# Influence of inclined magnetic field on peristaltic flow of a Jeffrey fluid with heat and mass transfer in an inclined symmetric or asymmetric channel.

код для вставкиСкачатьASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING Asia-Pac. J. Chem. Eng. 2012; 7: 33–44 Published online 22 July 2010 in Wiley Online Library (wileyonlinelibrary.com) DOI:10.1002/apj.488 Research article Influence of inclined magnetic field on peristaltic flow of a Jeffrey fluid with heat and mass transfer in an inclined symmetric or asymmetric channel S. Nadeem1 * and Safia Akram2 1 2 Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan Department of Electrical (Telecom) Engineering, Military College of Signals, National University of Sciences and Technology, Pakistan Received 3 February 2010; Revised 31 May 2010; Accepted 11 June 2010 ABSTRACT: In this article, we have investigated the peristaltic flow of a Jeffrey fluid with heat and mass transfer under the influence of inclined magnetic field in an inclined symmetric or asymmetric channel. The dissipation term in the energy equation is also taken into account. The governing equations are transformed from moving to fixed frame of reference and the resulting equations have then been simplified using the assumptions of long wavelength and low but finite Reynolds number approximation. The reduced equations have been solved numerically by Adomian decomposition method and the exact solutions have also been computed for stream function, temperature and concentration. The expression for pressure rise is computed numerically. Lastly, the physical features of various parameters have been discussed through graphs. 2010 Curtin University of Technology and John Wiley & Sons, Ltd. KEYWORDS: peristaltic flow; Jeffrey fluid; inclined magnetic field; Adomian; exact solution INTRODUCTION Peristaltic flows in symmetric and asymmetric channels have renewed the interest among the researchers due to its applications in physiology and especially in myometrial contractions.[1 – 7] The study of heat transfer analysis in connection with peristalsis has recently acquired a special status. Some recent applications in this area are hypothermia, laser therapy, cryosurgery, hemodialysis, transport of water from the ground to upper branches of trees, sanitary fluid transport, blood pumps in heart–lung machines and transport of corrosive fluids where contact of the fluid with the machinery parts is prohibited. Vajravelu et al .[8] have analyzed the peristaltic flow and heat transfer in a vertical porous annulus with long wavelength approximation. The influence of heat transfer and magnetic field on the peristaltic flow of a Newtonian fluid in a vertical annulus under a zero Reynolds number and long wavelength approximation has been studied by Mekheimer and Abd elmaboud.[9] Nadeem and Akram[10] have examined the heat transfer in a peristaltic flow with partial *Correspondence to: S. Nadeem, Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan. E-mail: snqau@hotmail.com 2010 Curtin University of Technology and John Wiley & Sons, Ltd. Curtin University is a trademark of Curtin University of Technology slip. They[11] also studied the slip effects on the peristaltic flow of a Jeffrey fluid in an asymmetric channel under the effect of an induced magnetic field. The effects of heat transfer on the peristaltic transport of MHD Newtonian fluid with variable viscosity using the Adomian decomposition method have been analyzed by Nadeem and Akbar.[12] In another study, Nadeem and Akbar[13] have discussed the influence of heat transfer on a peristaltic transport of Herschel–Bulkley fluid in a non-uniform inclined tube. The peristaltic transport of a Newtonian fluid in a vertical asymmetric channel with heat transfer and porous medium has been examined by Srinivas and Gayathri.