Simultaneous effects of Hall and convective conditions on peristaltic flow of couple-stress fluid in an inclined asymmetric channel

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1 PRAMANA c Indian Academy of Sciences Vol. 85, No. 1 journal of July 015 physics pp Simultaneous effects of Hall and convective conditions on peristaltic flow of couple-stress fluid in an inclined asymmetric channel THAYAT 1,, MARYAM IQBAL 1, HUMAIRA YASMIN 1,, FUAD E ALSAADI and HUIJUN GAO 3 1 Department of Mathematics, Quaid-I-Azam University 4530, Islamabad 44000, Pakistan Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 1589, Saudi Arabia 3 Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin, , People s Republic of China Corresponding author. qau011@gmail.com MS received 6 September 013; revised 4 February 014; accepted 16 June 014 DOI: /s ; epublication: 5 January 015 Abstract. A mathematical model is developed to analyse the peristaltic flow of couple-stress fluid in an inclined asymmetric channel with convective conditions. Soret and Dufour and Hall effects are taken into account. Analysis has been carried out in a wave frame of reference. Expressions for velocity, pressure gradient, temperature and concentration are constructed. Pumping and trapping phenomena are examined. Impact of sundry parameters on the velocity, temperature and concentration is discussed. Keywords. Couple stress fluid; convective conditions; Hall effect; inclined channel. PACS No d 1. Introduction The study of peristaltic pumping of fluids generated by a progressive wave of area concentration or expansion along the length of distensible tube is significant in physiological and industrial processes. Many physiological fluids like urine from kidney to bladder through ureters, food material through the digestive tract, semen in vas deferens, spermatozoa in ductus efferentes, blood circulation in small blood vessels, cilia movement, bile flow, etc. obey the principle of peristalsis. The process of peristaltic mechanism is also useful in heart lung machine, dialysis machine, roller and finger pumps and noxious fluid transport in nuclear industry. Ample investigations for peristalsis have been made in view of aforementioned applications. Such analyses have been carried out under one or more simplified assumptions of large wavelength, small amplitude ratio, small Reynolds Pramana J. Phys., Vol. 85, No. 1, July

2 T Hayat et al number and small wave number see recent attempts [1 11] and several related studies therein. The interaction of peristalsis with heat transfer is quite obvious in the oxygenation and hemodialysis processes. Simultaneous effects of heat and mass transfer cannot be ignored in heat convection due to blood flow through the pores of tissues and metabolic heat generation and external interactions such as electromagnetic radiation from cell phones. Magnetohydrodynamic MHD character of fluid is important in MHD power generators, metallurgical process, magnetic resonance imaging MRI, magnetic devices, magnetic particles as the drug carriers, in cancer tumour treatment and also to reduce bleeding during surgeries. Hence, several researchers discussed the peristaltic transportof fluidthrough MHD/heat and mass transfer aspects. A few representative contributions in this direction can be seen in [1 0]. When heat and mass transfer occur simultaneously in a moving fluid, the relations between the fluxes and the driving potentials are of more intricate nature. It has been observed that an energy flux can be generated not only by the temperature gradient but also by the concentration gradient. The energy flux caused by a concentration gradient is termed as the diffusion-thermo Dufour effect. On the other hand, mass fluxes can also be created by temperature gradients and this embodies the thermo diffusion Soret effect. In most of the studies related to heat and mass transfer process, Soret and Dufour effects are neglected their magnitude is smaller than the effects described by the Fourier s defines the thermal conductivity of a material and Fick s define the diffusion coefficient for species in the mixture laws. Note that the effects of Soret and Dufour are significant when density differences are present in the flow regime. Another important aspect is related to Hall effect. Such effect cannot be ignored when flow subjected to high magnetic field is considered. It has been noticed that peristalsis in the presence of Hall/Soret and Dufour effects is given very less attention. To our knowledge, only few studies [1 6] are available in this direction. On the other hand, in the past, the peristalsis mostly dealt with the heat transfer effect through either prescribed surface temperature or prescribed heat flux. Only limited information is available about the peristalsis with convective boundary conditions [7,8]. The purpose of the present investigation is three-fold: Firstly, to analyse the effect of inclined magnetic field on peristaltic flow of couple-stress fluid in an inclined channel, secondly to discuss the influence of Hall current and thirdly to examine the flow in an asymmetric channel with convective effects. It is important to consider asymmetric channel because uterine peristalsis resulting from myometrial concentrations can take place in both symmetric and asymmetric directions. Mathematical analysis is presented for large wavelength and low Reynolds number. Flow quantities of interest are analysed in detail for various pertinent parameters.. Mathematical modelling and solution We investigate the peristaltic flow of an incompressible couple-stress fluid in an inclined asymmetric channel. The fluid is electrically conducting in the presence of an inclined magnetic field. The induced magnetic field is neglected for small magnetic Reynolds number. Hence B = B 0 sinθ,b 0 cos θ, Pramana J. Phys., Vol. 85, No. 1, July 015

