Journal of Applied and Computational Mechanics
Magnetohydrodynamic Free Convection Flows with Thermal Memory over a Moving Vertical Plate in Porous Medium
Document Type : Research Paper
Authors
1 Department of Mathematics, Lahore Leads University, Lahore Pakistan
2 Abdus Salam School of Mathematical Sciences, GC University Lahore, Pakistan
3 Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, 44519, Egypt
4 Department of Civil Engineering, School of Engineering, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK
Abstract
The unsteady hydro-magnetic free convection flow with heat transfer of a linearly viscous, incompressible, electrically conducting fluid near a moving vertical plate with the constant heat is investigated. The flow domain is the porous half-space and a magnetic field of a variable direction is applied. The Caputo time-fractional derivative is employed in order to introduce a thermal flux constitutive equation with a weakly memory. The exact solutions for the fractional governing differential equations for fluid temperature, Nusselt number, velocity field, and skin friction are obtained by using the Laplace transform method. The numerical calculations are carried out and the results are presented in graphical illustrations. The influence of the memory parameter (the fractional order of the time-derivative) on the temperature and velocity fields is analyzed and a comparison between the fluid with the thermal memory and the ordinary fluid is made. It was observed that due to evolution in the time of the Caputo power-law kernel, the memory effects are stronger for the small values of the time t. Moreover, it is found that the fluid flow is accelerated / retarded by varying the inclination angle of the magnetic field direction.
Keywords
Main Subjects
1. F.A.L. Dullien, Introduction, in Porous Media (Second Edition). 1992, Academic Press: San Diego. p. 1-3.
2. M.A. Mujeebu, M.Z. Abdullah, M.Z.A. Bakar, A.A. Mohamad, M.K. Abdullah, Applications of porous media combustion technology – A review. Applied Energy, 86(9) (2009) 1365-1375.
3. D.A. Nield, A. Bejan, Convection in Porous Media, 2013, Springer, New York, NY.
4. J. Sallès, J.F. Thovert, P.M. Adler, Reconstructed porous media and their application to fluid flow and solute transport. Journal of Contaminant Hydrology, 13(1) (1993) 3-22.
5. S. Bories, Natural Convection in Porous Media, in Advances in Transport Phenomena in Porous Media, NATO ASI Series (Series E: Applied Sciences), Vol. 128. Springer, Dordrecht, 1987.
6. M.M. Rashidi, M. Ali, N. Freidoonimehr, B. Rostami, M. Anwar Hossain, Mixed convective heat transfer for MHD viscoelastic fluid flow over a porous wedge with thermal radiation, Advances in Mechanical Engineering, 2014 (2014) p. 735939.
7. M.M. Rashidi, E. Erfani, Analytical Method for Solving Steady MHD Convective and Slip Flow due to a Rotating Disk with Viscous Dissipation and Ohmic Heating, Engineering Computations, 29(6) (2012) 562–579.
8. C. Fetecau, N.A. Shah, D. Vieru, General solutions for hydromagnetic free convection flow over an infinite plate with Newtonian heating, mass diffusion and chemical reaction, Commun. Theor. Phys., 68 (2017) 768-782.
9. A. Bejan, K.R. Khair, Heat and mass transfer by natural convection in a porous medium. International Journal of Heat and Mass Transfer, 28(5) (1985) 909-918.
10. M. Narayana, A.A. Khidir, P. Sibanda, and P.V.S.N. Murthy, Soret Effect on the Natural Convection From a Vertical Plate in a Thermally Stratified Porous Medium Saturated With Non-Newtonian Liquid. Journal of Heat Transfer, 135(3) (2013) 032501-032510.
11. C. Fetecau, R. Ellahi, M. Khan, N.A. Shah, Combined porous and magnetic effects on some fundamental motions of Newtonian fluids over an infinite plate, Journal of Porous Media, 21(7) (2018) 589-605.
12. K.R. Cramer, S. Bai, and S. Pai, Magnetofluid Dynamics for Engineers and Applied Physicists. Scripta Publishing Company, 1973.
13. A.K. Acharya, G.C. Dash, and S.R. Mishra, Free Convective Fluctuating MHD Flow through Porous Media Past a Vertical Porous Plate with Variable Temperature and Heat Source. Physics Research International, 2014 (2014) p. 8.
14. A. Khan, R. Solanki, Initial unsteady free convective flow past an infinite vertical plate with radiation and mass transfer effects, Int. J. Appl. Mech. Eng., 22(4) (2017) 931-943.
