. The governing eqations the balance of mass, momentm, nanoparticle temperatre and nanoparticle volme fraction for an incompressible nanoflid nder the effect of chemical reaction, magnetic field, thermal radiation and heat sorce/sink parameter are given b (Akbar et al. b Mstafa et al. ) o U o X þ o V o Y ¼ ð4þ o q f ot þ U o o X þ V o U ¼ o P o Y o X þ o ðs ox X X Þþ o ðs oy X Y Þ r B U þðc Þq f agðt T Þþðq p q f Þb g ðc C Þ ð5þ o q f ot þ U o o X þ V o V ¼ o P o Y o Y þ o ðs ox X X Þþ o ðs oy X Y Þ ð6þ ðqc o T Þ f ot þ U o T o X þ V o T ¼ j o T o Y o X þ o T o Y þðqc o C o T Þ p D B o X o X þ o C o T oq r o Y o Y o Y þ Q " þ D T o T þ o # T ð7þ T m o X o Y o C ot þ U o C o X þ V o C o C ¼ D B o Y o X þ o C o Y þ D T o T T m o X þ o T o Y k ð C C Þ ð8þ where l is the coefficient of viscosit of the flid, the volmetric volme expansion coefficient c, T m is the flid mean temperatre, q f is the densit of the base flid, q p is the densit of the particle, j is the thermal condctivit of the nanoflids, o=ot represents the material time derivative, P is the pressre, T is the nanoparticle temperatre, C is the nanoparticle concentration, D B is the Brownian diffsion coefficient, D T is the themophoretic diffsion coefficient, k is the chemical reaction parameter, Q is the constant heat addition/absorption, the radioactive heat flx q r, a is the thermal expansion coefficient and b is the coefficient of expansion with concentration. Convective bondar conditions The appropriate bondar conditions are given b (Parti 994) pffiffiffi k U ¼ o U a o Y k o T h o Y ¼ h o C hðt T Þand k m o Y ¼ h m ðc C Þ at Y ¼ H ð9aþ pffiffiffi k U ¼ o U a o Y k o T h o Y ¼ h h T o C ð T Þand k m o Y ¼ h m ðc C Þat Y ¼ H ð9bþ where k is the permeabilit of the poros walls (Darc nmber), a is slip coefficient at the srface of the poros walls, h h and h m are the heat and mass transfer coefficients, respectivel, the thermal condctivit k h and the mass condctivit k m. Non-dimensional qantities We introdce the following non-dimensional qantities in Eqs. 4, 5, 6, 7 and 8, we get X x ¼ k ¼ Y d t ¼ ct k ¼ U c v ¼ V c d ¼ d k H h ¼ d h H ¼ d h ¼ T T a ¼ a T T d Sc ¼ v D B b ¼ a d m ¼ km d r ¼ C C R ¼ q f cd C C l G r ¼ ð C Þq f gad ðt T Þ Q d b ¼ cl ðt T Þmc p ¼ lc f j N b ¼ sd BðC C Þ N t ¼ sd TðC C Þ m T m sffiffiffiffi r M ¼ db ¼ 6r T3 K ¼ k l 3k lc f d B r ¼ ðq p q f Þgb d ðc C Þ cl ¼ h md s ¼ d pffiffi S k m lc k L ¼ da ¼ ow o op Rd w w x w x w ¼ p ¼ d P ckl ¼ h hd k h ox þ d o ð ox s xx and v ¼d ow ox : Þþ o o s x M ow o þ G rh þ B r r ðþ