3 Peristaltic flow of couple-stress fluid Figure 1. Schematic diagram of the problem. We consider the Cartesian coordinate system in the laboratory frame X, Ȳ, t. We have the also considered the heat and mass transfer processes in the presence of Soret and Dufour effects. The channel walls satisfy convective boundary conditions. Moreover, the effect of Hall current is taken into account. Thus, the generalized Ohm s law can be expressed as [ J = σ V B 1 ] J B, en where effect of electrical field is not considered and geometry of the problem is shown in figure 1. From the above equation J B = σ V B B σ [J B B]. 3 en Using velocity V = Ū, V,0 and eqs 1 and 3, we obtain [ σ B0 J B = 1 + m cos θū cos θ V sinθ, σ B0 1 + m sinθū cos θ V sinθ, σ B0 m ] 1 + m Ū cos θ V sinθ, 4 in which J is the current density, B 0 is the applied magnetic field, σ is the electrical conductivity, Ū, V are the velocity components in fixed frame, m=σ B 0 /en is the Hall parameter, n is the number density of electrons, e is the electric charge and θ is the inclination of the magnetic field. Pramana J. Phys., Vol. 85, No. 1, July

4 T Hayat et al Under the assumptions of long wavelength and low Reynolds number Re 0 [9,30] mathematical model of the given problem is dp dx + u y u η 4 y M Re cos θucos θ + cos θ+ sinα = 0, m Fr p = 0, 6 y γ u y + Br u + ηbr + Pr Du ϕ = 0, 7 y y y ϕ y + Sc γ Sr = 0. 8 y Clearly eq. 6 implies that p = py. The corresponding boundary conditions and wall geometries h 1 x and h x in the dimensionless form are given by u = 1, u y = 0, γ y + Bi 1γ = 0, ϕ= 0, at y = h 1 x = 1 + a sinπx, 9 u = 1, u y = 0, γ y Bi γ = 0, ϕ= 1, at y = h x = d b sinπx + φ, 10 where Bi 1 and Bi are the Biot numbers at the upper and lower walls of the channel, a = a 1 /d 1, b = a /d 1 and d = d /d 1 satisfy the condition a + b + ab cos φ 1 + d. 11 Detailed simplification of the mathematical model and its corresponding boundary conditions are given in Appendix A. The dimensionless average flux F in the wave frame is defined as h1 F = u dy. 1 h Dimensionless average flux in the laboratory σ and wave F framesarerelatedbythe following expression: σ = F d. 13 The non-dimensional expression for the rise in pressure per wavelength p λ is given by p λ = 1 The closed form solution of eq. 5 is 0 dp dx. 14 dx uy = C 1 e y/ M 1 + C e y/ M 1 + C 3 e y/ M + C 4 e y/ M H sec θ, 15 z 18 Pramana J. Phys., Vol. 85, No. 1, July 015

5 Peristaltic flow of couple-stress fluid Using eqs 1 and 15 we obtain dp/dx in the form dp dx = f h 1 + h + h 11 + m R sec θ tan θ FrM h 1 + m R sec θ tan θ FrM A4 e e h 1 A3 η e h A3 η A3 A3 h 1 η h η +e FrM A 3 1+m R sec θ tan θ η A 5 A 3 e A 1 e A 1+m R sec θ tan θ e A 1 + e A FrM A 4 /ηa 5 + e A 1 e A 1+m R A 3 + M η cos θ+m A 3 sec θ tan θ / e A 1 +e A FrM A 4 η A h1 1+m sec θ 6 / M h 1+m sec θ A3 A3 h 1 η h η 1+m A M 4 e e sec θ/ e h 1 A3 η +e A3 h η M A 3 η A 5 A 3 e A 1 e A 1 + m sec θ e A 1 + e A FrM A 4 /ηa 5 + e A 1 e A 1 + m A 3 + M η cos θ + m A 3 sec θ / e A 1 + e A M A 4 η A / Substituting eqs 5 and 15 into eq. 7, the closed form solution of γyis given by γy= C 5 + yc Pr Sc Sr Du Bre M1 +M y e M y 3+M 3 C + e M1 y 3 + M 3 C Pr Sc Sr Du Br y zc 1 C + C 3 C 4 cos θ M 1 M 1+Pr Sc Sr Du Br e y/ M 1 +M C C 3 M 1 M z cos θ Br M1 M M 1 M ey/ 1 + Pr Sc Sr Du Pramana J. Phys., Vol. 85, No. 1, July