[14] The combined effect of heat and mass transfer is mostly useful in the chemical industry and in reservoir engineering in connection with thermal recovery process, and may be found in salty springs in the sea.[15] To our knowledge, only three papers have discussed the combined effects of heat and mass transfer in peristaltic flows. Eldabe et al .[16] have analyzed the mixed convection heat and mass transfer in a non-Newtonian fluid at peristaltic surface with temperature-dependent viscosity. Heat and mass transfer of blood in a single lymphatic blood vessel with uniform magnetic field has been discussed by Ogulu.[17] Srinivas and Kothandapani[18] have studied the influence of heat and mass transfer on MHD peristaltic flow 34 S. NADEEM AND S. AKRAM Asia-Pacific Journal of Chemical Engineering through a porous space with compliant walls. For some relevant work of interest we refer the reader to.[19 – 25] The study of flows on an inclined plane is also very important. Elshehawey et al .[25] have examined the effects of inclined magnetic field on magnetofluid flow through porous medium between two inclined wavy porous plates. Various authors have studied different kinds of variable magnetic fields here.[25] They considered the magnetic field as a function of inclination angle. The aim of the present investigation is multidimensional: to highlight the heat and mass transfer in non-Newtonian fluids and inclined magnetic field on an inclined symmetric or asymmetric channel. To our knowledge, no attempt has been reported yet to discuss the inclined magnetic field in peristaltic phenomena even in the absence of heat and mass transfer. This study is useful in filling the gap in this direction. The governing equations of motion, energy and concentration are simplified using the assumptions of long wavelength approximation. The exact solution of the reduced equation has been arrived at. The expressions for velocity, pressure gradient, pressure rise, stream function, heat transfer and concentration are found and discussed graphically for different physical parameters. The equations governing the flow with inclined magnetic field and in the presence of gravity effects are[25] ∂V ∂U + =0 (3) ∂X ∂Y ∂U ∂U ∂U ∂p ∂ ρ +U +V =− + (SXX ) ∂t ∂X ∂Y ∂X ∂X ∂ + (SXY ) − σ B02 cos (U cos ∂Y − V sin ) − ρg sin α (4) ∂V ∂V ∂p ∂ ∂V ρ +U +V =− + (SYX ) ∂t ∂X ∂Y ∂Y ∂X ∂ + (SYY ) + σ B02 sin (U cos ∂Y − V sin ) − ρg cos α (5) The heat and mass transfer equations are ∂T ∂T ∂T C +U +V ∂t ∂X ∂Y = where a1 and b1 are the amplitudes of the waves, λ is the wavelength, d1 + d2 is the width of the channel, c is the velocity of propagation, t is the time and X is the direction of wave propagation, φ is the phase difference. φ varies in the range 0 ≤ φ ≤ π; φ = 0 corresponds to a symmetric channel with waves out of phase and for φ = π the waves are in phase. Furthermore, a1 , b1 , d1 , d2 and φ satisfy the condition a12 + b12 + 2a1 b1 cos φ ≤ (d1 + d2 )2 (2) 2010 Curtin University of Technology and John Wiley & Sons, Ltd. (6) ∂C ∂C ∂ 2C ∂ 2C ∂C +U +V = Dm + ∂t ∂X ∂Y ∂X 2 ∂Y 2 Dm KT ∂ 2 T ∂ 2T + + (7) Tm ∂X 2 ∂Y 2 MATHEMATICAL FORMULATION Let us consider the peristaltic flow of an incompressible non-Newtonian fluid (Jeffrey fluid) in a twodimensional channel having widths d1 and d2 , under the effect of a constant magnetic field. Both the magnetic field and channel are inclined at angles and α, respectively. The lower wall of the channel is maintained at temperature T1 while the upper wall has temperature T0 . The geometry of the wall surface is defined as 2π (X − ct) Y = H1 = d1 + a1 cos λ upper wall 2π (X − ct) + φ Y = H2 = −d2 − b1 cos λ lower wall (1) K 2 ∇ T + υ ρ where ∇2 = and = ∂2 ∂X 2 + ∂2 ∂Y 2 ∂ ∂ ∂ 1 +U +V 1 + λ2 1 + λ1 ∂t ∂X ∂Y 2 2 ∂U ∂V 2 ∂V ∂U 2 + , +2 + ∂X ∂Y ∂Y ∂X in which U , V are the velocities in the X and Y directions in a fixed frame, ρ is the constant density, p is the pressure, ν is the kinematic viscosity, σ is the electrical conductivity, g is the acceleration due to gravity, K is the thermal conductivity, C is the specific heat, T is the temperature, Dm is the coefficient of mass diffusivity, Tm is the mean temperature, KT is the thermal diffusion ratio and C is the concentration of fluid. The constitutive equation for the extra stress tensor S is[7] µ (8) S= (γ̇ + λ2 γ̈ ) 1 + λ1 Asia-Pac. J. Chem. Eng. 2012; 7: 33–44 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering PERISTALTIC FLOW OF A JEFFREY FLUID WITH HEAT AND MASS TRANSFER Reδ(y θx − x θy ) In the above equation, λ1 is the ratio of relaxation to retardation times, γ̇ the shear rate, λ2 the retardation time and dots denote the differentiation with respect to time. Introducing a wave frame (x , y) moving with velocity c away from the fixed frame (X , Y ) by the transformation = x = X − ct, y = Y , u = U − c, v = V , p(x ) = p(X , t) and 1 2 (δ xx Sc + yy ) + Sr(δ 2 θxx + θyy ) Reδ(y x − x y ) = x y u v , y= , u= , v= , λ d1 c c Sxx Sxy Syy (10) Making use of Eqns (3)–(10), we obtain the following equations in terms of the stream function ∂ (dropping the bars, u = ∂ ∂y , v = −δ ∂x ) Reδ(y xy − x yy ) Re sin α Fr Reδ 3 (−y xx + x xy ) + δx sin ) + = 0 on y = h1 , a 2 + b 2 + 2ab cos φ ≤ (1 + d )2 (11) (18) (19) Under the lubrication approach, Eqns (11)–(14) become ∂p ∂ 1 ∂ 2 − + − M 2 cos2 (y + 1) ∂x ∂y 1 + λ1 ∂y 2 ∂ ∂p ∂ + δ 2 (Syx ) + δ (Syy ) ∂y ∂x ∂y + M 2 δ sin ((y + 1) cos Re cos α Fr (17) where F is the flux in the wave frame, a, b, φ and d satisfy the relation −M 2 cos ((y + 1) cos + δx sin ) − δ θ = 1 on y = h2 , = 1 on y = h2 , ∂ ∂ ∂p + δ (Sxx ) + (Sxy ) =− ∂x ∂x ∂y =− F at y = h1 = 1 + a cos 2π x , 2 F = − at y = h2 = −d − b cos(2πx + φ)(15) 2 ∂ = −1 at y = h1 , ∂y ∂ = −1 at y = h2 (16) ∂y θ = 0 on y = h1 , = ρνC c2 Sd1 , Pr = , , S = C (T1 − T0 ) K µc C − C0 σ B0 d1 , = µ C1 − C0 ∂ ∂ λ2 cδ 2δ 1+ y = − x xy , 1 + λ1 d1 ∂x ∂y 1 λ2 cδ ∂ ∂ − x (yy − δ 2 xx ), = 1+ y 1 + λ1 d1 ∂x ∂y 2δ λ2 cδ ∂ ∂ − x xy . =− 1+ y 1 + λ1 d1 ∂x ∂y The corresponding boundary conditions are c2 T − T0 Fr = , θ= , gd1 T1 − T0 ρDm Kt (T1 − T0 ) µ Sr = , Sc = , Tm µ(C1 − C0 ) ρDm Ec = (14) where d1 d2 d 2p , d= , p= 1 , λ d1 µcλ ct H1 t = , h1 = , λ d1 H2 a1 b1 h2 = , a= , b= , d2 d1 d1 cd1 Re = , = , v cd1 δ= M = (13) 2 (4δ 2 xy + (yy − δ 2 xx )2 ) (9) Defining x= 1 (θyy + δ 2 θxx ) Pr Ec λ2 cδ + 1+ (1 + λ1 ) d1 ∂ ∂ − x y ∂x ∂y (12) 2010 Curtin University of Technology and John Wiley & Sons, Ltd. + Re sin α = 0 Fr (20) Asia-Pac. J. Chem. Eng. 2012; 7: 33–44 DOI: 10.1002/apj 35 36 S. NADEEM AND S. AKRAM ∂p =0 − ∂y (21) 1 ∂ 2θ Ec + 2 Pr ∂y (1 + λ1 ) ∂ 2 2 =0 ∂y 2 (22) ∂ 2θ 1 ∂ 2 + Sr =0 Sc ∂y 2 ∂y 2 (23) Elimination of pressure from Eqns (20) and (21) gives ∂2 ∂y 2 1 ∂ 2 1 + λ1 ∂y 2 − M 2 cos2 yy = 0 Ec 1 ∂ 2θ + 2 Pr ∂y (1 + λ1 ) ∂ 2 (24) 2 =0 (25) ∂ θ 1 ∂ + Sr 2 = 0 2 Sc ∂y ∂y (26) ∂y 2 2 2 It is observed that the results of Kothandapani et al .[7] can be recovered as a special case of our problem in the absence of heat and mass transfer and when = α = 0. The results of Srinivas et al .