International Journal of Scientific & Engineering Research, Volume 5, Issue 7, July ISSN

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1 International Journal of Scientific & Engineering Research Volume Issue 7 Jul- 69 ISSN 9-8 Peristaltic transport of MHD Carreau fluid under the effect of partial slip with different wave forms M Sukumar* and SVK Varma Abstract: This article investigates with the effects of magnetic field and partial slip on the peristaltic transport of Carreau fluid with different wave form the model in an asmmetric channel has been investigated The problem is simplified b using long wavelength and low Renolds number approximations The perturbation and numerical presented The expressions for pressure rise pressure gradient stream function magnetic force function current densit distribution have been computed The results of pertinent parameters have been discussed graphicall The trapping phenomena for different wave forms have been also discussed Ke words: peristaltic transport; Hartmann number; Weissenberg number; partial slip; shear stress; trapping; different wave forms Introduction Peristaltic pumping has been the object of scientific and engineering research in recent ears The word peristaltic comes from a Greek word Peristaltikos which means clasping and compressing The peristaltic transport is traveling contraction Recentl MHD peristaltic flows have acquired a lot of credence due to their wave along a tube-like structure and it results phsiologicall from neuron-muscular applications The effects of MHD on the peristaltic flow of Newtonian and nonproperties of an tubular smooth muscle Peristaltic motion of blood in animal or Newtonian fluids for different geometries have been discussed b man human bodies has been considered b man authors It is an important mechanism researchers [ ] with a view to understand some practical phenomena for transporting blood where the cross-section of the arter is contracted or such as blood pump machine and Magnetic Resonance Imaging (MRI) which expanded periodicall b the propagation of progressive wave It plas an is used for diagnosis of brain vascular diseases and all the human bod In indispensable role in transporting man phsiological fluids in the bod in various the studies [ ] the uniform MHD has been used There are a few situations such as urine transport from the kidne to the bladder through the ureter attempts in which induced magnetic field is used The are mentioned in the transport of spermatozoa in the ducts efferentes of the male reproductive tract and works of [6 ] The flow of Carreau fluid with partial slip under the effect of the movement of ovum in the fallopin tubes Some worms to make locomotion using magnetic field with different wave forms and ield stress on the pumping the mechanism of peristalsis Roller and finger pumps using viscous fluids also characteristics have been reported in their investigation the assumption for operate on this principle gastro-intestinal tract bile ducts and other glandular ducts the solution is that wavelength of the peristaltic wave is long A regular The principle of peristaltic transport has been exploited for industrial applications like perturbation technique is emploed to solve the present problem and sanitar fluid transport blood pumps in heart lungs machine and transport of solutions are expanded in a power of small Weissenberg number The corrosive fluids where the contact of the fluid with the machiner parts is prohibited analsis is made for the stream function axial pressure gradient and pressure Since the first investigation of Latham [] extensive analtical studies have been rise over a wavelength The influence of emerging parameters is shown on undertaken which involve such fluids Important studies to the topic include the pumping pressure gradient and trapping works in [ ] M Sukumar* Department of Mathematics SV Universit Tirupati- 7 India *Corresponding author sukumarphd@gmailcom S V K Varma Department of Mathematics SV Universit Tirupati- 7 India svijaakumarvarma@ahoocoin