6 T Hayat et al C 1 C 4 M 1 M z cos θ+ Br e y/ M 1 +M C 1 C 3 M 1 M + z cos θ + Br M 1 + M e y/ 1 + Pr Sc Sr Du C C 4 M 1 M + z cos θ M Pr Sc Sr Du Br e M y C 3 + M 3 zη cos θ M Pr Sc Sr Du Br e M1 y M 1 + M 1+Pr Sc Sr Du M1 +M C 1 + M 3 + zη cos θ, 17 where the constants C 5 and C 6 have the involvement of the Biot numbers Bi 1 and Bi. Substituting eq. 17 into eq. 8 we get the closed form solution of ϕy as follows: ϕy = C 7 + yc 8 + Br Sc Sr zy C 1 C + C 3 C 4 cos θ 1 + Pr Sc Sr Du Br e M y Sc Sr C 3 + M 3 zη cos θ 41 + M Pr Sc Sr Du +Br e M 1 y Sc Sre M 1 y C 1 3e M 1 y M 3 C 1 4e M 1 y zηc 1 +6e M 1 y M 3 zηc 1 +6e M 1 y z η C 1 + C 3M 3C 4zηC + 6M 3zηC +6z η C 8ey/ 3M 1 +M M 1 M M 3 ηc 1 C 3 + 8e y/ 3M 1 +M M 3 zηc 1 C 3 +8e y/ 3M 1 +M M 1 M M 3 zη C 1 C 3 +8e y/ M 1 +M M 1 M M 3 ηc C 3 + 8e y/ M 1 +M M 3 zηc C 3 8e y/ M 1 +M M 1 M M 3 zη C C 3 + 8e y/ 3M 1 M M 1 M M 3 ηc 1 C 4 +8e y/ 3M 1 M M 3 zηc 1 C 4 8e y/ 3M 1 M M 1 M M 3 zη C 1 C 4 8e y/ M 1 M M 1 M M 3 ηc C 4 + 8e y/ M 1 M M 3 zηc C 4 +8e y/ M 1 M M 1 M M 3 zη C C 4 e M1 M y C 4 3e M1 M y M 3 C 4 M1 M +4e y zηc4 + 6e M1 M y M 3 zηc4 6e M1 M y z η C4 + zηe M 1 y + 3M 3 + 4zηC M 3 + 4zηC +4e y/ 3M 1 +M M 3 1 +M 1 M ηc 1 C 3 4e y/ M 1 +M M M 1 M ηc C Pramana J. Phys., Vol. 85, No. 1, July 015

7 Peristaltic flow of couple-stress fluid 4e y/ 3M 1 M M M 1 M η C 1 C 4 + 4e y/ M 1 M M M 1 M ηc C 4 M1 M +e y + 3M 3 4zηC4 cos θ +z η e M 1 y C1 + C e M1 M y C4 cos 4θ /4M 1 M 1 + Pr Sc Sr Duη 1 + 4zη cos θ, 18 where the constants appearing in the solutions are given in Appendix B. 3. Graphical results and discussion 3.1 Pumping characteristics Figures 13 examine the effects of various pertinent flow parameters on the longitudinal pressure gradient dp/dx and pressure rise per wavelength p λ. Mathematical form of the pressure gradient is given in eq. 16. Note that eq. 14 involves the integration of dp/dx. The integral in eq. 14 is not solvable analytically. Hence Mathematica has been used for its computation. Effect of couple-stress parameter η on pressure gradient dp/dx against x is plotted in figure. It is realized that dp/dx increases at the centre of the channel while it decreases near the boundary walls. It is also interesting to note from figure that resistance or assistance from dp/dx for a non-newtonian fluid η = 0 is higher than the Newtonian fluid η 0. Figure 3 indicates that dp/dx decreases at the centre of the channel and it increases near the boundary walls with an increase of Hall parameter m. Figure 4 elucidates the decreasing behaviour of dp/dx at the centre of the channel and it has increasing behaviour near the channel walls with the increasing values of inclination angle θ for magnetic field. Figures 5 7 illustrate the variation of pressure rise per wavelength p λ vs. average flux σ. From figure 5 we observe that in peristaltic pumping region p λ > 0,σ > 0 the pressure rise increases while it decreases in the co-pumping region p λ < 0,σ >0 with an increase in the couple-stress parameter η. It means that the peristaltic pumping Figure. Plot of pressure gradient dp/dx for various values of η when a = 0.6, b = 0.7, d = 1.5, R = 0.5, φ = π/4, σ = 1, α = π/3, m = 0.03, Fr = 1., θ = π/3 and M = 4. Pramana J. Phys., Vol. 85, No. 1, July