[6] can be recovered when = λ1 = 0 and in the absence of heat and transfer analysis. Moreover, the results of Mishra and Rao[3] can be recovered as a special case of our problem in the absence of heat and mass transfer and when = M = α = 0 = λ1 . SOLUTION OF THE PROBLEM Exact solution The exact solution of Eqn (24) can be written as = A + By + A1 cosh M cos 1 + λ1 y + B1 sinh M cos 1 + λ1 y (27) Asia-Pacific Journal of Chemical Engineering √ −2 1 + λ1 +√2 1 + λ1 cosh (h1 − h2 )M 1 + λ1 cos λ1 ) cos +FM (1 + √ sinh (h√1 − h2 )M 1√+ λ1 cos , B= 2 1 + λ1 2 1 + λ1 −√ cosh (h1 − h2 )M 1 + λ1 cos +(h − h )M (1 + λ ) cos 2 1 √ 1 sinh (h1 − h2 )M 1 + λ1 cos √ √ √ (F + h1 − h2 ) 1 + λ1 cosh h1 M 1 + λ1 cos √ λ1 cos − cosh h2 M 1 + √ √ A1 = , 2 1 + λ1 2 1 + λ1 −√ cosh (h1 − h2 )M 1 + λ1 cos +(h − h )M (1 + λ ) cos 2 1 √ 1 sinh (h1 − h2 )M 1 + λ1 cos √ √ (F + h1 − h2 ) 1 + λ1 sinh h1 M 1 + λ1 cos √ λ1 cos − sinh√ h2 M 1 +√ B1 = −2 1 + λ1 +√2 1 + λ1 cosh (h1 − h2 )M 1 + λ1 cos −(h − h )M (1 + λ ) cos 2 1 √ 1 sinh (h1 − h2 )M 1 + λ1 cos (28) Solution by Adomian decomposition method In this section, the Adomian solutions will be determined for the velocity field. According to Adomian decomposition method, we write Eqn (24) in the operator form as Lyyyy = M 2 (1 + λ1 ) cos2 yy . Applying the inverse operator L−1 yyyy dydydydy we can write Eqn (29) as = C 0 + C1 y + C 2 1 y2 y3 + C3 + L−1 (yy ) (30) 2! 3! A yyyy 1 = M 2 (1 + λ ) cos2 and C , C , C , C are where A 1 0 1 2 3 functions of x . Now we decompose where A, B , A1 , B1 are functions of x . √ √ (h1 + h2 ) −2 1 + λ1 + 2 1 + λ1 √ cosh (h1 − h2 )M 1 + λ1 cos λ1 ) cos +FM (1 + √ sinh (h√ + λ1 cos 1 − h2 )M 1√ , A= −4 1 + λ + 4 1 + λ1 1 √ cosh (h1 − h2 )M 1 + λ1 cos −2(h − h )M (1 + λ ) cos 2 1 1 √ sinh (h1 − h2 )M 1 + λ1 cos 2010 Curtin University of Technology and John Wiley & Sons, Ltd. (29) [.] = = ∞ n . (31) n=0 Substituting into Eqn (30), we obtain y2 y3 = C0 + C1 y + C2 + C3 , ei θ 2! 3! 1 (n )yy dydydydy, n ≥ 0(32) n+1 = A Asia-Pac. J. Chem. Eng. 2012; 7: 33–44 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering PERISTALTIC FLOW OF A JEFFREY FLUID WITH HEAT AND MASS TRANSFER Therefore, 4 5 1 C y +C y , 1 = A 2 4! 3 5! 6 y y7 1 2 = 2 C2 6! + C3 7! , A .. . y 2n+2 y 2n+3 1 + C3 , n = n C2 (2n + 2)! (2n + 3)! A Integrating Eqn (39) over one wavelength, we get p = 0 n>0 (33) According to Eqn (31), the closed form of can be written as y = C0 + C1 y + AC2 cosh √ − 1 A √ y y (34) + A AC3 sinh √ − √ A A Now the Adomian solution (36) and exact solution (27) are exactly same, in which A, B , A1 , B1 are calculated using boundary conditions which are defined in Eqn (28). The flux at any axial station in the fixed frame is Q= (u + 1) h2 + dy = c1 = c2 = udy (37) h2 The average volume flow rate over one period (T = λ/c) of the peristaltic wave is defined as 1 T 1 T Qdt = (F + h1 + h2 ) Q= T 0 T 0 dt = F + 1 + d (38) The pressure gradient is obtained from the dimensionless momentum equation for the axial velocity as Re dp = −M 2 cos2 (B + 1) + sin α dx Fr (39) 2010 Curtin University of Technology and John Wiley & Sons, Ltd. 1 2h1 (4 + Brh2 (A21 8(h1 − h2 ) − B12 )(−h1 + h2 ) cos4 M 4 (1 + λ1 ) + BrM 2 cos2 − A21 + B12 h2 cosh[2h1 M 1 + λ1 cos ] + A21 + B12 h1 cosh[2h2 M 1 + λ1 cos ] + 2A1 B1 −h2 sinh[2h1 M 1 + λ1 cos ] . (42) + h1 sinh[2h2 M 1 + λ1 cos ] h1 dy = F + h1 + h2 1 −8 + 2(A21 − B12 ) 8(h1 − h2 ) Br(h12 − h22 )M 4 (1 + λ1 ) cos4 + BrM 2 cos2 (A21 + B12 ) cosh 2h1 M 1 + λ1 cos − (A21 + B12 ) cosh 2h2 M 1 + λ1 cos + 2A1 B1 sinh 2h1 M 1 + λ1 cos − sinh 2h2 M 1 + λ1 cos , h2 h1 (40) Using solution (27) in Eqn (25), the exact solution of the resulting equation satisfying the boundary conditions can be written as Br θ = − M 3 1 + λ1 cos4 (A21 − B12 )M 1 + λ1 y 2 4 A1 B1 sec2 sinh[2My cos 1 + λ1 ] + M 1 + λ1 2 A1 + B12 cosh[2My cos √ 1 + λ1 ] sec2 + + c1 y + c2 , (41) 2M 1 + λ1 y = A + By + A1 cosh √ A y (35) + B1 sinh √ A = A + By + A1 cosh M cos 1 + λ1 y + B1 sinh M cos 1 + λ1 y (36) h1 dp dx dx where which can be put in the simplest form 1 Using solution (41) in Eqn (26), the exact solution of the resulting equation satisfying the boundary conditions can be written as = ScSrBr 3 M 1 + λ1 cos4 