4 36 Appl Nanosci (6) 6: Rd 3 op w w x þ w x w ¼ o þ o d ox s o x þ d o s ðþ Rd oh ot þ w oh ox dw oh x ¼ d o h o ox þ o h o h o þ o þ N b d or oh ox ox þ or " oh þ N t d oh þ oh # o o ox o ð3þ Rd Sc or ot þ w or ox þ dw or x o ¼ d o r ox þ o r o þ N t d o h N b ox þ o h o ð4þ where s xx ¼ d þ dk c þ k d s x ¼ þ dk c þ k d s ¼ d þ dk c ow o þ k d o ox ow o ow o o w o ox ox o oxo ow o ow o o w o ox ox o ow ox o o o d o w ox o w oxo and continit eqation is atomaticall satisfied. Using the long-wavelength approximation, neglecting the wave nmber, one can find from Eqs.,, 3 and 4 that op ox ¼ o o op o ¼ þ o w þ k o o h o þ N b M ow o þ G rh þ B r r or oh þ N t o o o r o þ N t o h cr ¼ : N b o ð8þ Frther, Eq. 6 indicates p 6¼ pðþ: From Eqs. 5 and 6, we have o o w o þ k o oh þb ¼ o M o w o þ G oh r o þ B or r o ¼ ð5þ ð6þ ð7þ ð9þ where M is the Hartmann nmber, K is the permeabilit parameter, R is the Renolds nmber, p is the dimensionless pressre, a and b are amplitdes of left and right walls, respectivel, Sc is the Schmidt nmber, d is wave nmber, m is the non-niform parameter, Renolds nmber R, m is the nanoflid kinematic viscosit, stream fnction w, h is the dimensionless temperatre, r is the dimensionless rescaled nanoparticle volme fraction, Prandtl nmber, b is the non-dimensional heat sorce/sink parameter, G r is the local temperatre Grashof nmber, B r is the local nanoparticle Grashof nmber, N b is the Brownian motion parameter, is the heat transfer Biot nmber, is the mass transfer Biot nmber, L is the slip parameter,n t is the thermophoresis parameter, the chemical reaction parameter c, Jeffre parameter k and the thermal radiation parameter : From Eq. 9a, 9b, we get, w ¼ F ow o ¼ L o w o oh o ¼ h and or o ðaþ ¼ r at ¼ h w ¼ F ow o ¼L o w o oh o ¼ ð hþ or o ¼ ð rþ at ¼ h ðbþ in which h and h represent the dimensionless form of the srfaces of the peristaltic walls, h ¼mx a sinðpðx tþþ/þ and h ¼ þ mx þ b sinðpðx tþþ: ðþ It is observed that the instantaneos average volme rate of the flow Fðx tþ periodic in ðx tþ (Srivastava et al. 983 Srivastava and Srivastava 988 Kothandapani and Srinivas 8c Gpta and Sheshadri 8 Kothandapani and Prakash 5c) as Fðx tþ ¼H þ a sin pðx tþþb sin½pðx tþþ/ ðþ in which F ¼ Z h h d: Soltions procedre The comptations of Eqs. 7, 8 are made throgh homotop pertrbation method (HPM) (He 3 Wazwaz Abbasband 6) with appropriate bondar condition Eq. a, b. For that, we write Hðh ~pþ ¼ð ~pþ ðlðhþlðh ÞÞ þ ~p oh or LðhÞþA o o þ A! oh þa 3 o ð3þ Hðr ~pþ ¼ð ~pþ ðlðrþlðr ÞÞ þ ~p LðrÞþ N t o h cr : ð4þ N b o Let s write h ¼ h þ ~ph þ ~p h þ ð5þ

5 Appl Nanosci (6) 6: r ¼ r þ ~pr þ ~p r þ ð6þ The soltions of temperatre and nanoparticle volme fraction phenomena (for ~p! ) are constrcted as h ¼ A 3 þ A þ A 9 þ A 3 þ A 4 6 þ ð h Þþ ðh h Þþ þ A 4ðh h Þ þ A 4ðh Þðh Þ ð7þ r ¼ A 8 þ A 5 þða 7 þ A 4 Þ þ ð h Þþ ðh h Þþ ða 4 N t A 7 A 8 N b cþ þ A 6 N b þ A þ A 6c 4 4 þ A 5c 5 : ð8þ Using Eqs. 7 and 8, the exact soltions of Eqs. 9 and 5 and with corresponding bondar conditions Eq. a, b are obtained as wðþ¼a þ 4A 6 N 8 þ A 8 N 6 þ A N 4 þ A þ 6A 7 þ N A 9 N 6 þ e N A 3 þ A 4 e N þ A 6 þ N A 8 N 6 þ A N þ 3 A 7 N 4 