Free Convective Poiseuille Flow Through Porous Medium Between

Pramana – J. Phys. (2020) 94:69 © Indian Academy of Sciences https://doi.org/10.1007/s12043-020-1916-y

Free convective Poiseuille flow through porous medium between two infinite vertical plates in slip flow regime

PRIYA MATHUR1,∗ and S R MISHRA2

1Department of Mathematics, Poornima Institute of Engineering and Technology, Jaipur 302 022, India 2Department of Mathematics, Siksha O Anusandhan Deemed to be University, Khandagiri, Bhubaneswar 751 030, India ∗Corresponding author. E-mail: [email protected]

MS received 1 July 2019; revised 23 August 2019; accepted 9 December 2019

Abstract. The present study investigates the heat and mass transfer of magnetohydrodynamic (MHD) free convection through two infinite plates embedded with porous materials. In addition to that the combined effect of viscous dissipation, heat source/sink considered in energy equation and thermodiffusion effect is taken care of in the mass transfer equation. Using suitable non-dimensional variables, the expressions for the velocity, temperature, species concentration fields, as well as shear stress coefficient at the plate, rate of heat and mass transfer, i.e. Nusselt number (Nu) and Sherwood number (Sh) are expressed in the non-dimensional form. These coupled nonlinear differential equations are solved using perturbation technique and their behaviour is demonstrated via graphs for various values of pertinent physical parameters namely, Hartmann number (Ha), Reynolds number (Re), Schmidt number (Sc), Soret number (So), permeability parameter etc. In a particular case, the present result was compared with earlier established results and the results are found to be in good agreement. However, major findings are elaborated in the results and discussion section.

Keywords. Magnetohydrodynamics; free convection; Reynolds number; suction and injection; velocity slip; Soret effect.

PACS Nos 44.25.+f; 47.10.A−; 47.10.ad; 47.15.−x

1. Introduction free convective flow of laminar fluid past an oscillating plate where the the flow is through a porous medium. In the last few decades, the investigations carried out Free convective MHD flow of viscous fluid through a for various fluids flowing through porous channels have channel under the effects of magnetic field and mass dif- received considerable interest among the researchers fusion is studied by Ahmed and Kalita [8]. Both the Joule due to their importance in the field of engineering and and viscous dissipations are also taken into account technology, water hydrology, irrigation and filtration in their investigation. Inclusion of all these parame- processes in chemical engineering. It is also useful in ters leads to the conclusion that, uniform magnetic field biological field. The application of flow through porous reduces the velocity distribution due to its resistive prop- medium plays a vital role in various areas such as soil erty. Acharya et al [9] studied the free convective flow erosion, irrigation etc. where their mathematical forms of MHD viscous fluid through vertical porous plate after are also important [1Ð3]. The combined effects of grav- considering variable plate temperature and the effect ity force and force caused by the density differences of heat source/sink. But in their study, they have not between diffusion of thermal and species concentration considered the influence of thermal radiation and ther- were studied by many researchers to get a systematic mal diffusion. It is true to assume that thermal diffusion solution of the problem on free convection. Wide range is applicable when the concentration level is low. For of applications are used in both engineering and geo- the isotope separation the application of thermal diffu- physics. Moreover, Bejan and Khair [4], Trevisan and sion, i.e. Soret effect is important whereas in mixtures Benjan [5], Acharya et al [6], Choudhary and Jain [7] of gases such as H2 and He, with very light/medium have investigated the magnetohydrodynamic (MHD) molecular weight, the diffusion-thermo effect is not to