2 International Journal of Scientific & Engineering Research Volume Issue 7 Jul- 6 ISSN 9-8 Mathematical formulation The flow is unstead in the laborator frame ( X Y ) However if observed in Let us consider the peristaltic transport of an incompressible Carreau fluid under the effect of magnetic field in a two dimensional channel of width d + d The flow is generated b sinusoidal wave trains propagating with constant speed c along the asmmetric channel with partial slip The geometr of the wall surfaces is defined as π ( ) = + cos ( ) h X t d a X ct upper walls () λ π h( X t ) = d acos ( X ct) + φ lower wall () u v λ + = (8) x in which a and a are the amplitudes of the waves λ is the wave length c is the p τ τ wave speed φ ( φ π ) is the phase difference X and Y are the rectangular xx ρ u + v u= σbu (9) coordinates with X measured along the axis of the channel and Y is perpendicular x x x to X Let ( U V ) be the velocit components in fixed frame of reference ( X Y ) It p τ τ should be noted that φ = it corresponds to smmetric channel with waves out of ρ u + v v = () phase and for φ = π the waves are in phase Furthermore a a d dandφ satisf x x the condition The following non-dimension quantities are also defined a ( ) + a + aa cosφ d+ d () x u v c h x = u v t t h Equations of motion λ = d = c = c = λ = d h λ d d The constitutive equation for a Carreau fluid is h = τxx = τ τ = τ τ = τ n ( )( ( ) ) xx d ηc ηc ηc τ = η + η η + Γ γ γ () γ d γ = d Γc δ = We = d ρcd p = p R e = where τ is the extra stress tensor η is the infinite shear rate c λ d cλη η viscosit η is the zero shear-rate viscosit Γ is the time constant σ k a M= Bd Da = a = a b = d d = () n is the dimensionless power law index and γ is defined as η d d d d and the stream function ψ ( x ) is defined b γ= γγ ij ji = Π () i j u = v= δ () here Π is the second invariant of strain- rate tensor We consider in the x constitutive Eq () the case for whichη = and so we can write ( ) n τ = η γ + Γ γ (6) The above model reduces to Newtonian Model for n= or Γ= a moving coordinate sstem with the wave speed c (wave frame) ( ) x it can be treated as stead The coordinates and velocities in the two frames are x X ct = Y u x = U c v x = V (7) = ( ) ( ) where u and v indicate the velocit components in the wave frame The equations the following of a Carreau fluid are given b

3 International Journal of Scientific & Engineering Research Volume Issue 7 Jul- 6 ISSN a M= M= M= M= - b β= β= β= β= - c We= We= We=6 We=9 - d n=98 n= u u u u Fig : The axial velocit u with for a= b= d= Q = x= φ = π ; (a) We= n=98 β=; (b) We= n=98 M=; (c) M= n=98 β=; 6 (d) We= M= β= and δ Re and We are the wave Renolds and Weisssenberg numbers Using the above non-dimensional quantities have given in the Eqs (8) - () respectivel Under the assumptions of long wavelength and low Renolds number p δ τ τ xx Eqs () - () after using Eq (6) become δ Re = M () x x x x p ( n ) We ψ ψ = + ψ ψ ψ p τ τ δ Re = δ δ M ψ (9) x () x x x x p = () ( n ) ψ Where τxx = + We γ () The pressure p is an eliminating from Eqs (9)- () we finall get ( n ) We ψ ψ + M ψ = () ( n ) τ = + We γ δ (6) x ( n ) Rate of volume flow ψ τ = δ + We γ (7) In laborator frame the dimensional volume flow rate is γ = δ + δ + δ (8) x h( Xt ) Q = U ( X Y t ) dy () h ( X t )

4 International Journal of Scientific & Engineering Research Volume Issue 7 Jul- 6 ISSN 9-8 a M=7 b β= M=8 β= M=9 β= M= β= dp dx 6 8 c x We= We= We= We= dp dp d dx dx x x Fig : The pressure distribution dp dx with x for a= b= d= Q = φ = π ; (a) We= n=98 β=; (b) We= n=98 M=; (c) M= n=98 6 β=; (d) We= M= β= dp dx 6 8 Carreau Fluid (n=98) x Newtonian Fluid (n=) in which h and h are functions of X and t the above expression in the wave frame becomes h ( x ) q = u ( x ) d () h ( x ) where h and h are onl functions of x from Eqs (7) () and () we can write Q = q + ch ( x) ch ( x) () The time- averaged flow over a period T at a fixed position X is given as T Q = Qdt () T where θ F Q = q= (6) cd cd in which h( x) F = d = ψ ( h ( x) ) ψ ( h ( x) ) (7) h ( x) here h ( x) and ( x) h represent the dimensionless form of the surfaces of the peristaltic walls

5 International Journal of Scientific & Engineering Research Volume Issue 7 Jul- 6 ISSN 9-8 the resulting equations in terms of stream function can be written as ( ) π ( ) ( π φ) h x = + acos x (8) h x = d bcos x+ (9) on substituting () into () and performing the integration we obtain θ = F + cd+ cd () inserting Eq (6) into Eq () ields Q = q+ + d () P λ a M=7 M=8 M=9 M= P λ - - Q Q c We= We= We= d We= P λ b Newtonian Fluid (n=) β= β= β= β= P λ Q Q Fig : The pressure rise P λ with Q for a= b= d= φ = π ; (a) We= n=98 β=; (b) We= n=98 M=; (c) M= n=98 β=; (d) 6 We= M= β= In the wave frame the boundar conditions in terms of streams function ψ are Boundar conditions Carreau Fluid (n=98)

