Transient free convective MHD flow through porous medium in slip flow regime

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1 IOSR Journal of Mathematics (IOSR-JM) e-issn: , p-issn: X. Volume 11, Issue 5 Ver. 5 (Sep. - Oct. 2015), PP Transient free convective MHD flow through porous medium in slip flow regime H. K. Patel 1, R K. Singh 2, T. R. Singh 3 1 Sarvajanik College of Engineering and Technolog, Surat, India 2 Pacific Institute of Technolog, Surat, India 3 SVNIT, Surat, India. Abstract: The unstead free convective viscous incompressible flow of an electricall conducting fluid, under uniform magnetic field, through porous medium near vertical insulated wall with variable suction in slip flow has been investigated. The governing equation in non-dimensional form is solved with the help of Perturbation technique. An approximated solution for velocit, skin friction, temperature and Nusselt number are obtained. The effects of various non-dimensional parameters on velocit profile, skin friction and Nusselt number are shown graphicall. Nomenclature B o magnetic field C p specific heat at constant pressure Gr Grashof number velocit slip parameter K permeabilit of the porous medium M magnetic parameter Nu Nusselt number Pr Prandtl number t dimensionless time t dimensionless time u velocities along to the plate V 0 scale of suction velocit perpendicular to the plate, respectivel dimensionless ratio of viscosit T temperature of the fluid T temperature of the free steam T w temperature of the wall Greek smbols β 0 coefficient of thermal induction ε scalar constant (1) υ kinematic viscosit ρ fluid densit θ Ratio of temperature difference κ thermal conductivit μ dnamic viscosit of fluid effective viscosit of porous medium μ eff I. Introduction In recent deca, The fluid flow through porous media has attracted the attention of man researchers in recent ears because it is encountered in man industrial applications such as optimization of solidification processes of metals and allos, waste nuclear processing, dissemination control of chemical waste and pollutants, and design of MHD power generators. The Ingham and Pop [1990], and Nield and Bejan [1998] have worked on porous media. Siegel [1958], Berezovsk et al. [1977], and Martnenko et al. [1984] have studied the free convective flow on vertical flat plate whereas Sharma and Chaudhar [2003] investigated the effect of variable suction on transient free convective viscous incompressible flow past a vertical plate with periodic temperature variations in slip flow regime. Later on, Das et al. [1999] have discussed on transient free convection flow past an infinite vertical plate DOI: / Page

