Simulation of Heat and Chemical Reactions on Peristaltic Flow of a Williamson Fluid in an Inclined Asymmetric Channel

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1 Iran. J. Chem. Chem. Eng. Vol., No., Simulation of Heat and Chemical Reactions on Peristaltic Flow of a Williamson Fluid in an Inclined Asmmetric Channel Nadeem, Sohail* + Department of Mathematics, Quaid-i-Azad Universit 45, Islamabad 44, PAKISTAN Akram, Safia Department of Humanities and Basic Sciences, Militar College of Signals, National Universit of Sciences and Technolog, Rawalpindi 46, PAKISTAN Akbar, Noreen Sher DBS&H, CEME, Nathonal Universit of Sciences and Technolog, Islamabad, PAKISTAN ABSTRACT: This work concerns the peristaltic flow of a Williamson fluid model in an inclined asmmetric channel under combined effects of heat and mass transfer. The governing nonlinear partial differential equations are simplified under the lubrication approach and then solved analticall and numericall. The analtical results are computed with the help of regular perturbation and the numerical results are found b using shooting method. KEY WORDS: Williamson fluid, Peristaltic flow, Heat and mass transfer, Perturbation solution, Numerical solution. INTRODUCTION During the last few decades, the stud of non-newtonian fluids become more important because of its variet of applications in engineering and Biomechanics. A few of these applications include food miing and chme movement in the intestine, flow of plasma, flow of blood-a Bingham fluid, flow of nuclear fuel slurries, flow of liquid metals and allos, flow of mercur amalgams and lubrication with heav oils and greases. Due to compleit of non-newtonian fluids there are man models of non-newtonian fluids. An etensive stud of * To whom correspondence should be addressed. + snqau@hotmail.com -86/// 5/$/.5 these non-newtonian fluid models is available in the refs [-8]. To be more specific heat and mass transfer analsis in peristaltic mechanism plas important role in a number of scientificall and engineering applications. The combined effects of heat and mass transfer in a moving fluid usuall describes the relations between the flues and the driving potentials []. Further, the mass flues can also be created b temperature gradients and this is the Soret or thermal diffusion effect. In general, the thermal-diffusion and diffision-thermo effects are of smaller order of

2 Iran. J. Chem. Chem. Eng. Nadeem S. et al. Vol., No., magnitude than the effects described b Fourier's or Fick law and are often neglected in heat and mass transfer processes []. The effects of heat and mass transfer in peristaltic flows of Newtonian and non-newtonian problems are carried b Nadeem & Akbar in their papers [-]. To the beat of authors knowledge onl a limited attention has been focussed to the stud of combined effects of heat and mass transfer in the peristaltic flow problems. The present investigation discuss the combined effects of heat and mass transfer on the peristaltic flow of a Williamson fluid model in an asmmetric channel. The governing equations of Williamson fluid model along with heat and mass transfer equations are reduced under the assumptions of long wavelength. The reduced equations are then solved subject to the boundar conditions of asmmetric channel with the help of regular perturbation method and numericall b shooting technique. At the end, the phsical feature of the pertinent parameters of interest are discussed in detail. MATHEMATICAL FORMULATION Let us consider the peristaltic transport of an incompressible Williamson fluid in a two dimensional inclined channel of width d + d. The flow is generated b sinusoidal wave trains propagating with constant speed c along the channel walls. The geometr of the wall surface is defined as π Y = H = d + a cos ( X ct, upper wall λ π Y = H = d b cos ( X ct + ϕ, lower wall λ Where a and b are the amplitudes of the waves, λ is the wave length, d + d is the width of the channel, c is the velocit of propagation, t is the time and X is ( the direction of wave propagation, the phase difference ϕ varies in the range ϕ π, ϕ= corresponds to smmetric channel with waves out of phase and ϕ=π the waves are in phase, and further a, b, d, d and ϕ satisfies the condition ( a + b + a b cos ϕ d + d The equations governing the flow of a Williamson fluid in the presence of gravit effects are U V + = ( X Y U U U U V ρ + + t X Y = ( P τxx τ + + X X Y XY + ρg sin α V V V U V ρ + + t X Y = (4 P τxy τ + + Y X Y YY ρg cos α Heat and mass transfer equations are T T T C + U + V = t X Y K U V U V T + τ XX + τ XY + + τyy ρ X X Y Y C C C + U + V = t X Y C C DK T T D X Y T X Y T m where X Y = + (5 (6 In which U, V are the velocities in X and Y directions in fied frame, ρ is constant densit, P is the pressure, ν is the kinematic viscosit, g is the acceleration due to gravit, K' is the thermal conductivit, C' is the specific heat, T is the temperature, D is the coefficient of mass diffusivit, T m is the mean temperature, K T is the thermal diffusion ratio and C is the concentration of fluid. The constitutive equation for the etra stress tensor τ for Williamson fluid model is [4]. ( ( τ = µ + µ + µ Γγ γ, (7 In which - PI is the spherical part of the stress due to constraint of incompressibilit, τ is the etra stress tensor, µ is the infinite shear rate viscosit, µ is the zero shear rate viscosit, Γ is the time constant and γ is defined as γ = γ ijγ ji = Π (8 i j 4