6 International Journal of Scientific & Engineering Research Volume Issue 7 Jul- 6 ISSN 9-8 q ψ = at = h x () + β = = ( ) () ( ) q ψ = at = h ( ) x () + β = at = h ( x) () β = = at h ( x) () 6 Perturbation solution p n ψ = + M (6) For perturbation solution we expandψ q and p as x ψ = ψ + We ψ + O ( We ) (6) q ψ = at = h ( x ) (7) q = q + We q + O ( We ) q (7) ψ = at = h ( x ) (8) p = p + We p + O ( We ) (8) + β = at = h ( x) (9) Sstem of orderwe : M = β = at = h ( x) () (9) p = M () x 6 Solution for sstem of orderwe : q ψ = at = h ( x ) () Solving Eq (9) and then using the boundar conditions given in () - () we have ( ) q ψ = at = h x () at h x β = at = h ( x) () Sstem of order We ( n ) ψ M + = Where ( ) () ψ = C + C + C cosh M + C sinh M ()

7 International Journal of Scientific & Engineering Research Volume Issue 7 Jul- 6 ISSN a - - Fig : The shear stress τ τ τ with for a= b= d= Q = (d) We= M= β= q C = + M=7 M=8 M=9 M= x= φ = π ; (a) We= n=98 β=; (b) We= n=98 M=; (c) M= n=98 β=; 6 ( ) h h h h dp β Mqhcosh M + ( + M qβ ) h ( q + h h ) sinh M ( β ( )) h h h h M β ( h h) sinh M M ( h h) cosh M h h h h dp ( + M qβ ) sinh M + Mq cosh M P λ dx dx h h h h ( M β ( h h) ) sinh M M ( h h) cosh M C = h+ h ( q+ h h) sinh M C = h h h h M β ( h h) sinh M M ( h h) cosh M h+ h ( q+ h h) cosh M C = h h h h M β ( h h) sinh M M ( h h) cosh M The expressions for the axial pressure gradient at this order is b β= β= β= β= - - c - We= - We= - We= We= τ h h h h M + M q sinh M + Mq cosh M = dx h h h h M h h sinh M M( h h)cosh M Integrating Eq () over per wavelength we get = () 6 Solution for sstem of order We Substituting the zeroth-order solution () and Eq () and solving the resulting sstem along with the corresponding boundar conditions Eqs (7)- () we obtain ψ = C + C + C cosh M + C sinh M where - d - - n=98 n= ( ) () M ( n ) C ( C + C ) cosh M + C ( C + C ) sinh M () 6 M ( n )( C C) (( C+ MC) coshm + ( MC + C) sinh M) 6 τ

8 International Journal of Scientific & Engineering Research Volume Issue 7 Jul- 66 ISSN 9-8 ( ) q h ql L8L sinh Mh q M C = L6 q + L8 L = ( n ) C ( h h) L7 ( h h) L ( C + C ) coshmh + C ( C + C ) sinhmh 7 L7 h h 6 cosh Mh M q M ( n ) h( C C )( Csinh Mh + Ccosh Mh) + ( cosh Mh cosh Mh) + L8+ L L L 6 M ( sinh Mh sinh Mh) L7 h h M ( n ) ( C( C C ) β MC( C C )) sinh Mh L = cosh Mh ql L8L 6 + ( C ( C + C ) + β MC ( C + C )) C6 = q L6 ( h h) ( h h) L7 L 7 M ( n ) ( ( C ( cosh Mh cosh Mh) q ( L L) ( ) + MhC ) + β M ( MhC + C) ) sinh Mh C C 6 (( MhC C) β M ( C MhC ) ) cosh Mh C7 = + L ( sinh Mh sin M h) L 7 ( h h ) M q C8 = + L8 L 7 ( h h) M L = ( n ) ( C( C + C ) coshmh+ C( C + C ) sinhmh) 6 M ( n ) h( C C )( Csinh Mh+ Ccosh Mh) 6 Fig : The stream lines for a= b= d= φ = We= n=98 β= M=; (a) Q = ; (b) Q = ; (c) Q =