þ A 9 6N þ 4 A 6 N 4 þ A 8 þ A 6 6 N 3N þ A 7 5 N ð9þ ðþ ¼A N A 3 e N A 4 e N þ A 6 5 N þ A 7 4 þ A 8 3 þ A 9 þ A þ 6A 7 þ 4A 6 N 6 þ N 4 4A 6 3 þ 3A 7 þ A 8 þ A 9 ð3þ op ox ¼A N A A 6 5 A þ G r h þ B r r p ffiffiffiffiffiffiffiffiffiffiffiffiffi A A 9 ð3þ where N ¼ M þ k : The coefficient of heat transfer at the right wall is given b Z ¼ h x h : Reslts and discssion ð3þ To discss the inflences of varios parameters of interest on flow variables qalitativel sch as the nanoparticles velocit, temperatre, volme fraction and heat transfer coefficient, we have prepared the Figs., 3, 4, 5, 6, 7, 8, 9, :.877 : -.9 :.889 : ,,, 3, 4, 5, 6, 7, 8, 9,,,, 3, 4, 5 and 6 at the fixed vales of x ¼ :5 and t ¼ : The effects of Brownian motion and thermal radiation parameters on the amplitde of velocit are displaed in Figs. and 3. It is viewed that the profiles of axial velocit are parabolic in natre and the amplitde of velocit field increases at the center part of the channel with increasing Brownian motion and thermal radiation parameters. Figre 4 presents the inflence of chemical reaction parameter c with constant vales of other parameters. It shows that the effect of increasing c leads to decrease in velocit of nanoflids in the core part of the channel. The behavior of velocit distribtion for varios valed of the Hartmann nmber (M) is shown in Fig. 5. It is observed that an increase in M cases a decrease in magnitde of axial velocit. The instillation of magnetic nanoparticles in N b =. N =. b N b =.3 a=,b=.3,m=.3,θ=.8, =, =3,G r =.7,N t =, =,β=3,λ =., L=.,B r =.9, =4,φ = π,γ=,m=. Fig. Axial velocit profile ðþ for N b : : = R =. n R = n a=.,b=.,m=,θ=.6, =, =3,G r =,N b =.,N t =,β=,λ =.3, L=.,B r =.7, =,φ = π/4,γ=.,m=.5. Fig. 3 Axial velocit profile ðþ for :.959 : 85

6 38 Appl Nanosci (6) 6: : -.56 : 59. γ = γ = γ = :.4 : : -.75 :.75 m = -. m = m =. :.9943 : a=,b=.3,m=.5,θ=., =.3,λ =., =,G r =,N b =., =.5,β=.5, L=.,B r =.7, =,φ = π/,n t =.,M=. - Fig. 4 Axial velocit profile ðþ for c M = M = M = 3 a=,b=.3,m=,θ=.4, =.,λ =, =,G r =.7,N b =., =.5,β=, L=,B r =, =.,φ = π/3,γ=,n t =.. Fig. 5 Axial velocit profile ðþ for M.3 a=.3,b=,m=.,θ=.5, B =.8,λ =, h P =,G =,N =.5,R =.,β=., r r b n L=.5,B =.3,B =,φ = π/4,γ=,m=. r m - - Fig. 7 Axial velocit profile ðþ for m..8. : -585 :.8597 L = L =.5 L =. - - :.8446 :.879 a=,b=.,m=.,θ=.7, =, =,G r =.,N b =, =.,β=,λ =., N t =.3,B r =, =,φ = π/3,γ=.8,m=3. Fig. 8 Axial velocit profile ðþ for L : -4 :.7958 λ = λ =.3 λ = : 65 : :.774 : : -354 :.748 N t = N t = N t = a=.,b=.,m=.3,θ=.7, B =,N =.3, h t P =.8,G =,N =.,R =.3,β=, r r b n L=.,B =,B =,φ = π/5,γ=,m=. r m a=.,b=.3,m=.3,θ=.5, =4, =,G r =.8,N b =.3, =3,β=,λ =.3, L=.3,B r =, =3,φ = π/6,γ=,m= Fig. 6 Axial velocit profile ðþ for k glioblastoma mltiforme (GBM) patients indced the ptake of nanoparticles in macrophages to a major extent, and the ptake was frther promoted b magnetic flid Fig. 9 Axial velocit profile ðþ for N t hperthermia (MFH) therap (Landeghem et al. 9). Figre 6 is prepared to std the effect of Jeffer parameter ðk Þ on the magnitde of velocit field. It is seen that an