8 T Hayat et al Figure 3. Plot of pressure gradient dp/dx for various values of m when a = 0.6, b = 0.7, d = 1.5, R = 0.5, φ = π/4, σ = 1, α = π/3, Fr = 1., θ = π/3,η= 0.1 and M = 4. Figure 4. Plot of pressure gradient dp/dx for various values of θ when a = 0.6, b = 0.7, d = 1.5, R = 0.5, Fr = 1., σ = 1, α = π/3, η = 0.1, φ = π/4,m= 0.03 and M = 4. Figure 5. Plot of pressure rise P λ for various values of η when a = 0.7, b = 1., d =, R = 0.5, Fr = 1., α = π/3, φ = π/4,m= 0.03, M = 4andθ = π/3. for the non-newtonian fluid η = 0 is greater than that for the Newtonian fluid η 0. Figure 6 reveals that the pressure rise decreases in the retrograde p λ > 0,σ < 0 region and it increases in the co-pumping region p λ < 0,σ >0 as the Hall parameter m increases. Figure 7 elucidates that the pressure rise decreases in the peristaltic pumping region p λ > 0,σ > 0 when there is an increase in the angle of inclination θ of the magnetic field and it increases in the co-pumping region p λ < 0,σ >0. 13 Pramana J. Phys., Vol. 85, No. 1, July 015

9 Peristaltic flow of couple-stress fluid Figure 6. Plot of pressure rise P λ for various values of m when a = 0.7, b = 1., d =, R = 0.5, Fr= 1., α = π/3, φ = π/4,η= 0.1, M = 4andθ = π/3. Figure 7. Plot of pressure rise P λ for various values of θ when a = 0.7, b = 1., d =, R = 0.5, Fr= 1., α = π/3, φ = π/4,m= 0.03, η = 0.1andM = Velocity behaviour In this subsection we discuss the effect of various physical parameters on the velocity profile uy. Mathematical form of the velocity is given in eq. 15. Figure 8 shows that the longitudinal velocity profile uy decreases when couple-stress parameter η increases. It is also interesting to note that for non-newtonian fluid η = 0 the longitudinal velocity Figure 8. Plot of velocity field uy for various values of η when a = 0.6, b = 0.7, d = 1.5, R = 0.5, Fr = 1., σ = 1, α = π/3, M = 4, φ = π/4,m= 0.03, θ = π/3, x = 0.5 anddp/dx = 1. Pramana J. Phys., Vol. 85, No. 1, July

10 T Hayat et al profile uy is lesser than that of Newtonian fluid η 0. As couple-stress parameter depends on the couple-stress viscosity and this couple-stress viscosity acts as a retarding agent which makes the fluid more dense resulting in a decrease in the velocity of the fluid. Figure 9 gives the information that for subcritical flow Fr < 1 the velocity profile uy has greater effect than that of critical Fr = 1 and supercritical Figure 9. Plot of velocity field uy for various values of Fr when a = 0.6, b = 0.7, d = 1.5, R = 0.5, η = 0.1, σ = 1, α = π/3, M = 4, φ = π/4,m= 0.03, θ = π/3, x = 0.5 anddp/dx = 1. Figure 10. Plot of velocity field uy for various values of m when a = 0.6, b = 0.7, d = 1.5, R = 0.5, η = 0.1, σ = 1, α = π/3, Fr = 1., φ = π/4, M = 4, θ = π/3, dp/dx = 1andx = 0.5. Figure 11. Plot of velocity field uy for various values of θ when a = 0.6, b = 0.7, d = 1.5, R = 0.5, η = 0.1, σ = 1, α = π/3, Fr = 1., φ = π/4, m = 0.03, M = 4, dp/dx = 1andx = Pramana J. Phys., Vol. 85, No. 1, July 015