0123456789().: V,-vol 69 Page 2 of 9 Pramana – J. Phys. (2020) 94:69 be neglected. Further, Baag et al [10] worked on the the work of Kalita and Ahmed [12], considering the flow of non-Newtonian fluid through a porous medium Soret effect on the flow and transfer characteristics. Sep- between infinite parallel plates where they have consid- aration of various components from a fluid mixture has ered that the suction is time-dependent. Makinde and many applications in environmental engineering, sci- Mishra [11] have illustrated the chemically reacting ence and technology and separation of isotopes from MHD fluid past a heated vertical plate embedded with their naturally occurring mixture. Under a temperature porous medium. Flow through porous medium bounded gradient, Soret effect is the tendency of a convective by two infinite vertical plates under the effects of ther- free fluid mixture to separate. It has also some impor- mal diffusion and magnetic field has been considered tant characteristics in the hydrodynamics instability by Kalita and Ahmed [12]. They have also considered of mixtures. Due to the dissipative term, the problem the effect of buoyancy on the Poiseuille flow of electri- becomes coupled and nonlinear. Perturbation technique cally conducting viscous fluid. It is well accepted for the is used to solve the non-dimensional governing equa- microscopic level that, for viscous fluid at a solid wall tions with suitable choice of perturbation parameter. there is ‘no slip’ which means that the solid boundary is The physical significance of the parameters are obtained fixed. It is also experimentally verified for many macro- and presented via graphs and the numerical compu- scopic flows, and there is no need to prove it physically. tations for the engineering coefficients are presented Long back, Navier proposed a general boundary condi- as a table. Finally, the validation of the present result tion which shows the feasibility of the no-slip boundary with that of the earlier study is obtained in a particular condition. In his assumption, he stated that the veloc- case. ity, Vx , is proportional to the shear stress at the surface (Navier [13] and Goldstein [14]). 2. Mathematical formulation dVx Vx = γ , Consider an electrically conducting incompressible dy steady viscous fluid flow past two infinite vertical porous where γ is the slip coefficient. In general γ = 0repre- plates embedded with porous medium. Here, both the sents no slip condition and when γ =∞fluid slip occurs plates are separated by a distance h apart. In addition at the wall where the length scale of the flow affects the to that the momentum equation is enhanced by incor- flow characteristics. From the aforesaid assumption, it porating the thermal buoyancy effect which causes a is clear that the velocity of the fluid is linearly related free convective flow. Viscous dissipation and thermo- to shear stress at the plate. Also, Yu and Amed [15] diffusion effects are also taken into account in energy in their study assumed slip boundary condition in the and mass transfer equation respectively. The flow is flow phenomena. The effect of fluid slippage at the wall along x-axis which is placed vertically. Both the plates for Couette flow was considered by Marque et al [16]. are at y = 0andy = h respectively. A uniform Khaled and Vafai [17] have studied a steady periodic transverse magnetic field is applied normal to the flow and transient velocity field considering slip boundary direction. Due to low magnetic Reynolds number (Re) condition where they obtained a closed form solution induced magnetic field can be ignored (figure 1). As for their problem. Choudhary and Jha [18] investigated the plate is in infinite length, all the physical quan- the MHD micropolar fluid flow under the influence of tities except pressure p are independent of x.From chemical reaction where flow is past a vertical plate. the above assumptions, the equation governing the flow In addition to that, they have imposed the slip conditions. Mahapatra and his co-workers [19Ð21]have studied the heat transfer phenomena on the flow of various fluids considering the effect of thermal radiation in different geometries. However, Goqo et al [22] investigated an unsteady Jeffery fluid flow over a shrink- ing sheet under the effect of thermal radiation. Further, heat transfer due to the interaction of magnetic field and thermal radiation in an unsteady Casson nanofluid over a stretching surface has been discussed by Oyelakin et al [23]. Recently, Ghiasi and Saleh [24] and Mehmood and Rana [25] have worked on the heat transfer proper- ties of various fluids in different geometries. Our aim is to study the thermodiffusion effect which is not considered in earlier studies. Here, we have extended Figure 1. Flow configuration. Pramana – J. Phys. (2020) 94:69 Page 3 of 9 69 phenomena are By implementing all the above assumptions into eqs (2)Ð ¯ (5)weget dV = ¯ 0(1)du 1 d2u dy − = + Gr θ + Gm ϕ dy Re dy2 ¯ 2 ¯ du d u ¯ ¯ ¯ ¯ ¯ u −V0 = ν + gβ(T − Ts) + gβ(C − Cs) − − Ha Re u (6) dy¯ dy¯2 α Re θ 2θ 2 ν ¯ σ 2 ¯ d 1 d Ec du u B0 u − = + + Qθ (7) − − (2) y y2 y K ρ d Pr Re d Re d 2 2 ¯ 2 ¯ 2 dφ 1 d φ So d θ dT κ d T ν du¯ − = + (8) −V = + + Q (T − T ) 2 2 0 2 0 (3) dy Sc Re dy Re dy dy¯ ρcp dy¯ cp dy¯ ⎫ dC¯ d2C¯ d2T¯ u = 0,θ= 1,φ= 1at y = 0, ⎬ −V¯ = D + D . (4) 0 ¯ M ¯2 T ¯2 ∂u (9) dy dy dy u = λ ,θ= m,φ= n y = , ⎭ ∂ at 1 The corresponding wall conditions are y ⎫ where Gr is the thermal Grashof number, Gm is the ¯ = , ¯ = ¯ , ¯ = ¯ ¯ = u 0 T T0 C C0 at y 0 ⎬ solutal Grashof number, Re is the Reynolds number, Ha ∂u . (5) is the Hartmann number, α is the porosity parameter, Pr u¯ = , T¯ = T¯ , C¯ = C¯ at y¯ = h ⎭ ∂y 1 1 is the Prandtl number, Ec is the Eckert number, Q is the heat source/sink parameter, Sc is the Schmidt number, ¯ From eq. (1) it is clear that V =−V0 = constant stands So is the Soret number and m, n are the constants. for suction/injection. Also, u¯ is the velocity component, T¯ , C¯ are the fluid temperature and concentration, ν is the kinematic viscosity, g is the acceleration due to gravity, 3. Solution of the problem β,β¯ are the volumetric expansion coefficients for thermal and solutal, K is the porosity, B0 is the magnetic To solve eqs (6)Ð(8) subject to condition (9), we use ρ field strength, is the fluid density, cp is the specific perturbation technique taking Ec as the perturbation / heat, Q is the heat source sink coefficient and DM is parameter as Ec 1 for most of the incompressible the mass diffusion coefficient. fluids. The expression for the flow profiles are assumed The suitable choice for the non-dimensional quanti- as ties is as follows: ⎫ = ( ) + ( ) + 2 ( ) +··· ⎫ u u0 y Ec u1 y Ec u2 y ⎬⎪ ¯ ¯ y¯ u¯ T − T ⎪ 2 . = , = ,θ= s , ⎪ θ = θ0(y) + Ec θ1(y) + Ec θ2(y) +··· (10) y u ¯ ⎪ ⎭⎪ h V0 T − Ts ⎪ 2 0 ⎪ φ = φ0(y) + Ec φ1(y) + Ec φ2(y) +··· ⎪ ¯ ¯ μ ⎪ C = Cs cp V0h ⎪ Substituting eq. (10) into eqs (6)Ð(8),andbyequating φ = , Pr = , Re = , ⎪ − κ ν ⎪ the like powers of Ec, i.e. the zeroth and first order of C0 Cs ⎪ ⎪ Ec, we get the following equations: K ⎪ α = ,λ= /h, ⎪ 2 ⎪ u + Re u − Au = Gr Re θ − Gm Re φ , (11) h ⎪ 0 0 0 0 0 ⎪ 2 2 ¯ ¯ ¯ ⎪ + − =− θ − φ , σ B h hgβ(T − T ) ⎬ u1 Re u1 Au1 Gr Re 1 Gm Re 1 (12) Ha = 0 , Gr = 0 s , ρ 2 V0 V0 ⎪ θ + Re Pr θ + Q Pr Re = 0, (13) ⎪ 0 0 ⎪ β(¯ ¯ − ¯ ) 2 ⎪ 2 hg C0 Cs V0 ⎪ θ + Re Pr θ + Q Pr Re =−Pr u , (14) Gm = , Ec = ,⎪ 1 1 0 V 2 c (T¯ − T¯ ) ⎪ 0 p 0 s ⎪ φ + φ =− θ , ⎪ 0 Sc Re 0 Sc So 0 (15) ν ( ¯ − ¯ ) ⎪ DT T0 Ts ⎪ φ + φ =− θ , Sc = , So = , ⎪ 1 Sc Re 1 Sc So 1 (16) D ν(C¯ − C¯ ) ⎪ M 0 s ⎪ ⎪ where the higher order coefficients of Ec are neglected ¯ ¯ ¯ ¯ ⎪ 2 T1 − Ts C1 − Cs ⎪ and A = (1/α) + Ha Re . In particular, the differentia- m = , n = . ⎭ ¯ ¯ ¯ ¯ tion with respect to y is presented in the form of primes. T0 − Ts C0 − Cs 69 Page 4 of 9 Pramana – J. Phys. (2020) 94:69