7 Appl Nanosci (6) 6: increase in k spports the amplitde of velocit of the flid near the channel walls, and then, the sitation is changed in the middle of the channel. The effect of m on amplitde of velocit distribtion is displaed in Fig. 7. It is expressed that the behavior of axial velocit near the channel walls and at center is not similar, also the maximm velocit of the nanoparticles alwas occrs at the heart part of the channel, decaing smoothl to zero at the peripher (channel wall). Perhaps, the axial velocit of the nanoflid in a niform channel is higher than non-niform channel. Figres 8 and 9 are inclded to std the effects of slip and thermophoresis parameters on the amplitde of velocit field. It is observed that an increase in L and N t spports the velocit of the flid near the channel walls, bt the sitation changes in the core part of the channel. The temperatre and volme fraction of nanoparticles distribtions are illstrated in Figs.,,, 3, 4, 5, 6, 7, 8, 9, and for varios vales of the parameters c and b. It is clear from Fig. θ a=.,b=,m=., =.75, φ = π/6, =,N b =, =,N t =3,β=3,γ =. - - Fig. Temperatre profile hðþ for.6 = =. =. θ γ = γ = γ = 4 a=.,b=.3,m=., =., φ = π/4,n t =.8,N b =, =, =,β =., =. - - θ = = = 3 a=.,b=.,m=.3, =, φ = π/5,n t =,N b =.5, =.8, =,β =,γ= Fig. 3 Temperatre profile hðþ for Fig. Temperatre profile hðþ for c 8 = = = = =.5 = θ 6 4 a=,b=.,m=.3, =, φ = π/3,n t =,N b =, =, =3,β = 4,γ=. - - θ.3. a=.,b=,m=.3, =, φ = π/,n t =,N b =, =.5, =,β=,γ =. - - Fig. Temperatre profile hðþ for Fig. 4 Temperatre profile hðþ for

8 33 Appl Nanosci (6) 6: β = β =. β =. a=.,b=,m=.,γ=., φ = π/3, =,N b =.5, =, =3,β=,N t =..7.3 θ.3 a=.3,b=,m=.3, =, φ = π/4,n t =.,N b =, =, =, =,γ = σ = = 3 = 5 Fig. 5 Temperatre profile hðþ for b Fig. 8 Nanoparticle volme fraction rðþ for a=,b=.3,m=., =, φ = π/4, =,N b =.5, =3, =,β=,n t = = = = a=.3,b=,m=., =, φ = π/4,γ=.8,n b =, =3, =,β=,n t =. σ. - σ Fig. 6 Nanoparticle volme fraction rðþ for c γ = γ =. γ = Fig. 9 Nanoparticle volme fraction rðþ for σ.7.3 a=.3,b=,m=., =3, φ = π/, =,N b =.3, =, =,γ=.3,n t =.7. σ -. - = =.5 =.5 a=.3,b=,m=.3,γ=., φ = π/3, =3,N b =.6, =, =,β=,n t = β = β =. β = Fig. 7 Nanoparticle volme fraction rðþ for b that decrease in chemical reaction lead to decrease in the temperatre of the nanoflid. The effects of increasing heat transfer Biot nmber on h are plotted in Fig.. We Fig. Nanoparticle volme fraction rðþ for notice that amplitde of the temperatre distribtion decreases as increases. The distribtion of temperatre is plotted against for the different vale of in Fig.. It