9 International Journal of Scientific & Engineering Research Volume Issue 7 Jul- 67 ISSN 9-8 Fig 6: The stream lines for a= b= d= Q = φ = We= n=98 β=; (a) M=; (b) M=; (c) M=9 M ( n ) ( C( C C ) β MC( C C )) sinh Mh L L L L sinh Mh+β M cosh Mh 6 = cosh Mh L ( C ( C + C ) + β MC ( C + C )) ( h h) ( sinh Mh sinh Mh) M ( n ) ( ( C ) ( )) ( ) + MhC + β M MhC + C sinh Mh The axial pressure gradient is given b C C 6 + (( MhC + C) + β M ( C + MhC ) ) cosh Mh dp M ql L8L ( sinh Mh sinh Mh) ( cosh Mh cosh Mh) = q L6 L = dx ( h h ) ( h h) L7 L 7 ( sinh Mh sinh Mh) ( L L)( cosh Mh cosh Mh) Integrating Eq () over are wavelength we get the pressure as L6= L L M ( sinh Mh sinh Mh) dp Pλ dx L = 7 L M ( cosh Mh cosh Mh)( sinh Mh + β M cosh Mh) + ( h h ) ( sinh Mh sinh Mh) M ( cosh Mh + β M sinh Mh ) ( )( ) =L dx () = (6) The perturbation series solution up to second order for stream functionψ velocit u pressure gradient dp/dx and pressure rise P λ ma be summarized as

10 International Journal of Scientific & Engineering Research Volume Issue 7 Jul- 68 ISSN 9-8 Fig 7: The stream lines for a= b= d= Q = φ = n=98 β= M=; (a) We=; (b) We=; (c) We= Fig 8: The stream lines for a= b= d= Q = φ = n=98 M=We=; (a) β=; (b) β=; (c) β=9 ψ ψ We (7) dp dp dp = + We dx dx dx (8) P = P We + P (9) λ λ λ The non-dimensional shear stress of the channel reduces to ( n ) ( ) τ = M Ccosh M M Csinh M We M Ccosh M + M Csinh M ( + ) ( ) 9 C C C cosh M 6 + M C7cosh M + M C8sinh M M ( n ) ) 6 C C C sinh M M ( n )( C C )(( C + MC) coshm + ( MC + C) sinh M) 6 (6) 7 Results and discussion:

11 International Journal of Scientific & Engineering Research Volume Issue 7 Jul- 69 ISSN 9-8 The stud of the pressure rise Q velocit u pressure gradient dp / dx and shear stress are presented and discussed for different phsical quantities of interest τ The pressure rise is an important phsical measure in the peristaltic mechanism The perturbation method on Weissenberg number restricted us for choosing the parameters for a Carreau fluid such that Weissenberg number is less than one According to the values of various parameters for Carreau fluid are: n = and Γ = 8 In Figs the axial velocit distribution is shown for different parameters partial slip β Hartmann number M and Weissenberg number ( We < ) In Figs (a)-(c) we found that the magnitude of the axial velocit decreases in the center and increases nearer at the walls of the channel with increasing the Hartmann number M partial slip β and Weissenberg numberwe Fig (d) shows a comparison of the Carreau fluid (n=98) and Newtonian fluid (n=) it is observed that the magnitude of velocit increases from Carreau fluid to Newtonian fluid Figs are plotted to see the effect of the parameters β M and We on the pressure gradient dp / dx Figs (b) and (c) show that the pressure gradient dp / dx decreases with increasing the partial slip β and Weissenberg We on the other hand in the wider part of the channel x [ ] and x [6 ] the pressure gradient is reall small that is the flow can easil pass without imposition of a large pressure gradient Besides in a narrow part of the channel x [ 6] a much pressure gradient is required to maintain β andwe it especiall near x = Fig (a) indicates that the pressure gradient dp / dx increases with increasing Hartmann number M and the maximum pressure gradient is also near x = Fig (d) indicated that the pressure gradient in Newtonian fluid (n=) is slightl high comparing with Carreau (n=98) Fig 9: The stream lines for a= b= d= Q = φ = n=98 M=We=; β=; (a) Sinusoidal; (b) Trapeziodal;(c) Square;(d) Triagle;(e) Sawtooth waves