The boundary conditions (9) reduce to (λ2−a1)y (λ2−a2)y ⎫ +a116e + a117e u0 = 0,θ0 = 1,φ0 = 1, u1 = 0, ⎪ (λ +λ ) −( + ) ⎪ +a e 1 2 y + a e a1 Sc Re y ⎪ 118 119 θ = 0,φ= 0, at y = 0, ⎪ 1 1 ⎬ −(a2+Sc Re)y (λ1−Sc Re)y +a120e + a121e (17) ∂u0 ⎪ (λ − ) u = λ ,θ= m,φ= n, u = 0,⎪ +a e 2 Sc Re y, 0 ∂y 0 0 1 ⎪ 122 ⎪ ⎭⎪ where θ1 = 0,φ1 = 0at y = 1. −Re + Re2 + 4A The solutions of eqs (12)Ð(16) subject to boundary con- λ = , 1 2 ditions (17) are as follows: − − θ ( ) = a1 y + a2 y, − − 2 + 0 y a3e a4e λ = Re Re 4A. 2 2 −Sc Re y −a1 y −a2 y φ0(y) = a8 + a7e + a5e + a6e , The other constants a1, a2,...,a125 are obtained but λ1 y λ2 y −Sc Re y u0(y) = a18e + a17e + a11 + a12e for the sake of brevity these are not mentioned.