11 Peristaltic flow of couple-stress fluid Fr > 1 flow. The velocity profile uy decreases with an increase of Froude number Fr. Figure 10 demonstrates that the increase the velocity profile uy increases as Hall parameter m increases. Figure 11 shows that with an increase in inclination angle θ of the magnetic field θ = 0, the longitudinal velocity uy has greater value in comparison to the case when θ = Temperature profile Figures 1 0 demonstrate the variation of temperature profile γyfor embedded physical parameters. From figure 1 it is clear that as couple-stress parameter η increases, the temperature profile γydecreases. In fact, the average kinetic energy decreases due to the decrease in velocity of the fluid and hence temperature decays. Figure 13 depicts that with an increase of inclination angle θ of the magnetic field, the temperature profile γyincreases. Similar observation is noticed for Hall parameter m see figure 14. It is found from figure 15 that the temperature profile γyhas greater effects for supercritical Figure 1. Plot of temperature profile γy for various values of η when a = 0.5, b = 0.5, d = 1, R = 0., M = 4, σ = 1, α = π/3, Fr = 1., φ = π/4,m= 0.03, θ = π/3, dp/dx =, x = 0.5, Du = 0.1, Bi 1 = 10, Bi = 10, Sc = 0.5, Sr = 0.6, Pr = andbr=. Figure 13. Plot of temperature profile γy for various values of θ when a = 0.6, b = 0.7, d = 1.5, R = 0., M = 4, σ = 1, α = π/3, Fr = 1., φ = π/4,m= 0.03, η = 0.1, dp/dx =, x = 0.5, Du = 0.1, Bi 1 = 10, Bi = 10, Sc = 0.5, Sr = 0.6, Pr = andbr=. Pramana J. Phys., Vol. 85, No. 1, July

12 T Hayat et al Figure 14. Plot of temperature profile γy for various values of m when a = 0.6, b = 0.7, d = 1.5, R = 0., M = 4, σ = 1, α = π/3, Fr = 1., φ = π/4,η= 0.1, θ = π/3, dp/dx =, x = 0.5, Du = 0.1, Bi 1 = 10, Bi = 10, Sc = 0.5, Sr = 0.6, Pr = andbr=. Figure 15. Plot of temperature profile γy for various values of Fr when a = 0.6, b = 0.7, d = 1.5, R = 0., M = 4, σ = 1, α = π/3, η = 0.1, φ = π/4,m= 0.03, θ = π/3, dp/dx =, x = 0.5, Du = 0.1, Bi 1 = 10, Bi = 10, Sc = 0.5, Sr = 0.6, Pr = andbr=. Figure 16. Plot of temperature profile γy for various values of Br when a = 0.6, b = 0.7, d = 1.5, R = 0., M = 4, σ = 1, α = π/3, Fr = 1., φ = π/4,m= 0.03, θ = π/3, dp/dx =, x = 0.5, Du = 0.1, Bi 1 = 10, Bi = 10, Sc = 0.5, Sr = 0.6, Pr = andη = Pramana J. Phys., Vol. 85, No. 1, July 015

13 Peristaltic flow of couple-stress fluid Figure 17. Plot of temperature profile γy for various values of Sr when a = 0.6, b = 0.7, d = 1.5, R = 0., M = 4, σ = 1, α = π/3, Fr = 1., φ = π/4,m= 0.03, θ = π/3, dp/dx =, x = 0.5, Du = 0.1, Bi 1 = 10, Bi = 10, Sc = 0.5, Br =, Pr = andη = 0.1. Figure 18. Plot of temperature profile γy for various values of Du when a = 0.6, b = 0.7, d = 1.5, R = 0., M = 4, σ = 1, α = π/3, Fr = 1., φ = π/4,m= 0.03, θ = π/3, dp/dx =, x = 0.5, Br =, Bi 1 = 10, Bi = 10, Sc = 0.5, Sr = 0.6, Pr = andη = 0.1. Figure 19. Plot of temperature profile γyfor various values of Bi 1 when a = 0.6, b = 0.7, d = 1.5, R = 0., M = 4, σ = 1, α = π/3, Fr = 1., φ = π/4,m= 0.03, θ = π/3, dp/dx =, x = 0.5, Br =, Du = 0.1, Bi = 10, Sc = 0.5, Sr = 0.6, Pr = andη = 0.1. Pramana J. Phys., Vol. 85, No. 1, July

14 T Hayat et al Figure 0. Plot of temperature profile γyfor various values of Bi when a = 0.6, b = 0.7, d = 1.5, R = 0., M = 4, σ = 1, α = π/3, Fr = 1., φ = π/4,m= 0.03, θ = π/3, dp/dx =, x = 0.5, Br =, Du = 0.1, Bi 1 = 10, Sc = 0.5, Sr = 0.6, Pr =. flow Fr > 1 and critical flow Fr = 1 than subcritical flow Fr < 1. The temperature profile γy increases with an increase of Froude number Fr. In figure 16 we see the effect of Br on γy.brinkman number Br = 0 means that the effect of viscous dissipation is not considered and when Brinkman number Br = 0 the effects of viscous dissipation are taken into account and then the temperature profile γy increases. It is noticed that temperature profile γy increases with an increase in the value Sr as shown in figure 17. It is also noticed from figure 18 that the temperature profile γy has an increasing behaviour for large values of Dufour number Du. Figure 19 elucidates that with an increase of Bi 1 the temperature profile γy decreases near the upper wall while it has no significant effect near the lower wall. On the other hand, with an increase in Bi the temperature profile decreases near the lower wall and it has no significant effect near the upper wall see figure 0. Also we have considered Biot numbers much larger than 0.1 because of the consideration of non-uniform temperature fields within the fluid. Figure 1. Plot of concentration profile ϕy for various values of η when a = 0.6, b = 0.7, d = 1.5, R = 0., M =, σ = 1, α = π/3, Fr = 1., φ = π/4, m = 0.03, θ = π/3, dp/dx =, x = 0.5, Br =, Sr = 0.6, Du = 0.1, Pr = 4and Sc = Pramana J. Phys., Vol. 85, No. 1, July 015