3 Iran. J. Chem. Chem. Eng. Simulation of Heat and Chemical Reaction on Peristaltic Flow... Vol., No., Here Π is the second invariant strain tensor. We consider the constitutive Eq. (7, the case for which µ = and Γγ <. The component of etra stress tensor therefore, can be written as [ ] τ = µ ( Γγ γ = µ ( + Γγ γ ( Introducing a wave frame (, moving with velocit c awa from the fied frame (X,Y b the transformation = X ct, = Y, u = U c, v = V, ( and P ( = P( X, t Defining u v ct h =,, u, v, t, h, λ = d = c = c = = d λ d a b h λ d = a =, b =, h =, τ = τ, d d d d c µ d d C T T τ = τ, τ = τ, Fr =, θ =, µ c µ c gd T T d ρcd Γc d γ d δ =, Re =, We =, P = P, γ =, λ µ d cλµ c ( ( c ρνc σ C C Ec =,Pr =,M = B d, Φ = C T T K C C υ Using the above non-dimensional quantities and the resulting equations in terms of stream function Ψ(dropping the bars, u δ Re Ψ Ψ Ψ = P τ τ Re sin Fr + δ + + α δ Re Ψ Ψ Ψ Ψ Ψ =, v = δ can be written as τ τ Fr P Re + δ + δ δ cos α ( ( ( = ( Reδ Ψ θ Ψ θ = θ + δ θ + (4 Pr ( Ec( δ τψ τψ + τψ δτψ Reδ( Ψ Φ ΨΦ = (5 ( Sr ( Sc δ Φ + Φ + δ θ + θ where [ ] Ψ τ = + We γ, (6 Ψ Ψ τ = [ + We γ ] δ, [ ] Ψ τ = δ + We γ, Ψ Ψ Ψ Ψ γ = δ + δ + δ In which δ, Re, We represent the wave, Renolds and Weissenberg numbers, respectivel. Under the assumptions of long wavelength δ<< and low Renolds number, neglecting the terms of order δ and higher, Eqs. ( to (6 take the form Re = + We + sin α = Fr P Ψ Ψ P = θ Ψ Ψ = Pr Ec We + Φ θ + Sr = Sc / (7 (8 ( ( Elimination of pressure from Eqs. (7 and (8, ield Ψ Ψ We + = θ Ψ Ψ = Pr Ec We + Φ θ + Sr = Sc ( ( ( The instantaneous volume flow rate in the fied frame is defined as 5