−a1 y −a2 y +a15e + a16e , 3.1 Coefﬁcient of skin friction

−a1 y −a2 y −2a1 y θ1(y) = a50e + a49e + a34e

− λ λ + 2a2 y + 2 1 y + 2 2 y a35e a36e a37e τ = du =[ ( ) + ( )], 0 u0 0 Ec u1 0 − −( + ) dy = +a e 2ScRey + a e a1 a2 y y 0 38 39 (λ − ) (λ + ) du +a e 1 a1 y + a e 1 a2 y τ = =[ ( ) + ( )]. 40 41 1 u0 1 Ec u1 1 dy y=1 (λ2−a1)y (λ2−a2)y +a42e + a43e Rate of heat transfer (λ1+λ2)y −(a1+Sc Re)y 3.2 + a44e + a45e −( + ) (λ − ) + a2 Sc Re y + 1 Sc Re y θ a46e a47e =− d =−[θ ( ) + θ ( )], Nu0 0 0 Ec 1 0 (λ − ) dy +a e 2 Sc Re y, y=0 48 −Sc Re y −a1 y −a2 y dθ φ1(y) = a69 + a68e + a51e + a52e =− =−[θ ( ) + θ ( )]. Nu1 0 1 Ec 1 1 dy y=1 −2a1 y −2a2 y 2λ1 y +a53e + a54e + a55e