15 Peristaltic flow of couple-stress fluid Figure. Plot of concentration profile ϕy for various values of Fr when a = 0.6, b = 0.7, d = 1.5, R = 0., M =, σ = 1, α = π/3, η = 0.1, φ = π/3, m = 0.03, θ = π/3, dp/dx =, x = 0.5, Br =, Sr = 0.6, Du = 0.1, Pr = 4and Sc = 1. Figure 3. Plot of concentration profile ϕy for various values of m when a = 0.6, b = 0.7, d = 1.5, R = 0., M =, σ = 1, α = π/3, Fr = 1., φ = π/3, η = 0.1, θ = π/3, dp/dx =, x = 0.5, Br =, Sr = 0.6, Du = 0.1, Pr = 4and Sc = 1. Figure 4. Plot of concentration profile ϕy for various values of Sr when a = 0.6, b = 0.7, d = 1.5, R = 0., M =, σ = 1, α = π/3, Fr = 1., φ = π/3, m = 0.03, θ = π/3, dp/dx =, x = 0.5, Br =, η = 0.1, Du = 0.1, Pr = 4and Sc = 1. Pramana J. Phys., Vol. 85, No. 1, July

16 T Hayat et al Figure 5. Plot of concentration profile ϕy for various values of θ when a = 0.6, b = 0.7, d = 1.5, R = 0., M =, σ = 1, α = π/3, Fr = 1., φ = π/3, m = 0.03, η = 0.1, dp/dx =, x = 0.5, Br =, Sr = 0.6, Du = 0.1, Pr = 4and Sc = 1. Figure 6. Plot of concentration profile ϕy for various values of Du when a = 0.6, b = 0.7, d = 1.5, R = 0., M =, σ = 1, α = π/3, Fr = 1., φ = π/3, m = 0.03, θ = π/3, dp/dx =, x = 0.5, Br =, η = 0.1, Sr = 0.6, Pr = 4and Sc = Concentration profile The purpose of this subsection is to see the salient features of concentration profile ϕy for different values of the involved parameters. Figure 1 shows that ϕy increases when η increases. In non-newtonian fluid η = 0, the concentration profile ϕy has greater effect than that of Newtonian fluids η 0. Figure displays that for subcritical flow Fr < 1 the concentration profile ϕy is greater than the critical flow Fr = 1 and supercritical flow Fr > 1. It means that as the Froude number Fr increases, the concentration profile ϕy decreases. It is revealed from figure 3 that the concentration profile ϕy decreases by an increase in Hall parameter m. Figure 4 shows that for larger Sr the concentration profile ϕy decreases. Figures 5 and 6 depict that concentration profile ϕy decreases when θ and Dufour number Du increase figures 7, 8 and Trapping Effect of Froude number Fr on trapping is illustrated in figure 7. It is observed that the size of the trapped bolus decreases when Fr increases and it disappears for supercritical 140 Pramana J. Phys., Vol. 85, No. 1, July 015

17 Peristaltic flow of couple-stress fluid Figure 7. Streamlines for a = 0.5, b = 0.7,R= 10, d=, φ = π/6, σ = 1, α= π/3,dp/dx =,m= 0.03,η= 0.1, θ = π/3,m= 3whenaFr= 1, bfr= 1.5, c Fr=. flow Fr>1 Fr = see panel c. It is concluded from figure 8 that when couple-stress parameter η increases, the bolus decreases in size. Figure 9 shows that by increasing the values of m the trapping bolus reduces in size. 4. Conclusions The peristaltic flow of couple-stress fluid in an inclined asymmetric channel with convective conditions was discussed. Effect of Soret thermodiffusion and Dufour diffusionthermo were also considered. Effect of Hall current was taken into account. The main findings of the present research are mentioned below. Pramana J. Phys., Vol. 85, No. 1, July