4 Iran. J. Chem. Chem. Eng. Nadeem S. et al. Vol., No., Q q H = U ( X,Y, t dy (4 H In the wave frame the rate of volume flow presented as h = u(, d (5 h In which h and h are the functions of onl. From Eqs. (, (4 and (5 we obtain Q = q + ch ( ch ( (6 The time mean flow over a period T at a fied position X is defined as T Q = Qdt T (7 Making use of (6 into (7 and after integrating, we get Q = q + cd cd (8 The dimensionless time-mean flows in the laborator frame and F in the wave frame are defined as = Q q, F cd = cd ( Therefore, Eq. (8 can be written as = F + + d ( in which h ( Ψ F = d = Ψ h ( Ψ(h ( h ( ( ( h ( = + a cos π, h ( = d bcos(π + ϕ ( The corresponding boundar conditions are F Ψ = at = h = + a cos π, ( F Ψ = at = h = d b cos( π + ϕ, Ψ = at = h, Ψ = at = h θ =, Φ = at = h (4 θ =, Φ = at = h Solution of the problem Since Eq. ( is highl non linear, therefore, the eact solution of Eq. ( seems to be impossible. Adopting the similar procedure as done b Nadeem & Akram [4], the solution of Eq. ( straight forward can be written as F + h h Ψ = ( (h + h + 6hh + (h h (5 F F + h( h hh (h ( h hh + (h h 4 We A + B + A + B + A!! 4! where A = (6 6 A hh h ( h (h h h (h h (h h 4! 6 4 A h h hh h +,! 4 Ahh = (h + h ( h (h h h (h h, (h h = A (h + h (h + hh + h ( h (h h h (h h, 4(h h! A = ( h (h h h (h h, (h h 4! F + h h = 88 (h h The aial pressure gradient is obtained from Eq. (7 ( + dp F h h Re = + sin α + d (h h Fr A ( (h h 4! We h (h h h (h h F + h h 44(h + h (h h (7 6

5 Iran. J. Chem. Chem. Eng. Simulation of Heat and Chemical Reaction on Peristaltic Flow... Vol., No., The solution of Eqs. ( and ( with the help of Eq. (5 can be written as a a θ = Br ( h h ( h h a a A 6 We ( h h ( h h a4 a5 + We + 4 ( h h ( h h a a a ( h h ( h h ( h h a A We + c + c ( h h a a Φ = SrScBr ( h h ( h h a a A 6 We ( h h ( h h a4 a5 + We + 4 ( h h ( h h a a a ( h h ( h h ( h h a A We + c + c4 ( h h ( ( ( (8 ( c = h h ( 6 h h + (4 67( h h 5 ( + + ( + ( + (h a a h h a a h h h a a h h ( + a h a h h + a h h a h + a h + h ( a h + h h + h Br + b BrWe + b BrWe + b BrWe 4 ( c = h (67 h h 67( h h ( ( ( h h h (a a h + h + ( ( ( a h h h + h h + h + 8 ( a h + h h + h Br b BrWe b BrWe + b BrWe ScSr ( ( ( c = h h (6 h h + (4 67( h h ( + + ( + ( 7 ( ( (h a a h h a a h h h a + a h h + a h a h h + a h h a h + a h + h a h + h h + h Br + b BrWe b BrWe b BrWe, ScSr ( ( ( c4 = h ( h h (6 h h + 67( h h 5 ( ( ( ( ( h (a a h + h + a h h h + h h + h + a h h h h Br b BrWe b BrWe b ScSrBrWe, 4 5 Where the constants appearing in Eqs. (4 and (4 are defined in appendi. EXPERSSIONS FOR DIFFERENT WAVE SHAPE The non-dimensional epressions for three considered wave form are given as Sinusoidal wave h ( = + a sin π, h ( = d b sin(π + ϕ. Multisinusoidal wave h ( = + a sin nπ, h ( = d b sin(nπ + ϕ. Triangular wave m+ ( ( m h ( = + a sin ( π( m 8, π m = m+ ( ( m h ( = d b sin ( π( m + ϕ 8 π m = 5 Trapezoidal wave π ( 8 ( m sin m h ( = + a sin ( π( m, π m = π ( 8 ( m sin m h ( = d b sin ( π( m + ϕ π m= 7