15. N. Marneni, S. Tippa, and R. Pendyala, Ramp temperature and Dufour effects on transient MHD natural convection flow past an infinite vertical plate in a porous medium. The European Physical Journal Plus, 130(12) (2015) p. 251.
16. Samiulhaq, S. Ahmad, D. Vieru, I. Khan, and S. Shafie, Unsteady Magnetohydrodynamic Free Convection Flow of a Second Grade Fluid in a Porous Medium with Ramped Wall Temperature. PLoS One, 9(5) (2014) p. e88766.
17. P.K. Pattnaik, T. Biswal, Analytical Solution of MHD Free Convective Flow through Porous Media with Time Dependent Temperature and Concentration. Walailak Journal of Science and Technology, 12(9) (2014) p. 14.
18. G.S. Seth, R. Sharma, B. Kumbhakar, Heat and Mass Transfer Effects on Unsteady MHD Natural Convection Flow of a Chemically Reactive and Radiating Fluid through a Porous Medium Past a Moving Vertical Plate with Arbitrary Ramped Temperature, Journal of Applied Fluid Mechanics, 9(1) (2016) 103-117.
19. S.A. Gaffar, V. Ramachandra Prasad, E. Keshava Reddy, MHD free convection flow of Eyring–Powell fluid from vertical surface in porous media with Hall/ionslip currents and ohmic dissipation. Alexandria Engineering Journal, 55(2) (2016) 875-905.
20. A. Sattar, Unsteady hydromagnetic free convection flow with hall current mass transfer and variable suction through a porous medium near an infinite vertical porous plate with constant heat flux. International Journal of Energy Research, 18(9) (1994) 771-775.
21. M.M. Rashidi, E. Momoniat, B. Rostami, Analytic approximate solutions for MHD boundary-layer viscoelastic fluid flow over continuously moving stretching surface by homotopy analysis method with two auxiliary parameters, Journal of Applied Mathematics, 2012, Article ID 780415, 19 pages.
22. A. Hussanan, Z. Ismail, I. Khan, A.G. Hussein, S. Shafie, Unsteady boundary layer MHD free convection flow in a porous medium with constant mass diffusion and Newtonian heating. The European Physical Journal Plus, 129(3) (2014) p. 46.
23. C. Fetecau, N.A. Shah, D. Vieru, General solutions for hydromagnetic free convection flow over an infinite plate with Newtonian heating, mass diffusion and chemical reaction, Commun. Theor. Phys., 68 (2017) 768-782.
24. A.J. Omowaye, A.I. Fagbade, A.O. Ajayi, Dufour and soret effects on steady MHD convective flow of a fluid in a porous medium with temperature dependent viscosity: Homotopy analysis approach. Journal of the Nigerian Mathematical Society, 34(3) (2015) 343-360.
25. N.A. Shah, A.A. Zafar, S. Akhtar, General solution for MHD free convection flow over a vertical plate with ramped wall temperature, Arabian Journal of Mathematics, 7(1) (2018) 49–60.
26. Samiulhaq, I. Khan, F. Ali, S. Shafie, MHD Free Convection Flow in a Porous Medium with Thermal Diffusion and Ramped Wall Temperature, Journal of the Physical Society of Japan, 81(4) (2012) 044401.
27. F. Ali, I. Khan, S. Shafie, N. Musthapa, Heat and Mass Transfer with Free Convection MHD Flow Past a Vertical Plate Embedded in a Porous Medium. Mathematical Problems in Engineering, 2013, p. 13.
28. B. Lavanya, A.L. Ratnam, Dufour and soret effects on steady MHD free convective flow past a vertical porous plate embedded in a porous medium with chemical reaction, radiation heat generation and viscous dissipation, Advances in Applied Science Research, 5(1) (2014) 127-142
29. R. U. Haq, F. A. Soomro, T. Mekkaoui, Q. M. Al-Mdallal, MHD natural convection flow enclosure in a corrugated cavity filled with a porous medium, Int. J. Heat Mass Transfer, 121 (2018) 1168-1178.
30. O. D. Makinde, P. Y. Mhone, On temporal stability analysis for hydromagnetic flow in a channel filled with a saturated porous medium, Flow, Turbulence and Combustion, 83(1) (2009) 21-32.