A21 − B12 M 1 + λ1 y 2 4 A1 B1 sec2 sinh[2My cos 1 + λ1 ] + M 1 + λ1 Asia-Pac. J. Chem. Eng. 2012; 7: 33–44 DOI: 10.1002/apj 37 S. NADEEM AND S. AKRAM + Asia-Pacific Journal of Chemical Engineering 2 A21 + B1 cosh[2My cos √ 1 + λ1 ] sec2 + c3 y + c4 2M 1 + λ1 5 (43) λ1 = 0.5 λ1 = 1.0 λ1 = 1.5 ∆p 1 (−8 − 2(A21 − B12 ) c3 = 8(h1 − h2 ) ScSrBr(h12 − h22 )M 4 (1 + λ1 ) cos4 + BrM 2 ScSr cos2 −(A21 + B12 ) cosh 2h1 M 1 + λ1 cos + (A21 + B12 ) cosh 2h2 M 1 + λ1 cos + 2A1 B1 − sinh 2h1 M 1 + λ1 cos + sinh 2h2 M 1 + λ1 cos , 0 -5 -1 0 1 Q 2 3 Figure 2. Variation of p with Q for different values of λ1 for fixed b = 0.5, d = 1.2, φ = π/3, = π/4, = π/4, M = 0.5, a = 0.9, Re = 1, α = 0.2, Fr = 0.8. 1 (2h1 (4 + ScSrBrh2 8(h1 − h2 ) 10 (A21 − B12 )(h1 − h2 ) cos4 M 4 (1 + λ1 )) + ScSrBrM 2 cos2 A21 + B12 h2 cosh[2h1 M 1 + λ1 cos ] − A21 + B12 h1 cosh[2h2 M 1 + λ1 cos ] + 2A1 B1 h2 sinh[2h1 M 1 + λ1 cos ] − h1 sinh[2h2 M 1 + λ1 cos ] . (44) 8 Fr = 0.5 6 Fr = 1.0 4 ∆p c4 = Fr = 1.5 2 0 -2 -4 -6 -1 6 Θ = π/8 2 0 1 Q 2 3 Figure 3. Variation of p with Q for different values of Fr for fixed b = 0.5, d = 1.2, φ = π/3, = π/4, M = 2, λ1 = 1, Re = 2, α = 1, a = 0.9. Θ=0 4 ∆p 38 Θ = π/4 NUMERICAL RESULTS AND DISCUSSION 0 -2 -4 -6 -1 -0.5 0 0.5 1 1.5 2 Q Figure 1. Variation of p with Q for different values of for fixed b = 0.5, d = 1.2, φ = π/3, a = 0.5, M = 2, λ1 = 1, Re = 1, α = 0.2 Fr = 0.8. 2010 Curtin University of Technology and John Wiley & Sons, Ltd. In this section, numerical results of the problem under discussion are discussed through graphs. The expression for pressure rise is calculated numerically using mathematics software Mathematica. Figs 1–4 are displayed to see the effects of pressure rise for various values of the angle of inclination , Jeffrey parameter λ1 , Froude number Fr and Hartmann number M . It is observed from Figs 1 and 2 that in the peristaltic pumping (p > 0, Q > 0), retrograde pumping (p > 0, Q < 0) and free pumping (p = 0) regions the pressure rise decreases with an increase in inclination angle and Jeffrey parameter λ1 , while in the Asia-Pac. J. Chem. Eng. 2012; 7: 33–44 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering PERISTALTIC FLOW OF A JEFFREY FLUID WITH HEAT AND MASS TRANSFER 6 3 M = 0.5 4 Θ=0 2 Θ = π/8 M = 1.0 2 1 Θ = π/4 dp\dx ∆p M = 1.5 0 -2 -1 -4 -2 -6 -1 0 1 Q 2 -3 3 0 0.2 0.4 0.6 0.8 1 x Figure 4. Variation of p with Q for different values of M for fixed b = 0.5, d = 1.2, φ = π/3, = π/4, a = 0.9, λ1 = 1, λ1 = 1, Re = 1.0, Fr = 0.8. Figure 6. Variation of dp/dx with x for different values of for fixed b = 0.5, d = 1.2, a = 0.5, M = 2, φ = π/3, λ1 = 1, α = 0.5, Q = 0.5, Fr = 0.8, Re = 0.5. 4 2 3 2 M = 0.5 1.5 M = 1.0 1 M = 1.5 0.5 λ1 = 0.0 0 λ1 = 2.0 -0.5 λ1 = 4.0 1 y dp/dx 0 0 -1 -1 -1.5 -2 0 0.2 0.4 0.6 0.8 1 x -2 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 u Figure 5. Variation of dp/dx with x for different values of M for fixed b = 0.5, d = 1.2, a = 0.9, φ = π/2, = π/8, λ1 = 1, α = 0.5, α = 0.5, Q = 0.5, Fr = 0.8, Re = 0.5. Figure 7. Velocity profile for different values of λ1 for fixed a = 0.7, b = 1.2, d = 2, x = 0, Q = 1, φ = π/2, = π/8, M = 1.0. copumping region (p < 0, Q > 0), the pressure rise increases with an increase in inclination angle and Jeffrey parameter λ1 . From Fig. 3, it can be observed that with an increase in Froude number Fr the pumping rate decreases in all the regions. It is also seen from Fig. 4 that with an increase in Hartmann number M , the pressure rise increases in peristaltic pumping, retrograde pumping and free pumping regions, while in the copumping region the pressure rise decreases. The pressure gradient for different values of M and against x is plotted in Figs 5 and 6. It is shown that for x ∈ [0, 0.2] and x ∈ [0.8, 1], the