2λ2 y −2Re Sc y 3.3 Rate of mass transfer +a56e + a57e −( + ) (λ − ) +a e a1 a2 y + a e 1 a1 y φ 58 59 =− d =−[φ ( ) + φ ( )], Sh0 0 0 Ec 1 0 (λ1−a2)y (λ2−a1)y dy = +a60e + a61e y 0 (λ2−a2)y (λ1+λ2)y dφ + a62e + a63e =− =−[φ ( ) + φ ( )]. Sh1 0 1 Ec 1 1 dy = −(a1+Sc Re)y −(a2+Sc Re)y y 1 +a64e + a65e (λ − ) (λ − ) +a e 1 Sc Re y + a e 2 Sc Re y, 66 67 4. Result and discussion λ1 y λ2 y −a1 y u1(y) = a124e + a123e + a106e Free convective flow, heat and mass transfer effect of an −a2 y −2a1 y −2a2 y +a107e + a108e + a109e electrically conducting viscous fluid past two vertically λ λ infinite plates is investigated in porous medium. The +a e2 1 y + a e2 2 y 110 111 flow phenomena are affected by the inclusion of ther- −2Sc Re y −(a1+a2)y mal slip parameter. Perturbation technique is employed +a112e + a113e to solve the non-dimensional governing equations. For (λ1−a1)y (λ1−a2)y +a114e + a115e the validation of this model and to get physical insight Pramana – J. Phys. (2020) 94:69 Page 5 of 9 69

Figure 2. Influence of Ha on velocity profiles when Re = 1, Figure 3. Influence of α on velocity profiles when Re = 1, Gr = 2, Gm = 2, Pr = 0.71, Ec = 0.05, Q = 0, Sc = 0.22, Gr = 2, Gm = 2, Pr = 0.71, Ec = 0.05, Q = 0, Sc = 0.22, So = 0, m = 0, n = 0 and λ = 0. So = 0, m = 0, n = 0 and λ = 0. into the problem, the numerical computations for velocity, temperature and species concentration distributions, skin friction coefficient, rate of heat transfer in terms of Nusselt number (Nu) and rate of mass transfer in terms of Sherwood number (Sh) are calculated for different values of physical parameters used in the problem and these are demonstrated via graphs and table. Through- out our investigation, Pr = 0.71 which corresponds to air at 20◦C. The value of Gr for heat transfer has been chosen as 5 whereas Gm for mass transfer has been chosen as 2. The value of Sc is taken as 0.22 (hydrogen), 0.30 (helium), 0.60 (water), 0.78 (ammonia), 0.94 (car- bon dioxide) with air as the diffusion medium. In our investigation, Ec = 0.05, non-dimensional temperature and species concentration m = 2andn = 2 respectively are fixed at the plate y = h. The other parameters Figure 4. Influence of Re on velocity profiles when Ha = 1, like Re, So, Ha and Sc are chosen arbitrarily. In partic- α = 0.1, Gr = 2, Gm = 2, Pr = 0.71, Ec = 0.05, Q = 0, ular case, the present result agrees well with the earlier Sc = 0.22, So = 0, m = 0, n = 0 and λ = 0. established result of [12]. Figure 2 describes the influence of Ha on the velocity profile for fixed values of other pertinent parameters. When Ha = 0, the present result coincides with the parabolic in nature for every values of Ha, i.e. for both published result of [12]. However, with an increasing high and low values. Figure 3 displays the behaviour Ha, the velocity profile decreases. It is interesting to of porous matrix on the velocity profiles. From the observe that the profile rises up to the middle of the mathematical expression for porous matrix equation (6), channel, i.e. velocity profile reaches the maximum value it is clear that, high value of α represents the absence near towards the centre of the channel and afterwards it of porous matrix and low value represents its presence. decreases smoothly. Rapid growth is marked near the Hence, increasing porosity also decreases the veloc- first plate. Appearance of Ha is due to the inclusion of ity distribution. Like magnetic parameter, porosity also magnetic field strength and it produces Lorentz force is a resistive force which opposes the velocity profile which is a resistive force. It has a tendency to retard resulting in its retardation. Influence of Re on the veloc- the velocity profile and therefore, the profile decreases ity profiles is exhibited in figure 4. The profile is very at all points throughout the flow domain. The profile is much significant for various values of Re. For increased 69 Page 6 of 9 Pramana – J. Phys. (2020) 94:69