12 International Journal of Scientific & Engineering Research Volume Issue 7 Jul- 66 ISSN 9-8 Q < and > is called retrograde (or) backward pumping It is shows that shear stress τ with is calculated from Eq (6) and is shown in Figs for P λ there is a nonlinear relation P λ versus Q Fig (a) is a graph of the pressure rise P λ per wavelength versus he mean flow rate Q of the asmmetric channel for fixed values of other parameters We observed that an increases in the Hartmann number M result decrease in the peristaltic pumping rate and also in an increase in the pressure rise Figs (b) and (c) show the variation of pressure rise with flow rate Q for values of partial slip β and Weissenberg P λ number We respectivel We observe that the peristaltic pumping rate decrease with increase β andwe Fig (d) reveals that variation of P λ with Q for a Carreau fluid and Newtonian fluid It is an indicated that the Newtonian fluid peristaltic pumping rate is more than a Carreau fluid The variation of the axial different phsical parameters In Figs (a) and (c) we observed that the curves intersect at origin and the axial shear stress τ decreases with increasing the Hartmann number M and Weissenberg numberwe in the upper wall and an opposite behavior is observed in the lower wall of the channel The relation between the shear stress τ and at different values the partial slip parameter β is depicted in Fig (b) We observe that the curves intersect at the origin and τ increases with increasing the partial slip parameter β upper the shear stress wall of the channel while an opposite behavior is observed in lower wall of the channel It is observed from Fig (d) that a Carreau fluid shear stress τ is less Fig : The stream lines for a= b= d= Q = φ = n= M=We=; β=; (a) Sinusoidal; (b) Trapeziodal;(c) Square; (d) Triagle; (e) Sawtooth waves

13 International Journal of Scientific & Engineering Research Volume Issue 7 Jul- 66 ISSN 9-8 than to the Newtonian fluid in upper channel and also an opposite behavior in lower channel 7 Trapping phenomena Another interesting phenomenon in peristaltic motion is the trapping It is basicall the formation of an internall circulating bolus of fluid b closed stream lines The trapped bolus will be pushed ahead along the peristaltic waves The stream lines are calculated form Eq (7) and plotted in Figs - Figs and 7 shows the effects of the volume flow rate Q and Weissenberg number We on the stream lines it is found that the size of the bolus increases with increase Q andwe It is shown in Figs 6 that the size of bolus decreases with increasing the Hartmann number M while the bolus disappears for M =9 Fig8 is depicted for various values of the partial slip parameter β It is found that the volume of the trapping bolus decreases as the partial slip parameter β increases moreover the bolus disappears at β =9 Figs 9- compare for different wave forms like sinusoidal triangular trapezoidal square and sawtooth wave it is finall observed that the volume of trapping bolus of the a Carreau fluid(n=98) large in the upper channel and small in the lower channel but an opposite behavior in the case of the Newtonian fluid(n=) Conclusion We have theoreticall analzed the problem of peristaltic flow of an incompressible MHD Carreau fluid in asmmetric channel under the effect on partial slip with different wave forms The governing equations are transformed to stead non-dimensional differential equations These equations are solved analticall Interaction of various emerging parameters with peristaltic flow is discussed with the help of graphs On the basis of present analsis the following observations have been noted: The magnitude of the velocit field increases near the walls and decreases at the center of the channel when increasing the Hartmann number M and partial slip parameter β The pressure gradient decreases with increasing the partial slip parameter β and Weissenberg numberwe In the peristaltic pumping region the pressure rise decreases with increasingwe β In the and M The shear stress distribution decreases in the upper wall and increases in the lower wall of the channel with increasing M andwe The size of tapping bolus decreases with increase β and M while it disappears at M =9 and β =6 Appendix: Expressions for wave shapes The non-dimensional expressions for the five considered wave forms are given b the following equations: I Sinusoidal wave: h ( x) = + asin ( x) (A) h ( x) = d bsin ( x+ φ ) (A) II Triangular wave: m+ 8 ( ) h ( x) = + a sin[(m ) x] (A) π m= (m ) m+ 8 ( ) h ( x) = d b sin[(m ) x+ φ] π m= (m ) (A) III Square wave: m+ ( ) h ( x) = + a cos [(m ) x] (A) π m= (m ) m+ h ( ) ( x ) d b cos [( ) ] m (m ) m x = + φ (A6) π = IV Trapezoidal wave: π sin (m ) ( ) 8 (A7) h x = + a sin[(m ) x] π m= (m ) π sin (m ) ( ) 8 (A8) h x = d b sin[(m ) x φ] + π m= (m ) V Sawtooth wave: [ mπ x] 8 sin h ( x) = + a (A9) π m= m [ mπ x] 8 sin h ( x) = d b + φ (A) π m= m References [ ] TW Latham Fluid motion in peristaltic pumpmsthesis MIII Cambridge MA966 [ ] AM Siddiqui WH Schehawe Peristaltic flow of a second-order