pressure gradient is small, i.e. the flow can easily pass without imposition of a large pressure gradient, while in the region x ∈ [0.2, 0.8], the pressure gradient increases with an increase in M and decreases with an increase in , so a large pressure gradient is required to enable the flux to pass. In order to see the effects of the Jeffrey parameter λ1 , Hartmann number M and volume flow rate Q on axial velocity Figs 7–9 are plotted. It is observed from Figs 7 and 8 that increase of Jeffrey parameter λ1 and Hartmann number M result in increasing resistance to the flow in the central part of the channel. It is also observed from Fig. 9 that the magnitude of velocity profile decreases with an increase in volume flow rate Q. Figs 10–12 show the variation of temperature profile for different values of Brinkman number Br, inclination angle and Hartmann number M . It is seen from Figs 10 and 11 that the temperature profile increases with an increase in Brinkman number Br and inclination angle . It is seen 2010 Curtin University of Technology and John Wiley & Sons, Ltd. Asia-Pac. J. Chem. Eng. 2012; 7: 33–44 DOI: 10.1002/apj 39 S. NADEEM AND S. AKRAM Asia-Pacific Journal of Chemical Engineering 2 1.5 1.5 Br = 1.0 1 1 Br = 1.5 0.5 Br = 2.0 M = 0.5 y y 0.5 0 M = 1.0 -0.5 M = 1.5 0 -0.5 -1 -1 -1.5 -2 -1 -1.5 -0.8 -0.6 -0.4 u -0.2 0 0 0.2 0.4 0.6 0.2 Figure 8. Velocity profile for different values of M for fixed b = 1.2, d = 2, a = 0.7, = π/8, φ = π/2, λ1 = 1, Q = 2. 0.8 1 1.2 θ Figure 10. Variation of temperature profile for different values of Br for fixed b = 1.2, d = 1.5, a = 0.5, = π/8, φ = π/2, λ1 = 1, x = 0, Q = 2, Br = 2. 2 1.5 Q = 0.0 1 Q = 0.5 0.5 Q = 1.0 1.5 Θ=0 1 Θ = π/4 0.5 Θ = π/3 0 y y 40 0 -0.5 -1 -0.5 -1.5 -1 -2 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 u Figure 9. Velocity profile for different values of Q for fixed a = 0.5, b = 1.2, d = 2, x = 0, λ1 = 1, φ = π/2, = π/8, M = 1.0. from Fig. 12 that with an increase in Hartmann number M the temperature profile decreases. In order to see the effects of concentration on Brinkman number Br, Hartmann number M , Schmidt number Sc and Soret number Sr, Figs 13–15 are prepared. It is observed from Fig. 13 that the concentration decreases with an increase in Brinkman number Br. It is also observed from Fig. 14 that with an increase in Hartmann number M the concentration increases. It is also seen from Fig. 15 that with an increase in Schmidt number Sc and Soret number Sr, the concentration decreases. Trapping phenomena Another interesting phenomena in peristaltic motion is trapping. It is basically the formation of an internally 2010 Curtin University of Technology and John Wiley & Sons, Ltd. -1.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ Figure 11. Variation of temperature profile for different values of for fixed b = 1.2, d = 1.5, a = 0.5, M = 2, φ = π/2, λ1 = 1, x = 0, Q = 2, Br = 2. circulating bolus of fluid by closed stream lines. This trapped bolus pushed a head along peristaltic waves. The trapping phenomena is discussed for different values of λ1 , and M for both symmetric and asymmetric channels. It is observed from the figures that for a symmetric channel the trapping bolus is symmetric about the center line of the channel (see panels (a) and (c)), while in case of an asymmetric channel the bolus tends to shift toward left side of the channel due to phase angle (see panels (b) and (d)). Figure 16 shows the stream lines for different values of Jeffrey parameter λ1 . It is observed from the figure that the size of the trapping bolus increases with an increase in λ1 . Further, we also observed that the size of the trapping bolus is small in the asymmetric channel as compared with the Asia-Pac. J. Chem. Eng. 2012; 7: 33–44 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering PERISTALTIC FLOW OF A JEFFREY FLUID WITH HEAT AND MASS TRANSFER 1.5 1.5 M = 1.0 1 M = 1.0 1 M = 1.5 M = 1.5 0.5 M = 2.0 M = 2.0 y y 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 0 0.2 0.4 0.6 0.8 1 -1.5 -1 1.2 -0.5 0 Φ θ Figure 12. Variation of temperature profile for different values of M for fixed b = 1.2, d = 1.5, a = 0.5, = π/8, φ = π/2, λ1 = 1, x = 0, Q = 2, Br = 2. 