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Numerical investigation of CattanneoChristov heat flux in CNT suspended nanofluid flow over a stretching porous surface with suction and injection
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Conservation laws and symmetries of timedependent generalized KdV equations
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, H12 Islamabad, Pakistan 
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 $v_0$.
References:
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S. Amit, 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), 316321. 
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R. C. Chaudhary, 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), 505524. 
[3] 
S. S. Das, S. K. Sahoo, 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), 3342. 
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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), 2941. 
[5] 
S. Das, M. Jana, 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), 111. doi: 10.4236/ojfd.2011.11001. 
[6] 
S. S. Das, R. K. Tripathy, R. K. Padhy, 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), 591604. 
[7] 
S. S. Das, S. Mishra, 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), 583590. 
[8] 
S. S. Das, M. R. Saran, 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), 234240. 
[9] 
G. C. Dash, 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), 389404. 
[10] 
R. Ellahi, E. Shivanian, S. Abbasbandy, T. Hayat, Analysis of some magnetohydrodynamic flows of third order fluid saturating porous space, Journal of Porous Media, 18 (2015), 8998. doi: 10.1615/JPorMedia.v18.i2.10. 
[11] 
R. Ellahi, M. M. Bhatti, I. Pop, Effects of hall and ion slip on MHD peristaltic flow of Jeffrey fluid in a nonuniform rectangular duct, International Journal of Numerical Methods for Heat and Fluid Flow, 26 (2016), 18021820. doi: 10.1108/HFF0220150045. 
[12] 
R. Ellahi, E. Shivanian, S. Abbasbandy, T. Hayat, Numerical study of magnetohydrodynamics generalized Couette flow of EyringPowell fluid with heat transfer and slip condition, International Journal for Numerical Methods for Heat and Fluid Flow, 26 (2016), 14331445. doi: 10.1108/HFF0420150131. 
[13] 
K. Javaherdeh, M. M. Nejad, 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), 423431. doi: 10.1016/j.jestch.2015.03.001. 
[14] 
D. C. Kesavaiah, P. V. Satyanarayana, A. Sudhakaraiah and S. Venkataramana, Natural convection heat transfer oscillatory flow of an elasticoviscous fluid from vertical plate, International Journal of Research in Engineering and Technology, 02 (2013). 
[15] 
A. Khalid, I. Khan, A. Khan, 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), 309317. doi: 10.1016/j.jestch.2014.12.006. 
[16] 
F. C. Lai, 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), 11891194. 
[17] 
S. Mukhopadhyay, 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), 98105. doi: 10.1016/j.jestch.2014.10.001. 
[18] 
R. Muthucumaraswamy, B. Saravanan, Finite difference solution of unsteady flow past an oscillating semiinfinite vertical plate with variable surface temperature and uniform mass flux, International Journal of Applied Mechanics and Engineering, 19 (2014), 709724. doi: 10.2478/ijame20140049. 
[19] 
R. Muthucumaraswamy, K. Manivannan, Mass transfer effects on vertical oscillating plate with heat flux, Theoretical and Applied Mechanics, 34 (2007), 309322. 
[20] 
H. Poonia, 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), 263287. doi: 10.2298/TAM1004263P. 
[21] 
A. Raptis, J. Vlahos, Unsteady hydromagnetic free convective flow through a porous medium, Letters in Heat and Mass Transfer, 9 (1982), 5964. doi: 10.1016/00944548(82)900480. 
[22] 
S. Rashidi, M. Dehghan, R. Ellahi, M. Riaz, M. T. JamalAbad, 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), 128137. doi: 10.1016/j.jmmm.2014.11.020. 
[23] 
G. A. Sheikhzadeh, M. E. Qomi, N. Hajialigol, A. Fattahi, Numerical study of mixed convection flows in a liddriven enclosure filled with nanofluid using variable properties, Results in Physics, 2 (2012), 53. doi: 10.1016/j.rinp.2012.01.001. 
show all references
References:
[1] 
S. Amit, 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), 316321. 
[2] 
R. C. Chaudhary, 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), 505524. 
[3] 
S. S. Das, S. K. Sahoo, 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), 3342. 
[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), 2941. 
[5] 
S. Das, M. Jana, 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), 111. doi: 10.4236/ojfd.2011.11001. 
[6] 
S. S. Das, R. K. Tripathy, R. K. Padhy, 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), 591604. 
[7] 
S. S. Das, S. Mishra, 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), 583590. 
[8] 
S. S. Das, M. R. Saran, 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), 234240. 
[9] 
G. C. Dash, 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), 389404. 
[10] 
R. Ellahi, E. Shivanian, S. Abbasbandy, T. Hayat, Analysis of some magnetohydrodynamic flows of third order fluid saturating porous space, Journal of Porous Media, 18 (2015), 8998. doi: 10.1615/JPorMedia.v18.i2.10. 
[11] 
R. Ellahi, M. M. Bhatti, I. Pop, Effects of hall and ion slip on MHD peristaltic flow of Jeffrey fluid in a nonuniform rectangular duct, International Journal of Numerical Methods for Heat and Fluid Flow, 26 (2016), 18021820. doi: 10.1108/HFF0220150045. 
[12] 
R. Ellahi, E. Shivanian, S. Abbasbandy, T. Hayat, Numerical study of magnetohydrodynamics generalized Couette flow of EyringPowell fluid with heat transfer and slip condition, International Journal for Numerical Methods for Heat and Fluid Flow, 26 (2016), 14331445. doi: 10.1108/HFF0420150131. 
[13] 
K. Javaherdeh, M. M. Nejad, 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), 423431. doi: 10.1016/j.jestch.2015.03.001. 
[14] 
D. C. Kesavaiah, P. V. Satyanarayana, A. Sudhakaraiah and S. Venkataramana, Natural convection heat transfer oscillatory flow of an elasticoviscous fluid from vertical plate, International Journal of Research in Engineering and Technology, 02 (2013). 
[15] 
A. Khalid, I. Khan, A. Khan, 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), 309317. doi: 10.1016/j.jestch.2014.12.006. 
[16] 
F. C. Lai, 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), 11891194. 
[17] 
S. Mukhopadhyay, 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), 98105. doi: 10.1016/j.jestch.2014.10.001. 
[18] 
R. Muthucumaraswamy, B. Saravanan, Finite difference solution of unsteady flow past an oscillating semiinfinite vertical plate with variable surface temperature and uniform mass flux, International Journal of Applied Mechanics and Engineering, 19 (2014), 709724. doi: 10.2478/ijame20140049. 
[19] 
R. Muthucumaraswamy, K. Manivannan, Mass transfer effects on vertical oscillating plate with heat flux, Theoretical and Applied Mechanics, 34 (2007), 309322. 
[20] 
H. Poonia, 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), 263287. doi: 10.2298/TAM1004263P. 
[21] 
A. Raptis, J. Vlahos, Unsteady hydromagnetic free convective flow through a porous medium, Letters in Heat and Mass Transfer, 9 (1982), 5964. doi: 10.1016/00944548(82)900480. 
[22] 
S. Rashidi, M. Dehghan, R. Ellahi, M. Riaz, M. T. JamalAbad, 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), 128137. doi: 10.1016/j.jmmm.2014.11.020. 
[23] 
G. A. Sheikhzadeh, M. E. Qomi, N. Hajialigol, A. Fattahi, Numerical study of mixed convection flows in a liddriven enclosure filled with nanofluid using variable properties, Results in Physics, 2 (2012), 53. doi: 10.1016/j.rinp.2012.01.001. 
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