31. B. M. Shankar, I. S. Shivakumara, Magnetohydrodynamic stability of pressure-driven flow in an anisotropic porous channel: Accurate solution, Applied Mathematics and Computation, 321 (2018) 752-767.
32. B. M. Shankar, J. Kumar, I. S. Shivakumara, Magnetohydrodynamic stability of natural convection in a vertical porous slab. Journal of Magnetism and Magnetic Materials, 421 (2017) 152-164.
33. J.A.T. Machado, M.F. Silva, R.S. Barbosa, I.S. Jesus, C. Reis, #237, l. M., M.G. Marcos, A.F. Galhano, Some Applications of Fractional Calculus in Engineering. Mathematical Problems in Engineering, 2010, Article ID 639801, 34 pages.
34. C. Li, Y. Chen, J. Kurths, Fractional calculus and its applications. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 371 (1990) 1-16.
35. N.A. Shah, I. Khan, Heat transfer analysis in a second grade fluid over and oscillating vertical plate using fractional Caputo–Fabrizio derivatives. The European Physical Journal C, 76(7) (2016) p. 362.
36. I. Khan, N.A. Shah, Y. Mahsud, D. Vieru, Heat transfer analysis in a Maxwell fluid over an oscillating vertical plate using fractional Caputo-Fabrizio derivatives. The European Physical Journal Plus, 132(4) (2017) p. 194.
37. I. Khan, N.A. Shah, L.C.C. Dennis, A scientific report on heat transfer analysis in mixed convection flow of Maxwell fluid over an oscillating vertical plate. Scientific Reports, 7 (2017) p. 40147.
38. N.A. Shah, D. Vieru, C. Fetecau, Effects of the fractional order and magnetic field on the blood flow in cylindrical domains. Journal of Magnetism and Magnetic Materials, 409 (2016) 10-19.
39. M. Khan, S.H. Ali, and H. Qi, Exact solutions for some oscillating flows of a second grade fluid with a fractional derivative model. Math. Comput. Model., 49(7-8) (2009) 1519-1530.
40. S. Aman, I. Khan, Z. Ismail, M. Z. Salleh, Applications of fractional derivatives to nanofluids: exact and numerical solutions. Math. Model.Natural Phenomena, 13(1) (2018) 2-15.
41. Q. Al-Mdallal, K. A. Abro, I. Khan, Analytical solutions of fractional Walter’s B fluid with applications. Complexity, 2018, Article ID 8131329.
42. S. Aman, I. Khan, Z. Ismail, M. Z. Salleh, I. Tlili, A new Caputo time fractional model for heat transfer enhancement of water based graphene nanofluid: An application to solar energy, Results in Physics, 9 (2018) 1352-1362.
43. N.A Shah, Y. Mahsud, A. Ali Zafar, Unsteady free convection flow of viscous fluids with analytical results by employing time-fractional Caputo-Fabrizio derivative (without singular kernel). The European Physical Journal Plus, 132(10) (2017) p. 411.
44. D.K. Gartling, C.E. Hickox, A numerical study of the applicability of the Boussinesq approximation for a fluid-saturated porous medium. International Journal for Numerical Methods in Fluids, 5(11) (1985) 995-1013.
45. B. Stankovic, On the function of E. M. Wright, Publications de L'Institut Mathematique, Nouvelle serie, tome, 10(24) (1970) 113-124.
46. C. F. Lorenzo, T. T. Hartley, Generalized Functions for the Fractional Calculus, NASA/TP-1999-209424/REV1, 1999.
47. B. I. Henry, T. A. Langlands, P. Straka, An Introduction to Fractional Diffusion, in Dewar, R. L. and Detering F. eds., Complex Physical, Biophysical and Econophysical Systems, Proc. 22nd Canberra International Physics Summer School, The Australian National University, Canberra, 8-19 December 2002, edn. World Scientific Lecture Notes in Complex Systems, vol. 9, World Scientific, Hackensack, NJ, 37-89, 2010.
48. J. Hristov, Derivatives with non-singular kernels. From the Caputo-Fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models, In book Frontiers in Fractional Calculus, 2017, Edition 1st, Bentham Science Publishers, Editor Sachin Bhalekar, Chapter 10.
49. D. Baleanu, B. Agheli, M. M. Al Qurashi, Fractional advection differential equation within Caputo and Caputo–Fabrizio derivatives, Adv. Mech. Eng., 8(12) (2016) 1-8.
)