0.5 1 Figure 14. Concentration profile for different values of M for fixed b = 1.2, d = 1.5, a = 0.5, Br = 2, φ = π/2, λ1 = 1, Sc = Sr = 1, Q = 2, = π/8. 1.5 1.5 1 Br = 1.0 1 0.5 Br = 1.5 y 0.5 0 y Br = 2.0 0 -0.5 -0.5 -1 Sc = 1.0 , Sr = 0.5 Sc = 1.5 , Sr = 1.0 Sc = 2.0 , Sr = 1.5 -1 -1.5 -0.2 -1.5 -1 0 0.2 0.4 0.6 0.8 1 -0.5 0 Φ 0.5 1 1.2 Φ Figure 13. Concentration profile for different values of Br for fixed b = 1.2, d = 1.5, a = 0.5, M = 2, φ = π/2, λ1 = 1, Sc = Sr = 1, Q = 2, = π/8. Figure 15. Concentration profile for different values of Sc and Sr for fixed b = 1.2, d = 1.5, a = 0.5, M = 2, φ = π/2, λ1 = 1, Br = 2, Q = 2, = π/8. CONCLUSIONS symmetric channel. The stream lines for different values of and M are shown in Figs 17 and 18. It is observed from the figures that the size of the trapping bolus decreases with an increase in and M in both symmetric and asymmetric channels. The results of velocity field against different values of flow rate Q, Jeffrey parameter λ1 and inclination angle are tabulated in Tables 1–3. It is observed from these tables that with the increase in all these three parameters the magnitude of velocity decreases at different values of y. The comparison of the present solution is compared with those available in the literature when some of parameters are replaced by zero in our problem is given in Table 4. 2010 Curtin University of Technology and John Wiley & Sons, Ltd. We have presented the exact and new numerical solutions with the help of Adomian decomposition method of the peristaltic flow of a Jeffrey fluid with heat and mass transfer in an inclined symmetric and asymmetric channel under the influence of an inclined magnetic field. We conclude with the following observations: • It is observed that pressure rise decreases in all the pumping regions with the increase in inclination angle and Jeffrey parameter λ1 . • The pumping rate decreases with the increase in Froude number Fr. • With the increase in Hartmann number M , pressure rise increases in peristaltic pumping, retrograde Asia-Pac. J. Chem. Eng. 2012; 7: 33–44 DOI: 10.1002/apj 41 42 S. NADEEM AND S. AKRAM 1.5 Asia-Pacific Journal of Chemical Engineering 1.5 (a) 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 (b) -1 φ=0 -1.5 -0.5 1.5 0 0.5 -0.5 1.5 (c) 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 φ=0 -1.5 -0.5 0 φ = π/2 -1.5 0 0.5 (d) φ = π/2 -1.5 0.5 -0.5 0 0.5 Figure 16. Stream lines for different values of λ1 (a) and (b) for λ1 = 0.05, (c) and (d) for λ1 = 0.07. The other parameters are Q = 1.5, = π , M = 2.6, b = 0.6, a = 0.6, d = 1.0. 8 1.5 1.5 (a) 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 φ=0 -1.5 -0.5 1.5 (b) 0 φ = π/2 -1.5 0.5 -0.5 1.5 (c) 1 1 0.5 0.5 0 0 -0.5 -0.5 0 0.5 (d) f -1 -1 φ=0 -1.5 -0.5 0 φ = π/2 -1.5 0.5 -0.5 0 0.5 Figure 17. Stream lines for different values of (a) and (b) for = 5π , (c) and (d) for = 5π . 4 6 The other parameters are Q = 1.5, λ1 = 0.05, M = 2.6, b = 0.6, a = 0.6, d = 1.0. 2010 Curtin University of Technology and John Wiley & Sons, Ltd. Asia-Pac. J. Chem. Eng. 2012; 7: 33–44 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering 1.5 PERISTALTIC FLOW OF A JEFFREY FLUID WITH HEAT AND MASS TRANSFER (a) 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 φ=0 -1.5 -0.5 1.5 (b) 0 φ = π/2 -1.5 0.5 -0.5 (c) 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 0 0.5 (d) f -1 -1 φ=0 -1.5 -0.5 0 φ = π/2 -1.5 0.5 -0.5 0 0.5 Figure 18. Stream lines for different values of M (a) and (b) for M = 2.0, (c) and (d) for M = 2.1. The other parameters are Q = 1.5, λ1 = 0.1, = π , b = 0.6, a = 0.6, d = 1.0. 