9 Appl Nanosci (6) 6: σ.8. a=.,b=,m=, =4, φ = π/5, =,N b =.3, =,γ=.3,β=.,n t =. Z 4 3 x:.35 Z: β = β = β = x:.8 Z: = R = n R = n a=.3, b=., m=.3, =, φ = π/4, =, N t =, =, =.3, γ=,n b = Fig. Nanoparticle volme fraction rðþ for Z x:.3333 Z: is renowned that the temperatre decreases with increasing radiation parameter ð Þ: The effect of mass transfer Biot nmber ð Þ on h is shown in Fig. 3. It is seen that the temperatre profiles are almost parabolic in natre and get increases as increases. One can see from Fig. 4 that as the vale of Prandtl nmber increases, the temperatre profile also increases. Figre 5 clearl indicates that the effect of b on temperatre field. Here, the temperatre profile is increased when b is increased. Figres 6, 7, 8, 9, and reveal that the concentration of nanoflid decreases when chemical reaction, heat sorce/sink parameter, heat transfer Biot nmber and Prandtl nmber are increased and it increases when there is an increase in the vale of mass transfer Biot nmber and thermal radiation parameter. The impacts of varios phsical parameters of heat transfer coefficient are shown in Figs., 3, 4 and 5. It is observed that the heat transfer coefficient is in oscillator behavior which ma de to contraction and expansion of the walls. The absolte vales of heat transfer coefficient x = B = h B = h a=, b=., m=.4, =., φ = π/4, =, N t =, =, β=, γ=,n b =. Fig. Heat transfer coefficient profile ZðxÞ for x:.867 Z: Fig. 3 Heat transfer coefficient profile ZðxÞ for b Z - - = = R = n increases with increase of heat transfer Biot nmber ð Þ and heat sorce/sink parameter ðbþ, while it get decreased with increasing thermal radiation parameter ð Þ and chemical reactionðcþ. An added interesting phenomenon in the peristaltic transport is trapping, and it is mainl the formation of an internall circlating bols of flid b the closed streamlines. This trapped bols pshed ahead along peristaltic waves and also the variation of circlating bols is represented for varios pertinent parameters. Figre 6a, b illstrates the inflence of chemical reaction ðcþ on the streamlines for the fixed vales of other parameters. It is observed that the size of the trapped bols is decreased on both walls of the channel with the increase in c. The effects of slip parameter ðlþ on trapping are shown in Fig. 6b, c. It is noted that the circlating trapped bols decreases in size and nmber with increase in the slip parameter. Figre 6c, d displas that the inflence of flow rate ðhþ on the streamline for fixed vales of other parameters. When x x:.3833 Z: x x:.867 Z: a=.3, b=., m=, =, φ = π/5, =., N t =.3, =, β=, γ=,n b =. Fig. 4 Heat transfer coefficient profile ZðxÞ for c

