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August  2018, 11(4): 595-606. doi: 10.3934/dcdss.2018034

Exact solution of magnetohydrodynamic slip flow and heat transfer over an oscillating and translating porous plate

 1 National University of Sciences and Technology, College of Electrical and Mechanical Engineering, Peshawar Road, Rawalpindi, Pakistan 2 National University of Sciences and Technology, School of Natural Science, H-12 Islamabad, Pakistan

Received  December 2016 Revised  May 2017 Published  November 2017

Objective of this paper is to study natural convection MHD flow past over a moving porous plate with heat source in the porous medium. The motion of the plate is translating as well as oscillating and embedded in the porous medium. The exact solution of the governing equations, of the flow and heat transfer for this model is obtained. To study heat flux for our model we use Nusselt number. Comparisons of effects of magnetic parameter $M$, translation $a$ and heat source parameter $S$ on velocity and temperature profile is given. The effects of some other physical parameters like Prandtl number $P_r$, Grashof number for heat transfer $G_r$, Permeability parameter $K_p$, is presented graphically on the distributions of velocity and temperature. It is concluded that the fluid motion in the boundary layer increases with increase of $a$, $S$, $G_r$ and $K_P$. Whereas opposite behavior is observed for $M$ and $P_r$. The heat source parameter increases the temperature of fluid and on the other hand cooling effects occur due to $P_r$ and $v_0$.

Citation: Yasir Ali, Arshad Alam Khan. Exact solution of magnetohydrodynamic slip flow and heat transfer over an oscillating and translating porous plate. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 595-606. doi: 10.3934/dcdss.2018034
References:
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Jana, Radiation effect on natural convection near a vertical plate embedded in porous medium with ramped wall Temperature, Open Journal of Fluid Dynamics, 2011 (2011), 1-11. doi: 10.4236/ojfd.2011.11001. Google Scholar [6] S. S. Das, R. K. Tripathy, R. K. Padhy and M. Sahu, Combined natural convection and mass transfer effects on unsteady flow past an infinite vertical porous plate embedded in a porous medium with heat Source, International Journal of Energy and Environment, 3 (2012), 591-604. Google Scholar [7] S. S. Das, S. Mishra and P. Tripathy, Natural convection mass transfer hydromagentic flow past an oscillating porous plate with heat source in a porous medium, International Journal of Energy and Environment, 5 (2014), 583-590. Google Scholar [8] S. S. Das, M. R. Saran and B. Pradhan, Natural convection hydromagnetic flow and heat transfer past an infinite vertical porous plate embedded in a porous medium, Journal of Applied Engineering, 3 (2015), 234-240. 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Hayat, Numerical study of magnetohydrodynamics generalized Couette flow of Eyring-Powell fluid with heat transfer and slip condition, International Journal for Numerical Methods for Heat and Fluid Flow, 26 (2016), 1433-1445. doi: 10.1108/HFF-04-2015-0131. Google Scholar [13] K. Javaherdeh, M. M. Nejad and M. Moslemi, Natural convection heat and mass transfer in MHD fluid flow past a moving vertical plate with variable surface temperature and concentration in a porous medium, Engineering Science and Technology, an International Journal, 18 (2015), 423-431. doi: 10.1016/j.jestch.2015.03.001. Google Scholar [14] D. C. Kesavaiah, P. V. Satyanarayana, A. Sudhakaraiah and S. Venkataramana, Natural convection heat transfer oscillatory flow of an elastico-viscous fluid from vertical plate, International Journal of Research in Engineering and Technology, 02 (2013).Google Scholar [15] A. Khalid, I. Khan, A. Khan and S. Shafie, Unsteady MHD free convection flow of casson fluid past over an oscillating vertical plate embedded in a porous medium, Engineering Science and Technology, an International Journal, 18 (2015), 309-317. doi: 10.1016/j.jestch.2014.12.006. Google Scholar [16] F. C. Lai and F. A. Kulacki, Coupled heat and mass transfer by natural convection from vertical surface in porous medium, International Journal of Heat Mass Transfer, 34 (1991), 1189-1194. Google Scholar [17] S. Mukhopadhyay and I. C. Mandal, Magnetohydrodynamic (MHD) mixed convection slip flow and heat transfer over a vertical porous plate, Engineering Science and Technology, an International Journal, 18 (2015), 98-105. doi: 10.1016/j.jestch.2014.10.001. Google Scholar [18] R. Muthucumaraswamy and B. Saravanan, Finite difference solution of unsteady flow past an oscillating semi-infinite vertical plate with variable surface temperature and uniform mass flux, International Journal of Applied Mechanics and Engineering, 19 (2014), 709-724. doi: 10.2478/ijame-2014-0049. Google Scholar [19] R. Muthucumaraswamy and K. Manivannan, Mass transfer effects on vertical oscillating plate with heat flux, Theoretical and Applied Mechanics, 34 (2007), 309-322. Google Scholar [20] H. Poonia and R. C. Chaudhary, MHD free convection and mass transfer flow over an infinite vertical porous plate with viscous dissipation, Theoretical and Applied Mechanics, 37 (2010), 263-287. doi: 10.2298/TAM1004263P. Google Scholar [21] A. Raptis and J. Vlahos, Unsteady hydromagnetic free convective flow through a porous medium, Letters in Heat and Mass Transfer, 9 (1982), 59-64. doi: 10.1016/0094-4548(82)90048-0. Google Scholar [22] S. Rashidi, M. Dehghan, R. Ellahi, M. Riaz and M. T. Jamal-Abad, Study of stream wise transverse magnetic fluid flow with heat transfer around an obstacle embedded in a porous medium, Journal of Magnetism and Magnetic Materials, 378 (2015), 128-137. doi: 10.1016/j.jmmm.2014.11.020. Google Scholar [23] G. A. Sheikhzadeh, M. E. Qomi, N. Hajialigol and A. Fattahi, Numerical study of mixed convection flows in a lid-driven enclosure filled with nanofluid using variable properties, Results in Physics, 2 (2012), 5-3. doi: 10.1016/j.rinp.2012.01.001. Google Scholar