8 Table 1. Comparison of velocity for different values of Q for fixed a = 0.5, b = 1.2, d = 2, x = 0, λ1 = 1, φ = π/2, = π/8, M = 1.0. y −2.0 −1.5 −1 −0.5 0.0 0.5 1.0 1.5 u(x , y) for Q = 0.0 u(x , y) for Q = 0.5 u(x , y) for Q = 1.0 −1 −1 −1 −0.883698 −0.767396 −0.651094 −0.826482 −0.652965 −0.479447 −0.803054 −0.606108 −0.409162 −0.803054 −0.606108 −0.409162 −0.826482 −0.652965 −0.479447 −0.883698 −0.767396 −0.651094 −1 −1 −1 Table 2. Comparison of velocity for different values of λ1 for fixed a = 0.5, b = 1.2, d = 2, x = 0, Q = 1, φ = π/2, = π/8, M = 1.0. y −2.0 −1.5 −1 −0.5 0.0 0.5 1.0 1.5 u(x , y) for λ1 = 0.0 u(x , y) for λ1 = 2 u(x , y) for λ1 = 4 −1 −1 −1 −0.666221 −0.638719 −0.619705 −0.477239 −0.481724 −0.486237 −0.392004 −0.423035 −0.44407 −0.392004 −0.423035 −0.44407 −0.477239 −0.481724 −0.486237 −0.666221 −0.638719 −0.619705 −1 −1 −1 Table 3. Comparison of velocity for different values of for fixed a = 0.5, b = 1.2, d = 2, x = 0, Q = 1, φ = π/2, λ1 = 1, M = 1.0. y −2.0 −1.5 −1 −0.5 0.0 0.5 1.0 1.5 u(x , y) for = 0.0 u(x , y) for = π /6 u(x , y) for = π /4 −1 −1 −1 −0.646577 −0.654479 −0.663391 −0.480225 −0.478901 −0.477608 −0.414243 −0.405341 −0.39523 −0.414243 −0.405341 −0.39523 −0.480225 −0.478901 −0.47608 −0.646577 −0.654479 −0.663391 −1 −1 −1 2010 Curtin University of Technology and John Wiley & Sons, Ltd. Asia-Pac. J. Chem. Eng. 2012; 7: 33–44 DOI: 10.1002/apj 43 44 S. NADEEM AND S. AKRAM Asia-Pacific Journal of Chemical Engineering Table 4. Comparison of velocity in case of Newtonian and non-Newtonian fluid. y −1.5 −1 −0.5 0.0 0.5 1.0 1.5 Ours Kothandapani et al .[7] Srinivas et al .[6] Mishra[3] −1 −1 −1 −1 −0.537189 −0.511971 −0.549315 −0.58333 −0.362456 −0.3843444 −0.354278 −0.333333 −0.317708 −0.354894 −0.299941 −0.25 −0.362456 −0.381344 −0.354278 −0.333333 −0.537189 −0.381344 −0.549315 −0.58333 −1 −1 −1 −1 pumping and free pumping regions while it decreases in the copumping region. • In the center of the channel, the pressure gradient increases with an increase in M and decreases with the increase in . • The temperature profile increases with the increase in Br and and decreases with an increase in M . • The concentration field increases with an increase in M and decreases with an increase in Sc and Sr. • The size of the trapping bolus decreases with the increase in and M . • A nice comparison of our results and available results available in the limiting case are also presented in the table 4. NOMENCLATURE U , V Velocity components in X and Y directions in fixed frame u, v Velocity components in x and y directions in wave frame ρ Constant density p Pressure σ Electrical conductivity C Thermal diffusion ratio K Thermal conductivity M Hartmann number Br Brinkmann number Pr Prandtl number Re Reynolds number Sc Schmidt number Sr Soret number g Acceleration due to gravity Dm Coefficient of mass diffusivity KT Thermal diffusion T Temperature of fluid in dimension form θ Temperature of fluid in dimensionless form C Concentration of fluid in dimension form Concentration of fluid in dimensionless form φ Amplitude ratio a1, b1, Amplitude of waves λ Wavelength c Velocity of propagation Q Volume flow rate δ Long wavelength 2010 Curtin University of Technology and John Wiley & Sons, Ltd. Stream function Fr Froude number α, Inclination angle of channel and magnetic field respectively Temperature at upper wall T0 T1 Temperature at lower wall S Extra stress tensor REFERENCES [1] T.W. 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