10 33 Appl Nanosci (6) 6: Z - - x:.3 Z: a=., b=.3, m=., =4, φ = π/3, =3, N t =, =, =., β=.3,n b =. x:.7667 Z: Fig. 5 Heat transfer coefficient profile ZðxÞ for x γ = γ = γ = 3 increase in H size of the trapped bols occrring at the walls increases. Figre 6d, e are plotted for varios vales of Hartmann nmber ðmþ on the streamlines. We notice that while Hartmann nmber is increased, the size of internal bols decreases. Figre 6e, f indicates that the trapped bols size is increased inside and nmber as nonniform parameter is increased. The comparisons between the analtical soltions obtained b sing HPM and nmerical soltions solved b emploing MATLAB throgh BVP command have also been made. It has been observed from Table that nmerical soltion in respect of the temperatre of the flid greatl agrees with the analtical soltion for the entire vales of width of the channel. Moreover, it has been noticed that or reslts in the limiting cases ðg r ¼ B r ¼ m ¼ Þ are in ver good agreement Fig. 6 Streamlines when a ¼ :3 b ¼ : / ¼ p=4 N t ¼ :5 N b ¼ :8 ¼ : ¼ :3 G r ¼ B r ¼ :5 b ¼ :3 ¼ ¼ :5 k ¼ t ¼ : a c ¼ :4 L ¼ : H ¼ :6 M ¼ m ¼ :5 b c ¼ 3 L ¼ : H ¼ :6 M ¼ m ¼ :5 c c ¼ 3 L ¼ :3 H ¼ :6 M ¼ m ¼ :5 d c ¼ 3 L ¼ :3 H ¼ :7 M ¼ m ¼ :5 e c ¼ 3 L ¼ :3 H ¼ :7 M ¼ 3m ¼ :5 and f c ¼ 3 L ¼ :3 H ¼ :7 M ¼ 3 m ¼ :4

11 Appl Nanosci (6) 6: Table Comparison between analtical soltion and nmerical soltion for different vale of when a ¼ :3 b ¼ : m ¼ : / ¼ p= ¼ N t ¼ :3 N b ¼ : ¼ b ¼ : c ¼ : ¼ : ¼ : x ¼ :3 and t ¼ :3 Present soltion Nmerical soltion Absolte error with the previos stdies (Kothandapani and Srinivas 8c) when slip parameter approaches to zero. Conclding remarks In this paper, a mathematical model to std the peristaltic transport of an electricall condcting nanoflid in a vertical tapered smmetric channel in the presence of slip effect, chemical reactions, thermal radiation and heat sorce/sink parameters has been presented. Under the assmptions of long-wavelength and low-renolds nmber, analtic soltions have been derived for the amplitde of velocit, temperatre, nanoparticles volme fraction and stream fnction. Interaction of varios emerging parameters with peristaltic transport is discssed. The main reslts can be smmarized as follows: The velocit of nanoflid decreases at the central part of channel when L and N t are increased as expected. The nanoparticles temperatre and volme fraction distribtions are decreased with the increase in chemical reaction and heat transfer Biot nmber. The volme of trapped channel decreases with increasing chemical reaction and slip parameter, bt it shows the opposite behavior with the non-niform parameter. Open Access This article is distribted nder the terms of the Creative Commons Attribtion License which permits an se, distribtion, and reprodction in an medim, provided the original athor(s) and the sorce are credited. Appendix A ¼ N b A ¼ N t A 3 ¼ b A 5 þ þ þ c ¼ ðh h Þþ A 6 ¼ cðh Þ ðh h Þþ A B h A 4 ¼A 3 ð ðh h ÞþÞ A ð ðh h ÞþÞð ðh h ÞþÞ A 5 ðh h Þ A 7 ¼ 6ð ðh h ÞÞ A 6ðh þ h Þ A 5ðh þ h þ h h Þ 6 A 8 ¼ h h ð3a 6 þ A 5 ðh þ h ÞÞ 6 þ ðh h ÞðA 6 þ A 5 ðh þ h ÞÞ 4 A 5ðh h Þðh h Þ ð ðh h ÞÞ A 9 ¼ A A 4 ðh þ h Þ ðh h Þþ A 7 A ðh h Þþ A 4 ðh þ h Þ ð ðh h ÞþÞ A 4 A ¼A ðh h Þþ þ A 6 ðh h Þþ A A 4 ðh h Þþ A A A 5 ¼ ðh h Þþ4 A ¼ A 9ðh þ h Þþ6A ðh þ h Þþ4A ðh 3 þ h3 Þ6A 9 ðh h ÞA ðh 3 h3 ÞA ðh 4 h4 Þ ðh h Þþ4