6 Iran. J. Chem. Chem. Eng. Nadeem S. et al. Vol., No., Table : Comparison of velocit profile for (a We=., (b We=.6 other parameters are d=, b=.7, a=.7,ϕ =π/, Re =.4 Fr=.5, s =, =.. R Perturb sol, We=. Numer sol, We=. Perturb sol, We=.6 Numer sol, We= u(, u(, Fig. : Comparison of velocit profile for (a We =., (b We =.6 other parameters are d=, b=.7, a =.7, ϕ =π/, Re =.4, Fr =.5, α =., =, =.. Numerical solution To get the numerical solution of the velocit profile we solved the Eq. ( with the relevant boundar conditions given in Eq. ( b emploing shooting method. The comparitive stud is also made to see the validit of the results. RESULTS AND DISCUSSION In this section numerical and graphical results of the problem under consideration are discussed. Figs. (a and (b and Table shows the comparison of velocit profile. The epression for the pressure rise and pressure gradient is calculated using a mathematics software Mathematica. In order to see the variation of pressure rise with volume flow rate for different values of width of channel a, Froude number Fr, amplitude ratio ϕ and Renolds number Re, Figs. to 5 are presented. It is observed from Fig. that the pumping rate increases with an increase in width of the channel a in the retrograde ( P >, < and peristaltic pumping ( P >, > regions, while pumping rate decreases in the copumping ( P <, > region with an increase in width of the channel a. From Fig. it is seen that with an increase in Froude number Fr, the pumping rate decreases in all the regions. Fig. 4 shows the variation of pressure rise with volume flow rate for different values of amplitude ratio ϕ. It is observed that in the retrograde ( P >, < and peristaltic pumping ( P >, > regions, the pumping rate decreases with an increase 8

7 Iran. J. Chem. Chem. Eng. Simulation of Heat and Chemical Reaction on Peristaltic Flow... Vol., No., P P Fig. : Variation of P with for different values of a for fied d=., b=., ϕ=π/, We=., Re=.4, Fr=.5, α=.. Fig. 4: Variation of P with for different values of ϕ for fied d=., b=., a=.7, We=., Re=.4, Fr=.5, α= P P Fig. : Variation of P with for different values of Fr for fied d=, a=.7, b=., ϕ=π/, We=., Re=.5, α=.. Fig. 5: Variation of P with for different values of Re for fied d=, a=.7, b=., ϕ=π/, We=., Fr=.5, α=.. in amplitude ratio ϕ, while the behavior is quite opposite in the copumping ( P <, > region, here pumping rate increases with an increase in amplitude ratio ϕ. It is depicted from Fig. 5 that the pressure rise increases with an increase in Renolds number Re in all the regions. The pressure gradient for different values of width of the channel a inclination angle α, Froude number Fr and amplitude ratio ϕ are shown in Figs. 6 to. Figs. to show the temperature profile for different values of Brinkmann number Br, volume flow rate and Weissenberg number We. It is observed from Figs. and that the temperature profile increases with an increase in Brinkmann number Br and volume flow rate. Fig. shows the temperature profile for different values of Weissenberg number We. It is observed that the temperature profile increases with an increase in Weissenberg number We when [-.5,], while the behavior of temperature profile is opposite when [,.5], here temperature profile decreases with an increase in Weissenberg number We. Futhermore, the maimum temperature occurs at =. The concentration profile for different values of Brinkmann number Br, volume flow rate, Soret number Sr, Schmidt number Sc and Weissenberg number We. It is observed that the concentration profile decreases with an increase in Brinkmann number Br, volume flow rate Soret number Sr and Schmidt number Sc (see Figs. to 5. It is observed from Fig. 6 that the concentration profile decreases with an increase Weissenberg number when [-.5,], while the behavior of concentration profile is opposite when [,.5], here concentration profile increases with an increase in Weissenberg number We. Futher the maimum concentration occurs at =.

