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 [1–3]. 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 con- ditions. Mahapatra and his co-workers [19–21]have studied the heat transfer phenomena on the flow of various fluids considering the effect of thermal radia- tion 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)– ¯ (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 .