Figure 5. Influence of λ on velocity profiles when Ha = 1, Figure 7. Influence of Pr on temperature profiles when α = 0.1, Re = 1, Gr = 2, Gm = 2, Pr = 0.71, Ec = 0.05, Ha = 1, α = 0.1, Re = 1, Ec = 0.05, Q = 0, Sc = 0.22, q = 0, Sc = 0.22, So = 0, m = 0.5 and n = 0.5. So = 0, m = 0.5, n = 0.5 and λ = 0.1.

which is fundamentally important and relevant in a wide range of applications such as MEMS, biological sys- tems, microfluidic devices etc. One of the most vital advantages of slippage is the reduction of flow resis- tance in the microchannel. Figure 6 shows the effects of buoyant force (Gr and Gm) on the velocity profiles. Here, Gr > 0andGm> 0 are considered as the case of cooling of the plate. By increasing both thermal and mass buoyancy, the velocity profile increases. Hence, it is concluded that in the case of cooling, the ambient temperature is more than that of the plate temperature and therefore due to rise in temperature it overshoots the velocity profiles. The effect of Pr on the temperature profile is shown in figure 7. Pr is the ratio of the dynamic viscosity with respect to the thermal diffusivity. So, mathematically it is very clear that, increase Figure 6. Influence of Gr and on velocity profiles when Ha = 1, α = 0.1, Re = 1, Pr = 0.71, Ec = 0.05, q = 0, in Pr leads to decrease in thermal diffusivity which Sc = 0.22, So = 0, m = 0.5, n = 0.5 and λ = 0.1. retards the fluid temperature. As a consequence, the profile presented in the corresponding figure retards. For higher Pr, the profile behaves asymptotically. Figure 8 exhibits the influence of Q on the temperature profiles Re values, the reverse effect is encountered compared for fixed values of other pertinent parameters. In the to figures 2 and 3. The velocity profile increases with absence of heat source, the present result agrees with an increase in Re. A similar observation is marked as that of [12]. Moreover, increasing heat source enhances described earlier that the profile attains its maximum the fluid temperature whereas reverse effect is observed value at the middle of the channel and the trend of for the sink. In the presence of heat source, some heat the curve is parabolic in nature. Figure 5 depicts the energy is stored up which helps to rise in tempera- variation of velocity distributions for different values ture of the profiles. The influence of non-dimensional of slip flow parameter. It is interesting to observe that, temperature condition, m, on the temperature profile is there is no significant variation near the first plate but displayed in figure 9. The fluid temperature attains its the variation is more at the second plate at y = h. minimum value in the absence of temperature condition When the slip parameter increases, the velocity pro- and further, by increasing m the profile increases sig- file decreases. Slip is an interfacial boundary condition nificantly and it is marked significantly. The increase Pramana – J. Phys. (2020) 94:69 Page 7 of 9 69

Figure 8. Influence of Q on temperature profiles when Figure 10. Influence of Sc on concentration profiles when Ha = 1, α = 0.1, Re = 1, Pr = 0.71, Ec = 0.05, Ha = 1, α = 0.1, Re = 1, Pr = 0.71, Ec = 0.05, Gr = Gc = 2, Sc = 0.22, So = 0, m = 0.5, n = 0.5 Gr = Gc = 2, So = 0, n = 0.5 and λ = 0.1. and l = 0.1.