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References:
 [1] S. Amit and R. K. Srivastava, Heat and mass transfer effects on flow past an oscillating infinite vertical plate with variable temperature through porous media, Research Journal Recent Science, 2 (2013), 316-321. Google Scholar [2] R. C. Chaudhary and A. Jain, Combined heat and mass transfer effects on MHD free convection flow past an oscillating plate embedded in porous medium, Romanian Journal of Physics, 52 (2007), 505-524. Google Scholar [3] S. S. Das, S. K. Sahoo and G. C. Dash, Numerical solution of mass transfer effects on unsteady flow past an accelerated vertical porous plate with suction, Bulletin Malaysian Mathematical Science Society, 29 (2006), 33-42. Google Scholar [4] K. Das, Exact solution of MHD free convection flow and mass transfer near a moving vertical plate in presence of thermal radiation, African Journal Of Mathematical Physics, 8 (2010), 29-41. Google Scholar [5] S. Das, M. Jana and R. N. Jana, Radiation effect on natural convection near a vertical plate embedded in porous medium with ramped wall Temperature, Open Journal of Fluid Dynamics, 2011 (2011), 1-11. doi: 10.4236/ojfd.2011.11001. Google Scholar [6] S. S. Das, R. K. Tripathy, R. K. Padhy and M. Sahu, Combined natural convection and mass transfer effects on unsteady flow past an infinite vertical porous plate embedded in a porous medium with heat Source, International Journal of Energy and Environment, 3 (2012), 591-604. Google Scholar [7] S. S. Das, S. Mishra and P. Tripathy, Natural convection mass transfer hydromagentic flow past an oscillating porous plate with heat source in a porous medium, International Journal of Energy and Environment, 5 (2014), 583-590. Google Scholar [8] S. S. Das, M. R. Saran and B. Pradhan, Natural convection hydromagnetic flow and heat transfer past an infinite vertical porous plate embedded in a porous medium, Journal of Applied Engineering, 3 (2015), 234-240. Google Scholar [9] G. C. Dash and S. S. Das, Hall effects on MHD flow along an accelerated porous flat plate with mass transfer and internal heat generation, Mathematical Engineering in Industry, 7 (1999), 389-404. Google Scholar [10] R. Ellahi, E. Shivanian, S. Abbasbandy and T. Hayat, Analysis of some magnetohydrodynamic flows of third order fluid saturating porous space, Journal of Porous Media, 18 (2015), 89-98. doi: 10.1615/JPorMedia.v18.i2.10. Google Scholar [11] R. Ellahi, M. M. Bhatti and I. Pop, Effects of hall and ion slip on MHD peristaltic flow of Jeffrey fluid in a non-uniform rectangular duct, International Journal of Numerical Methods for Heat and Fluid Flow, 26 (2016), 1802-1820. doi: 10.1108/HFF-02-2015-0045. Google Scholar [12] R. Ellahi, E. Shivanian, S. Abbasbandy and T. Hayat, Numerical study of magnetohydrodynamics generalized Couette flow of Eyring-Powell fluid with heat transfer and slip condition, International Journal for Numerical Methods for Heat and Fluid Flow, 26 (2016), 1433-1445. doi: 10.1108/HFF-04-2015-0131. Google Scholar [13] K. Javaherdeh, M. M. Nejad and M. Moslemi, Natural convection heat and mass transfer in MHD fluid flow past a moving vertical plate with variable surface temperature and concentration in a porous medium, Engineering Science and Technology, an