14 International Journal of Scientific & Engineering Research Volume Issue 7 Jul- 66 ISSN 9-8 fluid in tubes Journal of Non Newtonian Fluid Mechanics (99)7 8 [] AH Shapiro Y Jaffrin SL Weinberg Peristaltic pumping with long wave length sat low Renolds number Journal of Fluid Mechanics 7(969) [] M Mishra A Ramachardra Rao Peristaltic transport of Newtonian fluid in an asmmetric channel ZAMP () - [] MV Subba Redd A Ramachandra Rao S Sreenadh Peristaltic motion of a power-law fluid in an asmmetric channel Int J Non-Linear Mech (7) -6 [6] K Vajravelu S Sreenadh R Hemadri redd K Murugesan Peristaltic transport of a Casson fluid in contact with a Newtonian Fluid in a circular Tube with permeable wall International Journal of Fluid Mechanics Research Vol 6(9)() - [ 7 ] S Nadeem Safia Akram Heat transfer in a peristaltic flow of MHD fluid with partial slip Commun Nonlinear Sci Numer Simulat () [ 8 ] K Vajravelu S Sreenadh K Rajanikanth Chanhoon Lee Peristaltic transport of a Williamson fluid in asmmetric channels with permeable walls Nonlinear Analsis: Real World Applications () 8-8 [ 9 ] M Mishra A Ramachandra Rao Nonlinear and curvature effects on peristaltic flow of a viscous fluid in an asmmetric channel Acta Mechanica 68() -9 [] T Haat Nasir Ali Zaheer Abbas Peristaltic flow of a mocropolar fluid in a channel with different wave forms Phs [6] A Ebaid Effects of magnetic field and wall slip condition on the peristaltic transport of a Newtonian fluid in asmmetric channel Phsics letters A 7(8) 9-99 [7] S Srinivas M Kothandapani The influence of heat and mass transfer of MHD peristaltic flow through a porous space with complaints walls Applied Mathematics and Computation (9)97-8 [8] S Nadeem S Akram Heat transfer in a peristaltic flow of MHD fluid with partial slip Communications in Nonlinear Science and Numerical Simulation () [9] K Rajanikanth S Sreenadh Y Rajesh adav and A Ebaid MHD Peristaltic flow of a Jeffre fluid in an asmmetric channel with partial slip Advances in applied Science Research (6) 7-76 [] Prasana Hariharan V Seshadri Rupak K Banerjee Peristaltic transport of non-newtonian fluid in a diverging tube with different wave forms Mathematical and Computer Modeling 8(8)998-7 [] K Rajnikanth Y Rajesh adav and SSreenadh The Heat Transfer on the Peristaltic Flow of an MHD Fluid in an Inclined Channel in a Porous International Journal of Fluid Mechanics () ;pp8-96 [] Y Wang T Haat N Ali M Oberlack Magnetohdrodnamics peristaltic motion of Sisko fluid in a smmetric or asmmetric channel Phsica A 87(8) 7-6 [] M Kothandapani S Srinivas Peristaltic transport of a Jeffre fluid under the effect of magnetic field in an asmmetric channel Int J Nonlinear Mech (8) 9-9 [] T Haat Niaz Ahmad N Ali Effects of an endoscope and magnetic field on the peristalsis involving Jeffre fluid Communications in Nonlinear Science and Numerical Simulation Lett A 7 (8) - (8) 8 9 [] L M Srivastava V P Srivastava Peristaltic transport of [] K Rajanikanth S Sreenadh and A Ebaid Peristaltic flow of a in an blood: Casson model II J Fluid Mech (98) 9-6 asmmetric channel with partial slipinternational Journal of [] T Haat S Noreen Peristaltic transport of fourth grade fluid Engineering Sciences & Research Technolog () () with heat transfer and induced magnetic field ComptesRendusMécanique8() 8 8 [] Kh S Mekheimer Effect of magnetic field on peristaltic flow of a couple stress fluid Phs Lett A 7(8) 7-78 [] S Srinivas R Gaathri M Kothandapani The influence of slip conditions wall properties and heat transfer on MHD peristaltic transport Computer Phsics Communications 8(9) [] AM Abd-Alla GA Yaha SR Mahmoud HS Alosaimi Effect of the rotation magnetic field and initial stress on peristaltic motion of micropolar fluid Meccanica 7() 6