# American Institute of Mathematical Sciences

• Previous Article
Numerical investigation of Cattanneo-Christov heat flux in CNT suspended nanofluid flow over a stretching porous surface with suction and injection
• DCDS-S Home
• This Issue
• Next Article
Conservation laws and symmetries of time-dependent generalized KdV equations
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:
 [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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). [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. [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. [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. [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. [19] R. Muthucumaraswamy and K. Manivannan, Mass transfer effects on vertical oscillating plate with heat flux, Theoretical and Applied Mechanics, 34 (2007), 309-322. [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. [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. [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. [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.

show all references

##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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). [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. [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. [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. [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. [19] R. Muthucumaraswamy and K. Manivannan, Mass transfer effects on vertical oscillating plate with heat flux, Theoretical and Applied Mechanics, 34 (2007), 309-322. [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. [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. [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. [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.
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 $S$
 [1] Guofu Lu. Nonexistence and short time asymptotic behavior of source-type solution for porous medium equation with convection in one-dimension. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1567-1586. doi: 10.3934/dcdsb.2016011 [2] Edoardo Mainini. On the signed porous medium flow. Networks & Heterogeneous Media, 2012, 7 (3) : 525-541. doi: 10.3934/nhm.2012.7.525 [3] Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355 [4] Ansgar Jüngel, Ingrid Violet. Mixed entropy estimates for the porous-medium equation with convection. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 783-796. doi: 10.3934/dcdsb.2009.12.783 [5] Michela Eleuteri, Jana Kopfov, Pavel Krej?. Fatigue accumulation in an oscillating plate. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 909-923. doi: 10.3934/dcdss.2013.6.909 [6] Kaouther Ammar, Philippe Souplet. Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 665-689. doi: 10.3934/dcds.2010.26.665 [7] Anna Marciniak-Czochra, Andro Mikelić. A nonlinear effective slip interface law for transport phenomena between a fracture flow and a porous medium. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1065-1077. doi: 10.3934/dcdss.2014.7.1065 [8] Yaqing Liu, Liancun Zheng. Second-order slip flow of a generalized Oldroyd-B fluid through porous medium. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2031-2046. doi: 10.3934/dcdss.2016083 [9] Marie Henry, Danielle Hilhorst, Robert Eymard. Singular limit of a two-phase flow problem in porous medium as the air viscosity tends to zero. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 93-113. doi: 10.3934/dcdss.2012.5.93 [10] Wen Wang, Dapeng Xie, Hui Zhou. Local Aronson-Bénilan gradient estimates and Harnack inequality for the porous medium equation along Ricci flow. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1957-1974. doi: 10.3934/cpaa.2018093 [11] Zhiqiang Yang, Junzhi Cui, Qiang Ma. The second-order two-scale computation for integrated heat transfer problem with conduction, convection and radiation in periodic porous materials. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 827-848. doi: 10.3934/dcdsb.2014.19.827 [12] Changfeng Gui, Huaiyu Jian, Hongjie Ju. Properties of translating solutions to mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 441-453. doi: 10.3934/dcds.2010.28.441 [13] Noreen Sher Akbar, Dharmendra Tripathi, Zafar Hayat Khan. Numerical investigation of Cattanneo-Christov heat flux in CNT suspended nanofluid flow over a stretching porous surface with suction and injection. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 583-594. doi: 10.3934/dcdss.2018033 [14] Guillermo Reyes, Juan-Luis Vázquez. The Cauchy problem for the inhomogeneous porous medium equation. Networks & Heterogeneous Media, 2006, 1 (2) : 337-351. doi: 10.3934/nhm.2006.1.337 [15] Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393 [16] R.E. Showalter, Ning Su. Partially saturated flow in a poroelastic medium. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 403-420. doi: 10.3934/dcdsb.2001.1.403 [17] Paul Deuring, Stanislav Kračmar, Šárka Nečasová. Linearized stationary incompressible flow around rotating and translating bodies -- Leray solutions. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 967-979. doi: 10.3934/dcdss.2014.7.967 [18] Hongjie Ju, Jian Lu, Huaiyu Jian. Translating solutions to mean curvature flow with a forcing term in Minkowski space. Communications on Pure & Applied Analysis, 2010, 9 (4) : 963-973. doi: 10.3934/cpaa.2010.9.963 [19] Michiel Bertsch, Carlo Nitsch. Groundwater flow in a fissurised porous stratum. Networks & Heterogeneous Media, 2010, 5 (4) : 765-782. doi: 10.3934/nhm.2010.5.765 [20] Verena Bögelein, Frank Duzaar, Ugo Gianazza. Very weak solutions of singular porous medium equations with measure data. Communications on Pure & Applied Analysis, 2015, 14 (1) : 23-49. doi: 10.3934/cpaa.2015.14.23

2017 Impact Factor: 0.561

## Tools

Article outline

Figures and Tables