18 T Hayat et al a b c Figure 8. Streamlines for a = 0.5, b = 0.7,R= 10,d=, φ = π/6, σ = 1, α = π/3, dp/dx =,m= 0.03, Fr = 1., θ = π/3,m= 3whena η = 1, b η = 1.5, c η =. 1 Pressure gradient dp/dx shows the increasing and decreasing behaviour at the centre of the channel and near the boundary walls respectively when η increases. For increasing Hall parameter m, the pressure gradient dp/dx increases near the boundary walls and decreases at the centre of the channel. 3 Pressure rise p λ decreases in the retrograde region and increases in the co-pumping region when the Hall parameter m increases. Similar behaviour is observed for inclination angle θ of the magnetic field. 4 Magnitude of the velocity profile uy increases when there is an increase in the inclination angle θ of the magnetic field and Hall parameter m. 5 Velocity profile uy decreases with increasing values of couple-stress parameter η. 14 Pramana J. Phys., Vol. 85, No. 1, July 015

19 Peristaltic flow of couple-stress fluid Figure 9. Streamlines for a = 0.5, b = 0.7,R= 10,d=, φ = π/6, σ = 1, α = π/3, dp/dx =,M= 3,η= 0.1, θ = π/3, Fr = 1. whena m = 0.5, b m = 1, c m = Temperature profile γy increases when Hall parameter m and Froude number Fr are enhanced. 7 With an increase in the Biot numbers Bi 1,Bi the temperature profile γydecreases. 8 Concentration profile ϕy is a decreasing function of Fr, m and Du. 9 The size of the trapping bolus decreases with the increase in Froude number Fr and Hall parameter m while it increases when the angle of inclination of the channel increases. Acknowledgements The authors are grateful to the reviewer for the useful suggestions. This paper was funded by the Deanship of Scientific Research DSR, King Abdulaziz University, Jeddah Pramana J. Phys., Vol. 85, No. 1, July

20 T Hayat et al under grant no. 8-13/1434HiCi. The authors, therefore, acknowledge with thanks DSR technical and financial support. Appendix A The conservation laws for the governing flow problems are as follows: Ū X + V Ȳ = 0, ρ t +Ū X + V Ȳ Ū = P X + μ Ū X + Ū Ȳ η 4 Ū X + 4 Ū 4 X Ȳ + 4 Ū Ȳ 4 A1 ρ t +Ū X + V Ȳ ρξ σ B m cos θ Ū cos θ V sinθ + gρ sinα, V = P Ȳ + μ V X + V Ȳ η 4 V X + 4 V 4 X Ȳ + 4 V Ȳ 4 A + σ B m sinθ Ū cos θ V sinθ gρ cos α, t +Ū X + V [ T T =κ Ȳ X + T +μ Ȳ Ū A3 V + Ȳ X Ū + Ȳ + ] [ V + η X V X + V Ȳ ] Ū Ū + X + + ρdk T Ȳ c s t + Ū X + V C C = D Ȳ X + C + DK T Ȳ T m The boundary conditions and geometry of the channel walls are given by Ū = 0, κ T Ȳ = η 1 T T a, C = C 0, 144 Pramana J. Phys., Vol. 85, No. 1, July 015 C C X +, Ȳ T X + T Ȳ A4. A5

21 at at Peristaltic flow of couple-stress fluid [ ] Ȳ = H π 1 X, t = d 1 + a 1 sin λ X c t, A6 Ū = 0, κ T Ȳ = η T a T, C = C 1, [ ] Ȳ = H π X, t = d a sin λ X c t+ φ, A7 where P,C, T, k, ξ, T m, D, ρ and K T are respectively the pressure, concentration, temperature, thermal conductivity, specific heat at constant pressure, mean temperature of the medium, coefficient of mass diffusivity, fluid density and thermal-diffusion ratio. Further c s is the concentration susceptibility, η is the couple-stress fluid parameter and α is the inclination of the channel. Here η 1 and η are the heat transfer coefficients at the upper and lower walls respectively, T is the temperature of the fluid, T a is the ambient temperature, C 0 and C 1 are the concentration fields of the upper and lower walls respectively, d 1 and d are the distances of the upper and lower walls of the channel from the centerline Ȳ = 0, a i i = 1, are the wave amplitudes at the upper and lower walls, λ is the wavelength, t is the time and φ is the phase difference varying in the range 0 φ π. Here φ = 0 corresponds to symmetric channel with waves out of phase and φ = π the waves in phase, ρg sinα and ρg cos α are the X and Ȳ components of the body force in view of the inclined channel and H 1 and H are the upper and lower walls of the channel. Moreover, a i and d i i = 1, satisfy the condition a 1 + a + a 1a cos φ d 1 + d. Using the transformations x = X c t, ȳ = Ȳ, ū x,ȳ = Ū X,Ȳ, t c v x,ȳ = V X,Ȳ, t, p x,ȳ = P X,Ȳ, t, the equations in the wave frame x,ȳ are given by ū x + v ȳ = 0, ρ ū x + v ū= p ū ȳ x +μ ū 4ū x + η ȳ x + 4 A8 A9 A10 4 ū ū x ȳ + 4 ȳ 4 σ B0 cos θ [ū+ c cos θ v sinθ] + gρ sinα, 1 + m ρ ū x + v v= p ȳ ȳ +μ v v x + η 4 v ȳ A11 x + 4 v 4 x ȳ + 4 v ȳ 4 + σ B0 sinθ [ū + c cos θ v sinθ] gρ cos α, 1 + m A1 Pramana J. Phys., Vol. 85, No. 1, July