8 Iran. J. Chem. Chem. Eng. Nadeem S. et al. Vol., No., 4 5 dp/d - dp/d Fig. 6: Variation of dp/d with for different values of a for fied d=., Re=.4, b=., ϕ=π/4, We=., Fr=.5, =.4, α=.. Fig. : Variation of dp/d with for different values of ϕ for fied d=., Re=.4, b=., Fr=.5, We=., =.4, α=., a= dp/d Fig. 7: Variation of dp/d with for different values of α for fied d=, Re=.5, b=., ϕ=π/4, We=., Fr=.5, a=.4, =.4. Fig. : Temperature profile for different values of Br for fied d=.5, b=., We=., ϕ=π/, =, a=.5, = dp/d Fig. 8: Variation of dp/d with for different values of Fr for fied d=, Re=, b=., ϕ=π/4, We=., =.4, α=., a=.4. Fig. : Temperature profile for different values of for fied d=.5, b=., We=., Br=.8, ϕ=π/, a=.5, =.

9 Iran. J. Chem. Chem. Eng. Simulation of Heat and Chemical Reaction on Peristaltic Flow... Vol., No., Fig. : Temperature profile for different values of We for fied d=.5, b=., Br=.8, =, ϕ=π/, a=.5, =. Fig. 5: Concentration profile for different values of Sr and Sc for fied d=.5, b=., =, Br=., ϕ=π/, a=.5, We=., = Fig. : Concentration profile for different values of Br for fied d=.5, b=., Sc=.8, Sr=.6, =, ϕ=π/, a=.5, We=., =. Fig. 6: Concentration profile for different values of We for fied d=.5, b=., Sc=.8, Sr=.6, Br=., =, ϕ=π/, a=.5, = Fig. 4: Concentration profile for different values of for fied d=.5, b=., Sc=.8, Sr=.6, Br=., ϕ=π/, a=.5, We=., =. Stream lines for different values of volume flow rate, Weissenberg number We and width of the channel are shown in Figs. 7 to. It is observed from Fig. 7 that the size of the trapping bolus increases with an increase in volume flow rate. It is observed from Fig. 8 that the size of the trapped bolus decreases in the upper half of the channel, while the size of the trapping bolus increases in the lower half of the channel with an increase in Weissenberg number We. Fig. shows the stream lines for different values of width of the channel a. It is observed the size of the trapping bolus increases in the upper half of the channel and decreases in the lower half of the channel with an increase in width of the channel a. Fig. shows the stream lines for different wave forms.

10 Iran. J. Chem. Chem. Eng. Nadeem S. et al. Vol., No.,.5 (a.5 (a (b.5 (b (c.5 (c Fig. 7: Stream lines for different values of Fig. (a =.7, Fig. (b =.7, Fig. (c =.7. The other parameters are d =., b =.5, ϕ =.5, a =.5, We =.. Fig. 8: Stream lines for different values of We Fig. (a We =, Fig. (b We=., Fig. (c We=.4. The other parameters are d =., b =.5, ϕ =.5, a =.5, =.7.