2 Transient free convective MHD flow through porous medium in slip flow regime with periodic temperature variation. Sahin [2010] studied oscillator three dimensional flows and heat and mass transfer through a porous medium in presence of periodic suction. Uwanta et al. [2011] have examined heat and mass transfer flow through a porous medium with variable permeabilit and periodic suction. Mishra et al. [2014] have found some result on transient free convection flow past a vertical plate through porous medium with variation in slip flow regime obtained b perturbation method. Research works in the magnetohdrodnamics (MHD) have been advanced significantl during the last four decades in sciences and engineering disciplines after the pioneer work of Hartmann [1937] in liquid metal duct flows under the influence of a strong external magnetic field. This fundamental investigation provides basic knowledge for the development of several MHD devices such as MHD pumps, generators, breaks, and flow meters. Further, Cramer and Pai [1973] investigated several results on MHD convective flows. Jha [1990] have analticall studied the natural convective flow along a vertical infinite plate under a constant magnetic field. Singh and their collaborators have carried out a number of studies on hdromagnetic free convection covering several aspects such as effects of moving boundaries, rotation, asmmetrical heating, porous boundar, etc. [Chandran et al [1993, 1996, 1998, 2001]]. Kim [2000] presented the unstead MHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suction. In recent ears, Singh et al. [2010], and Singh & Singh [2012] have studied effect of induced magnetic field on natural convection in vertical wall and concentric annuli respectivel whereas Singh & Singh [2012] considered magnetohdrdnamic free convective flow past on semi infinite permeable vertical wall. Thermal radiation and oscillating plate temperature effects on MHD unstead flow past a semi-infinite porous vertical plate under suction and chemical reaction investigated b Sharma and Deka [2012]. Recentl, Singh et al. [2014] have studied effect of induced magnetic field on mixed convective flow. In present stud, we have discussed effect of magnetic field on unstead free convective transient flow and heat transfer of a viscous incompressible fluid near a vertical flat plate embedded in a porous medium. The governing equations are solved b using perturbation technique. The behaviour of velocit, Nusselt numbers and skin friction for the non-dimensional parameters such as permeabilit(k), slip parameter(h),ratios of viscosities (), Grashof number(gr) and Hartmann number(m) are shown graphicall. II. Mathematical model Consider a viscous, incompressible, electricall conducting fluid flow through a porous media occuping a semi-infinite region bounded b a vertical wall with slip boundar condition. Surface of wall is considered along X-axis in upward direction and Y-axis is considered direction perpendicular to wall. A uniform magnetic field of strength B 0 is applied perpendicular to the wall. The magnetic Renolds number is assumed to be small enough to neglect the induced magnetic field. Then governing equations, after neglecting viscous dissipation and under usual Boussinesq's approximation, can be given as follows (Mishra et al. [2014]): u V t εe n t u = gβ T T + μ eff 2 u u σb 2 0 ρ k ρ u (1) 2 μ f ρ T V t εe n t T = k 2 u ρ C p 2 (2) The boundar conditions are as follows u = L u T = T w at = 0 u 0, T T at (3) In non-dimensional process of the governing equations, the following dimensionless quantities are used = V 0, t = t V 0 u, u = υ υ V n = υn 0 V 2, θ = T T 0 T w T, Gr = gβυ (T w T ) V 2, = μ eff, K = K V 2 0, Pr = ρc p 2 0 u 0 μ f υ 2 k ρ V 2 0. The governing equations in the non-dimensional form are as follows: u u 1 + εe nt u t 2 + Grθ Nu (4) θ θ 1 + εe nt 2 θ t Pr 2 (5) The boundar conditions in dimensionless form are as follows: u = u, θ = 1, at = 0, u 0, θ 0, at. (6) where = L v 0 /v III. Solution of the problem For the small amplitude oscillations (ε 1), the velocit u and temperature θ near the wall using perturbation technique u, t = u 0 + εe nt u 1 () (7) DOI: / Page

3 Transient free convective MHD flow through porous medium in slip flow regime θ, t = θ 0 + εe nt θ 1 () (8) Substituting Eqs. 7 & 8 in Eqs. 4-6 to get zeroth order, first order and the boundar conditions, respectivel. Zeroth order equations 2 u du 0 2 θ Pr Nu d 0 + Grθ 0 = 0 (9) dθ 0 = 0 (10) d Boundar conditions for the zeroth order u 0 = u 0, θ 0 = 1, at = 0 u 0 0, θ 0 0, at. (11) B solving equations subjected to boundar conditions, the solution for zero order is obtained as follows: u 0 = C 1 e m 1 + A 1 e Pr, (12) θ 0 = e Pr, (13) where C 1 = PrA 1, A Gr 1+m 1 1 =, m (Pr 2 Pr N) 1 = First order equations 2 u du 1 2 θ Pr + 4N 2 Nu d 1 + Grθ 1 = 0 (14) dθ 1 = 0 (15) d Boundar conditions for the first order u 1 = u 0, θ 1 = 1 at = 0 u 1 0, θ 1 0 at (16) B solving equations subjected to boundar conditions, the solution for zero order is obtained as follows: u 1 = C 3 e m 3 + A 5 e Pr + A 6 e m 1 + A 7 e m 2 (17) θ 1 = Pr n e Pr C 2 e m 2 (18) where A 2 = A 1Pr Gr C 2 A 3 = m 1C 1, C 2 = Pr, A 4 = GrC 2 A 5 = A 2 Pr 2 Pr (N n) A 7 = A 4 m 2 2 m 2 (N n), m n 2 = Pr + Pr 2 4nPr 2,, 2., A 6 = A 3 m 2 1 m 1 (N n), C 3 = m 1A 6 +m 2 A 7 +Pr A 5 m 4, m 1+m 3 = (N n) 3 2 m 4 = A 5 + A 6 + A 7. The Nusselt number is Nu = θ =0 = Pr + εc 2 (Pr + m 2 ) (19) The skin-friction (τ) at the wall is, τ = du 0 + ε du 1 e nt, d =0 d =0 (20) τ = m 1 C 1 + PrA 1 ε m 3 C 3 + PrA 5 + m 1 A 6 + m 2 A 7 e nt (21) IV. Result and discussion: In order to obtained the effects of various non-dimensional parameters such as Permeabilit (K), velocit slip parameter (), Grashof Number(Gr), Ratio of Viscosit () and Prandtl number (Pr) on velocit profile and various results are shown graphicall. In Figure 1, the velocit profile are shown graphicall for different values of permeabilit (K) & magnetic field(m), when other parameters are kept constant ( = 1, Gr = 5, = 1, Pr = 0.71, t = 1, n = 1 & ε = 0.1). The results show that velocit increases with increases in value of Permeabilit(K) and it decrease with magnetic field(m)., 2, DOI: / Page