1 So = 0.1 0.95 So = 1.0 So = 2.0 0.9 So = 3.0

0.85

0.8

(y) 0.75

0.7

0.65

0.6

0.55

0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y

Figure 11. Influence of So on concentration profiles when Figure 9. Influence of m on temperature profiles when Ha = 1, α = 0.1, Re = 1, Pr = 0.71, Ec = 0.05, Ha = 1, α = 0.1, Re = 1, Pr = 0.71, Ec = 0.05, Gr = Gc = 2, Sc = 0.22, m = 0.5, n = 0.5 and λ = 0.1. Gr = Gc = 2, Sc = 0.22, So = 0, n = 0.5 and λ = 0.1. in figure 11. It is seen that increasing So favours enhancement in fluid concentration. It occurs due to the in m means the difference between the mean temper- thermo-diffusion process when the temperature gradient ature and the surface temperature decreases leading to diffuses in species concentration. Figure 12 shows the the enhancement in the plate temperature resulting in behaviour of non-dimensional concentration n on the an increase in the fluid temperature. Figure 10 dis- concentration profile and observation is similar to that plays the variation of Sc on the concentration profiles. of non-dimensional temperature parameter m described Sc is the ratio of viscous diffusivity with respect to earlier. mass diffusivity. As Sc increases, the mass diffusivity Finally, the numerical computations for rate of shear decreases which causes a decrease in concentration dis- stress, rate of heat and mass transfer are presented in tribution. Hence, it is concluded that heavier species is table 1.WhenHa= 0, α = 10, Q = 0andSo= 0, the favourable for retardation in fluid concentration. The present result coincides with the results of [12]which influence of So on the concentration profile is shown shows the conformity of the procedure adopted for the 69 Page 8 of 9 Pramana – J. Phys. (2020) 94:69

Table 1. Rate of shear stress, heat and mass transfer coefﬁcients.

Ha α Gr Gm QmSo n τ0 τ1 Nu0 Nu1 Sh0 Sh1 0 10 2 2 0 0.5 0 0.5 1.6754 −1.121 0.6852 0.3533 0.557 0.447 11.5753 −1.0709 0.687 0.352 0.557 0.447 21.4889 −1.0284 0.6885 0.351 0.557 0.447 0.5 1.497 −1.0324 0.6883 0.3511 0.557 0.447 0.1 1.0738 −0.8405 0.6939 0.3472 0.557 0.447 32.0901 −1.3982 0.6779 0.3588 0.557 0.447 52.9228 −1.9554 0.6585 0.3736 0.557 0.447 32.1002 −1.4057 0.6777 0.359 0.557 0.447 52.9502 −1.9755 0.6575 0.3743 0.557 0.447 −0.5 1.8507 −1.2484 −0.4644 1.086 0.557 0.447 0.5 2.0308 −1.3801 −1.5761 1.8137 0.557 0.447 0.1 1.5229 −0.8548 1.2469 0.6246 0.557 0.447 0.2 1.561 −0.9214 1.1065 0.5567 0.557 0.447 0.1 1.6761 −1.1215 0.6852 0.3533 0.552 0.4501 0.2 1.6768 −1.122 0.6852 0.3533 0.5486 0.4532 0.1 1.5405 −0.868 0.6879 0.3501 1.0026 0.8046 0.2 1.5742 −0.9313 0.6873 0.3509 0.8912 0.7152

5. Concluding remarks

Investigation of the present problem leads to the following conclusions. 1. Stronger magnetic field as well as the presence of porous matrix lead to a decrease in the velocity profiles. 2. The velocity of the fluid decelerates due to the slip flow whereas buoyant forces accelerate it significantly. 3. Heat source favours the enhancement in the fluid temperature whereas reverse effect is observed for the sink. 4. Rate of heat transfer shows its dual character at both the plates for the variation of buoyant forces. Figure 12. Influence of n on concentration profiles when 5. Heat source also favours the enhancement of the Ha = 1, α = 0.1, Re = 1, Pr = 0.71, Ec = 0.05, rate of heat transfer but an increase in So decreases Gr = Gc = 2, Sc = 0.22, m = 0.5 and λ = 0.1. Sh, i.e. the rate of mass transfer coefficient.

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