11 Iran. J. Chem. Chem. Eng. Simulation of Heat and Chemical Reaction on Peristaltic Flow... Vol., No., (a (b (c Fig. : Stream lines for different values of a Fig. (a a =.47, Fig. (b a =.48, Fig. (c a =.4. The other parameters are d =., b =.5, ϕ =.5, We =., = The results of pressure rise for differnt values of amplitude ratio ϕ and Froude number Fr are tabulated in Tables and. It is observed from the tables that the pressure rise decreases for small values of amplitude ratio ϕ and increases for large values of amplitude ratio ϕ. Moreover the pressure rise decreases for all values of volume flow rate with an increase in Froude number Fr. The results of velocit field against different values of volume flow rate and Weissenberg number We are tabulated in Tables 4 and 5. It is observed from the tables that the magnitude value velocit profile decreases with an increase in volume flow rate, while the velocit profile increases with an increase in Weissenberg number We for small values of and for large values of the behavior is quite opposite. Nomenclature U,V Velocit components in X and Y direction in fied frame u,v Velocit components in and directions in wave frame ρ p C' K' M Br Pr Re Sc Sr g D m K T T Smbols θ C ϕ a,b λ c Q δ Constant densit pressure Thermal diffusion ratio Thermal conductivit Hartmann number Brinkmann number Prandtl number Renolds number Schmidt number Soret number Acceleration due to gravit Coefficient of Mass diffusivit Thermal Diffusion Temperature of Fluid in dimension from Temperature of fluid in dimensionless form Concentration of fluid in dimension form Concentration of fluid in dimensionless form Amplitude ratio Amplitude of waves Wave length Velocit of propagation Volume flow rate Long wave length

12 Iran. J. Chem. Chem. Eng. Nadeem S. et al. Vol., No., Table : Variation of pressure rise with volume flow rate for different values of ϕ for fied d =., b =., a =.7, We =., Re =.4, Fr =.5, α =.. P f o r ϕ = ϕ = π/4 ϕ = π/ (a.5 (b (c.5 (d Fig. : Stream lines for different wave forms. Fig. (a sinusoidal wave, Fig. (b Multisinusoidal wave, Fig. (c triangular wave, Fig.(d trapezoidal wave. The other parameters are d =., b =.5, ϕ =.5 a =.5, =.7, We =.. 4

13 Iran. J. Chem. Chem. Eng. Simulation of Heat and Chemical Reaction on Peristaltic Flow... Vol., No., Table : Variation of pressure rise with volume flow rate for different values of Fr for fied d =, a =.7, b =., ϕ = π/, We =., Re =.5 α =.. P f o r Fr =.8 Fr =. Fr = Table 4: Velocit profile for different values of for fied d =, b =.7, a =.7, We =., =, ϕ =. u(, f o r = =. =

14 Iran. J. Chem. Chem. Eng. Nadeem S. et al. Vol., No., Table 5: Velocit profile for different values of We for fied d =, b =.7, a =.7, =.5, =, ϕ =. u(, f o r We = We =. We = Ψ Stream function Fr Froude number α, Inclination angle of channel and magnetic field respectivel T Temperature at upper wall Temperature at lower wall T τ Etra stress tensor Appendi ( b = h ( a a h + a h + a h + h ( ( a6 h a7 + a8h 4 + 5ah 8 h + ( a6 a7h a8h 5 ah ( a6 a7h ah h ( a7 5ah 8 h 4 ah 7 h 5 a8h h 6 hh 7 ( a8 ah 4 ( a8 5ah 4 h 8 ah h + ah h 5ahh + ah + a4 ( h + h a5 ( h + hh + h ( ( ( b = 56A h h h + h h h h + h ( h + hh + h ( ( ( ( b = 5A h h h + h h + h h + h = ( 4 + ( b h a h a a h a h a h h ( a6 h ( a7 + a8h 4 + 5ah 8 h + ( ( ( ( ( a a h + a h + a h h + a + a h h a h h h a a h h h a + a h h a 5a h h a h h + a h h 5a h h ( a h a h + h 5 ( ( b = 56A h h h h h h + h h + h h + h ( b = 5A h h + h 5 ( h 6 + h 5 h + h 4 h + h h + h h 4 + hh 5 + h 6 6