4 u u Transient free convective MHD flow through porous medium in slip flow regime M=1,K=3 M=1,K=2 M=1,K=1 M=5,K=3 M=5,K=2 M=5,K=1 Fig.1 Velocit profile for different values of permeabilit (K) & magnetic parameter (M) In Fig. 2, velocit profile are shown graphicall for different values slip parameter () and magnetic field (M), while other parameters, K = 1, Gr = 5, = 1, Pr = 0.71, t = 1, n = 1 and ε = 0.1, are kept constant. We have observed that magnitude of velocit decreases with increases of slip parameter () as well as magnetic parameter. For large value of magnetic field, we have observed that there is no significant effect of increasing slip parameter on velocit magnitude M=1,h=1 M=1,h=2 M=1,h= M=5,h=1,2,3 Fig.2. Velocit profile for different values of slip parameter () & magnetic parameter (M) Fig.3 displa velocit profile with K = 1, Gr = 5, h = 1, Pr = 0.71, t = 1, n = 1 & ε = 0.1 for different values of as Ratio of viscosities and magnetic parameter M. The results shows that velocit increases with increases in value of as Ratio of viscosities () and decrease with increases magnetic parameter M. DOI: / Page

5 u u Transient free convective MHD flow through porous medium in slip flow regime M=1,= M=1,=2 M=1,=1 M=5,=3 M=5,=2 0.5 M=5,=1 Fig.3. Velocit profile for different values of Ratios of viscosities magnetic parameter (M) In Fig. 4, the velocit profile are shown graphicall for different values of as Grashof number (Gr) where as other parameters, K = 1, = 1, h = 1, Pr = 0.71, t = 1, n = 1 & ε = 0.1, are kept constant. The results shows that velocit increases with increases in value of as Grashof number (Gr) where as it decreases with magnetic field. 5 M=5,Gr= M=1,Gr=15 M=5,Gr=10 M=1,Gr=10 1 M=5,Gr=5 M=1,Gr= Fig.4. Velocit profile for different values of Grash of number (Gr) & magnetic parameter M. Fig. 5 depicts variation skin-friction for different values M & ε with K = 1, = 1, h = 1, Pr = 7, Gr = 5 and n = 1. Since velocit having decreasing tendenc with magnetic field as it increases, then skin friction also having decreasing tendenc with increase of magnetic field. DOI: / Page

6 Nuselt Transient free convective MHD flow through porous medium in slip flow regime M=1, =0.1 M=1, =0.2 M=1, =0.3 M=5, = M=5, = M=5, = t Fig.5. Skin friction for different values of scalar constant ε and magnetic parameter M =0.2 = =0.4 = = =0.6 0 Pr Fig.6. Nusselt numbers for different values of scalar constant ε In Fig. 6, Nusselt numbers are shown graphicall for different values ε when other parameters (Pr = 0.71 & n = 1) are kept constant. The results show that Nusselt number decreases with increases in value of ε. V. Concluding remark As per the solution of problem, the velocit is decreasing as magnetic field increase. B this theme it will be small step to control salinit and can be produced pollution free energ. Acknowledgement: Authors are thankful to GUJCOST to provide financial support to carr research work. References: [1]. Berezovsk, A. A., Martnenko, O. G. and Yu, A., Sokovishin, Free convective heat transfer on a vertical semi infinite plate, J. Engng. Phs., 33, 32-39, [2]. Chandran, P., Sacheti, N. C., and Singh, A. K., Effects of rotation on unstead hdrodnamic Couette flow, Astrophsics and Space Science, 202, 1 10, [3]. Chandran, P., Sacheti, N. C., and Singh, A. K. Hadromagnetic flow and heat transfer past a continuousl moving porous boundar, International Communication in Heat and Mass Transfer, 23, , [4]. Chandran, P., Sacheti, N. C., and Singh, A. K. Unstead hdromagnetic free convection flow with heat flux and accelerated boundar motion, Journal of Phsical Societ of Japan, 67, , [5]. Chandran, P., Sacheti, N. C., and Singh, A. K. An undefined approach to analtical solution of a hdromagnetic free convection flow, Scientiae Mathematicae Japonicae, 53, , DOI: / Page