International Journal, 18 (2015), 423-431. doi: 10.1016/j.jestch.2015.03.001. Google Scholar [14] D. C. Kesavaiah, P. V. Satyanarayana, A. Sudhakaraiah and S. Venkataramana, Natural convection heat transfer oscillatory flow of an elastico-viscous fluid from vertical plate, International Journal of Research in Engineering and Technology, 02 (2013).Google Scholar [15] A. Khalid, I. Khan, A. Khan and S. Shafie, Unsteady MHD free convection flow of casson fluid past over an oscillating vertical plate embedded in a porous medium, Engineering Science and Technology, an International Journal, 18 (2015), 309-317. doi: 10.1016/j.jestch.2014.12.006. Google Scholar [16] F. C. Lai and F. A. Kulacki, Coupled heat and mass transfer by natural convection from vertical surface in porous medium, International Journal of Heat Mass Transfer, 34 (1991), 1189-1194. Google Scholar [17] S. Mukhopadhyay and I. C. Mandal, Magnetohydrodynamic (MHD) mixed convection slip flow and heat transfer over a vertical porous plate, Engineering Science and Technology, an International Journal, 18 (2015), 98-105. doi: 10.1016/j.jestch.2014.10.001. Google Scholar [18] R. Muthucumaraswamy and B. Saravanan, Finite difference solution of unsteady flow past an oscillating semi-infinite vertical plate with variable surface temperature and uniform mass flux, International Journal of Applied Mechanics and Engineering, 19 (2014), 709-724. doi: 10.2478/ijame-2014-0049. Google Scholar [19] R. Muthucumaraswamy and K. Manivannan, Mass transfer effects on vertical oscillating plate with heat flux, Theoretical and Applied Mechanics, 34 (2007), 309-322. Google Scholar [20] H. Poonia and R. C. Chaudhary, MHD free convection and mass transfer flow over an infinite vertical porous plate with viscous dissipation, Theoretical and Applied Mechanics, 37 (2010), 263-287. doi: 10.2298/TAM1004263P. Google Scholar [21] A. Raptis and J. Vlahos, Unsteady hydromagnetic free convective flow through a porous medium, Letters in Heat and Mass Transfer, 9 (1982), 59-64. doi: 10.1016/0094-4548(82)90048-0. Google Scholar [22] S. Rashidi, M. Dehghan, R. Ellahi, M. Riaz and M. T. Jamal-Abad, Study of stream wise transverse magnetic fluid flow with heat transfer around an obstacle embedded in a porous medium, Journal of Magnetism and Magnetic Materials, 378 (2015), 128-137. doi: 10.1016/j.jmmm.2014.11.020. Google Scholar [23] G. A. Sheikhzadeh, M. E. Qomi, N. Hajialigol and A. Fattahi, Numerical study of mixed convection flows in a lid-driven enclosure filled with nanofluid using variable properties, Results in Physics, 2 (2012), 5-3. doi: 10.1016/j.rinp.2012.01.001. Google Scholar
Velocity profile against $y$ for different values of $a$
Velocity against $y$ for different values of $M$
Velocity profile against $y$ for different values of $S$
Velocity profile against $y$ for different values of $G_r$
Velocity profile against $y$ for different values of $K_p$
Velocity profile against $y$ for different values of $P_r$
Velocity profile against $y$ for different values of $\upsilon_0$
Temperature profile against $y$ for different values of $\upsilon_0$
Temperature profile against $y$ for different values of $P_r$
Temperature profile against $y$ for different values of $S$
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