12 334 Appl Nanosci (6) 6: A 3 ¼ h h ða ðh þ h h þ h ÞþA ðh þ h Þþ6A 9 Þ ðh h Þðh h ÞðA þ A ðh þ h ÞÞ ð ðh h ÞÞ þ ðh h ÞðA ðh þ h h þ h Þþ3A ðh þ h Þþ6A 9 Þ A 3 ¼ A 33A 37 A 34 A 36 A 3 A 36 A 33 A 35 A 4 ¼ A 3A 37 A 34 A 35 A 3 A 36 A 33 A 35 A 5 ¼ A 6 N 6 þ A 8 N 4 þ A N A 6 ¼ A 7 N 4 þ A 9 6N A 7 ¼ A 6 N 4 þ A 8 N A 8 ¼ A 7 N A 9 ¼ A 6 3N A 4 ¼ A 4N t ðh þ h Þ cð5a 5 ðh 4 N þ h4 ÞþA 6ðh 3 þ h3 ÞþA 7A 8 ðh þ h 8 ÞA 5 ðh 5 h5 Þ b 5A 6 ðh 4 h4 Þ6A 7A 8 ðh h Þ ðð ðh h ÞþÞÞ A 5 ¼ h h A 5 N b cðh 3 N þ h h þ h h þ h3 Þ b þ 5A 6 N b cðh þ h h þ h Þ6ðA 4N t A 7 A 8 N b cþ þ ðh h Þ A 5 N b cðh 3 48BcN þ h h þ h h þ h3 Þ b þ 4A 6 N b cðh þ h h þ h Þ4ðA 4N t A 7 A 8 N b cþ cðh h Þðh h Þ 4ðBcðh h ÞÞ 3A 5h þ 4A 5h h þ A 6 h þ 3A 5 h þ A 6h A 6 ¼ A 5B r cð þ k Þ A 7 4 ¼ ða G r þ A 6 B r cþð þ k Þ A 8 6 ¼ ða 5B r þ A G r Þð þ k Þ A 9 ¼ðþk ÞG r ða 4 þ A 9 ÞþB r ð þ k Þ A 6 þ A 7 A 8 c A 4N t A ¼ðþk ÞG r A A 4ðh þ h Þ þ ðh h Þþ þ B r ð þ k Þ A 7 þ A 4 þ ðh h Þþ A ¼ F A h A 3 e Nh A 4 e Nh A 5 h A 6h 3 N b A 7 h 4 A 8h 5 A 9h 6 ða þ A 9 h ÞN 4 þða 8 þ 6A 7 h ÞN þ 4A 6 N 8 A ¼ A 9N þ 6A 7 þ A 5 L A 3 þ A 3 e Nh ðn þ LN Þ N 6 A 4 e Nh ðn LN Þ A 3 ¼ 6A 9 h 5 þð5a 8 3A 9 LÞh 4 þð4a 7 A 8 LÞh 3 þð3a 6 A 7 LÞh þða 5 6A 6 LÞh A 3 ¼ 6A 9 h 5 þð5a 8 þ 3A 9 LÞh 4 þð4a 7 þ A 8 LÞh 3 þð3a 6 þ A 7 LÞh þða 5 þ 6A 6 LÞh A 3 ¼ e Nh e Nh e Nh ðln þ NÞðh h Þ A 33 ¼ e Nh e Nh e Nh ðn LN Þðh h Þ A 34 ¼ A 3 ðh h ÞA 5 ðlðh h Þþh h ÞF A 6 ðh 3 h3 ÞA 7ðh 4 h4 ÞA 8ðh 5 h5 Þ A 9 ðh 6 h6 Þ A 35 ¼ e Nh ðn LN Þe Nh ðn þ LN Þ A 36 ¼ e Nh ðn LN Þe Nh ðn þ LN Þ A 37 ¼ A 3 A 3 4A 5 L: References Abbasband S (6) Nmerical soltions of integral eqations: homotop pertrbation method and Adomian decomposition method. Appl Math Comp 73:493 5 Akbar NS, Nadeem S () Endoscopic effects on peristaltic flow of a nanoflid. Commn Theor Phs 56: Akbar NS, Nadeem S () Nmerical and analtical simlation of the peristaltic flow of Jeffre flid with Renold s model of viscosit. Int J Nmer Methods Heat Flid Flow : Akbar NS, Nadeem S, Haat T, Hendi AA (a) Simlation of heating scheme and chemical reactions on the peristaltic flow of an Ering Powell flid. Int J Nmer Meth Heat Flid Flow : Akbar NS, Nadeem S, Haat T, Hendi AA (b) Peristaltic flow of a nanoflid with slip effects. Meccanica 47:83 94 Akbar NS, Nadeem S, Haat T, Hendi AA (c) Peristaltic flow of a nanoflid in a non-niform tbe. Heat Mass Transf 48: Akbar NS, Nadeem S, Khan ZH (4) Thermal and velocit slip effects on the MHD peristaltic flow with carbon nanotbes in an

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