7 Transient free convective MHD flow through porous medium in slip flow regime [6]. Cramer, K. R. and Pai, S. I., Magnetofluid Dnamics for Engineering and Applied Phsicists, Mc Graw-Hill, New York, [7]. Das, U. N., Deka, R.K. and Soundalgekar, V. M., Transient free convection flow past an infinite vertical plate with periodic temperature variation, J. Heat Transfer (ASME), 121, , [8]. Jha, B. K., MHD free-convection and mass transform through a porous media, Astrophsics and Space Science, 175, , [9]. Hartmann, J., Hg-dnamics I theor of the laminar flow of an electricall conductive liquid in a homogenous magnetic field, Det Kongelige Danske Videnskabernes Selskab Mathematisk-fsisk Meddeleser, XV, 1 27, [10]. Ingham, B. D., and Pop, I., Convection in porous media, (2 nd Ed.) Springer, New York, [11]. Kim J., Unstead MHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suction had studied, International J. Engg. Sci, , [12]. Martnenko, O. G., Berezovsk, A. A. and Yu. A., So-kovishin, Laminar free convection from a vertical plate, Int. J. Heat Mass Transfer, 27, , [13]. Mishra, A.K., Abdullahi, A., and Manjak, N. H, Transient free convection flow past a vertical plate through porous medium with variation in slip flow regime, International Journal of Advanced Technolog and Engineering Research, Vol. 4(2), Jan [14]. Nield, D. A., and Bejan, A., Transport phenomena in porous media, Oxford (1998). [15]. Sharma P. K, and Chaudhaur, R. C., Effect of variable sunction on transient free convective viscous incompressible flow past a vertical plate with periodic temperature variations in slip-flow regime, Emirates Journal for Engineering Research, 8(2), 33-38, [16]. Shahin, A., Oscillator three dimensional flow and heat and mass transfer through a porous medium in presence of periodic sunction Emirates Journal for Engineering Research, 15(2), 49-61, [17]. Siegel R., Transient free convection from a vertical flat plate, Transactions of the ASME, 30, 349, [18]. Singh, R. K., Singh, A. K., Chandran, P., Sacheti, N. C.: Effect of Induced magnetic field on hdromagnetic free convective flow between vertical walls, Heat and Mass transfer, 46, [19]. Singh, R. K., Singh, A. K.: Effect of induced magnetic field on natural convection in vertical concentric annuli, Acta Mechanica Sinica, 28(2), , [20]. Singh, R. K., Singh A. K.: MHD free convective flow past a semi-infinite vertical permeable wall, Applied Mathematics and Mechanics, 33(9), , [21]. Sharma, S. and Deka, R. K., Thermal radiation and oscillating plate temperature effects on MHD unstead flow past a semi-infinite porous vertical plate under suction and chemical reaction. International Journal available at Vol. 292, 33-52, [22]. Singh R. K., Singh T. R., and Kataria H. R., Induced magnetic field effects on mixed convection, IOSR Journal of Mathematics (IOSR-JM), 10(6), 76-85, [23]. Uwanta, I. J., Hanmza, M. M., and Ibrahim, A. O., Heat and Mass Transfer Flow through a Porous Medium with Variable Permeabilit and Periodic Suction, International Journal of Computer Applications, Volume 36, No.2, , DOI: / Page