15 Iran. J. Chem. Chem. Eng. Simulation of Heat and Chemical Reaction on Peristaltic Flow... Vol., No., a ( ( ( ( = 6 F + h h h + h + A h h We ( ( ( ( ( ( F + h ( h B h h We a = 5( 44( F + h h + a = 6 F + h h h + h + A h h We ( ( ( ( 6 h h F + h h 4B + A h h We + 6 ( AA + B ( h h We a ( ( ( = AWe F + h h + B h h We a ( ( ( ( 4 = 6 F+ h h h + h + A h h We ( ( ( ( 5 ( ( F + h ( h B h h We a6 = 8( 6( F+ h h ( h + h + A ( h h We a = 6 F + h h h + h + A h h We ( 88( F + h ( h + 6 h h ( F + h + h 6 ( 8B + A ( h + h We + ( AA + B ( h h We a7 = ( ( F+ h h B ( h h We ( 44 F h h 6 h h F h h ( ( ( ( 4B + A ( h + h We + ( AA + B ( h h We a8 = 56AWe ( 88( F + h h + ( ( ( ( = ( ( + + ( 6 h h F + h h 8B + A h h We + a 4A We F h h B h h We Received : Apr., ; Accepted :Oct. 6, REFERENCES [] Abd El Hakeem Abd El Nab, A.E.M. El Miser, M.F., Abd El Kareem, Separation in the Flow Through Peristaltic Motion of a Carreau Fluid in Uniform Tube, Phsica A, 4, p. (4. [] Mekheimer Kh. S., Peristaltic Flow of Blood under Effect of a Magnetic Field in a Non-Uniform Channels, Appl. Math. Comp., 5, p. 76 (8. [] Kothandapani M., Srinivas S., Peristaltic Transport of a Jeffre Fluid under the Effect of Magnetic Field in an Asmmetric Channel, Int. J. Non-LinearMech., 4, p. 5 (8. [4] Nadeem S., Akram Safia, Peristaltic Flow of a Williamson Fluid in an Asmmetric Channel, Commun. Nonlinear, Sci. Numer. Simulat. 5, p. 75 (. [5] Nadeem S., Akram Safia, Peristaltic Transport of a Hperbolic Tangent Fluid Model in an Asmmetric Channel, Z. Naturforsch., 64a, p. 55 (. [6] Haat T., Ellahi R., Asghar S., Unstead Magnetohdrodnamic Non-Newtonian Flow Due to Non-Coaial Rotations of a Disk and a Fluid at Infinit, Chem.l Eng. Commun., 4, p. 7 (7. [7] Rao A.R., Mishra M., Peristaltic Transport of a Ppower-Law Fluid in a Porous Tube, J. Non- Newtonian Fluid Mech.,, p. 6 (4. [8] Yıldırım A., Sezer S.A., Effects of Partial Slip on the Peristaltic Flow of a MHD Newtonian Fluid in an Asmmetric Channel, Math. and Comput. Model. 5, p. 68 (. [] Srinivas S., Kothandapani M., The Influence of Heat and Mass Transfer on MHD Peristaltic Flow Through a Porous Space with Compliant Walls, Appl. Math. Comput., Doi.6/j.amc. (. [] Eckert E.R.G., Drake R.M., "Analsis of Heat and Mass Transfer", McGraw-Hill, New York, (7. [] Nadeem S., Noreen Sher Akbar, Influence of Radiall Varing MHD on the Peristaltic Flow in an Annulus with Heat and Mass Transfer, Taiw. Institute. Chem.Enging. Doi:.6/ j.jtice (. [] Nadeem S., Noreen Sher Akbar, Naheeda Bibi, Sadaf Ashiq, Influence of Heat and Mass Transfer on Peristaltic Flow of a Third Order Fluid in a Diverging Tube, Commun. Nonlinear. Sci. Numer. Simulat., Doi:.6/j.cnsns. (. [] S. Nadeem, Noreen Sher Akbar, Influence of Heat and Mass Transfer on a Peristaltic Motion of a Jeffre-Si Constant Fluid in an Annulus, Heat Mass Transf., DOI :.7 / s 585 7, (. 7