22 T Hayat et al ρξ ū x + v T T = κ ȳ x + T ȳ ū +μ + x +η [ v x + v + ρdk T T m v ū + ȳ ȳ + v x ū ] + ȳ x + ū ȳ, A13 C X + C Ȳ ū x + v C C = D ȳ x + C + DK T T ȳ T m x + T. A14 ȳ Introducing the non-dimensional quantities x= x λ,y= ȳ d 1,u= ū c,v= v cδ,p= d 1 cμλ p, h 1= h 1 d 1,h = h d 1, S= d 1 μc S, η= η μd1,δ= d 1 λ, Re= ρcd 1 μ, Fr= c gd 1,κ= k 0c μd 1, = d 1 μc, γ = T T a T a, Ec = c, Pr = ξμ ξt a k,ϕ= C C 0, Sc = μ C 1 C 0 ρd, Sr = ρdk T T a μt m C 1 C 0, M = σ B0 d 1, Br = Pr Ec, Pe = RePr, Bi 1 = η 1d 1 μ κ, Bi = η d 1 κ, Du = ρdk TC 1 C 0 μξc s T a, A15 the incompressibility condition is automatically satisfied and eqs A11 A14 take the form δre u x +v u= p y x + u δ x + u y η δ 4 4 u x + 4 u 4 δ x y δ 3 Re u x +v v= p y + 4 u M cos θ u cos θ vδ sinθ + cos θ y m Re sinα +, A16 Fr y +δ δ v v ηδ x + δ 4 4 v 4 v y x 4 +δ x y + 4 v y 4 + M 146 Pramana J. Phys., Vol. 85, No. 1, July 015 δ δ sinθ u cos θ δv sinθ + cos θ 1 + m Re cos α, A17 Fr

23 Peristaltic flow of couple-stress fluid δpe u x + v γ = δ γ y x + γ y + Br u δ v x y δ u x + u y [ u v δ + δ x y ] + ηbr [δ δ v x + v y ] + Pr Du δ ϕ x + ϕ y, A18 δre u x + v ϕ = 1 δ ϕ y Sc x + ϕ + Sr δ γ y x + γ, y A19 where Re is the Reynolds number, Fr is the Froude number, Pe is the Peclett number, Br is the Brinkman number, Pr is the Prandtl number, Du is the Dufour number, Sc is the Schmidt number and Sr is the Soret number. Appendix B Here constants appearing in the solutions are mentioned as follows: z = M 1 + m, H = M 1 + m cos θ Re dp sinα + Fr dx, A 1 = h 1 A4 /η, A = h A4 /η, A 3 = 1 + A 5, A 4 = 1 + A 5, 1 + m A 5 = M η M η cos θ, A 1 + m 6 = 1 + m M η cos θ. M 1 = zη cos θ, M = η M 3 = 1 4zη cos θ, zη cos θ, η and the constants C i i = 1 8 in eqs can be obtained easily by using the boundary conditions 9 and 10. References [1] Kh S Mekheimer, Y Abd elmaboud and A I Abdellateef, Appl. Biol. Biomech. 10, [] Kh S Mekheimer and Y Abd elmaboud, Appl. Math. Model. 35, [3] D Tripathi, Transp. Porous Med. 9, [4] S K Pandey and M K Chaube, Math. Comput. Model. 5, [5] R Ellahi, A Riaz and S Nadeem, I. J. Phys. 87, [6] R Ellahi, A Riaz and S Nadeem Appl. Nanosci. 4, [7] M Kothandapani and S Srinivas, Phys. Lett. A 37, Pramana J. Phys., Vol. 85, No. 1, July

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