American Journal of Applied Mathematics and Statistics
ISSN (Print): 2328-7306 ISSN (Online): 2328-7292 Website: Editor-in-chief: Mohamed Seddeek
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American Journal of Applied Mathematics and Statistics. 2016, 4(5), 161-168
DOI: 10.12691/ajams-4-5-4
Open AccessArticle

Numerical Analysis on MHD Natural Convection within Trapezoidal Cavity Having Circular Block

Muhammad Sajjad Hossain1, , M. A. Alim2 and Kazi H Kabir3

1Department of Arts and Sciences, Ahsanullah University of Science & Technology (AUST), Dhaka-1208, Bangladesh

2Department of Mathematics, Bangladesh University of Engineering & Technology (BUET), Dhaka-1000, Bangladesh

3Department of Mathematics, Mohammadpur Kendriya College, Dhaka, Bangladesh

Pub. Date: October 28, 2016

Cite this paper:
Muhammad Sajjad Hossain, M. A. Alim and Kazi H Kabir. Numerical Analysis on MHD Natural Convection within Trapezoidal Cavity Having Circular Block. American Journal of Applied Mathematics and Statistics. 2016; 4(5):161-168. doi: 10.12691/ajams-4-5-4


In this paper, we have studied MHD natural convection within trapezoidal cavity having circular block with uniformly heated bottom wall with inclination angles (ф). To investigate the effects of uniform heating with the circular block a Galerkin finite element method is studied and also used for solving the Navier-Stokes equations for different angles Φs. Here left and right walls are considered as cold and upper wall is considered as thermal insulated in a trapezoidal cavities. Rayleigh number (Ra) from 103 to 105, Hartmann number (Ha = 20) and Prandtl number (Pr) from 0.026 to 0.7 with various tilt angles Ф = 450, 300 and 00 (square) are concerned with the fluid. By different sets of governing equations along with the corresponding boundary conditions are used to set the physical problems. Results are shown in terms of streamlines, isotherms, heat flux and heat transfer rates for different Ra and Pr. It is seen that for different angles Φs conduction dominant region changes for different Pr when Ra increases. Local and average nusselt numbers are also used for heat transfer analysis for different irrespective Φs.

MHD natural convection Galerkin finite element method trapezoidal cavity uniform heating circular block

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[1]  J.M. Garrpeters, The neutral stability of surface-tension driven cavity flows subject to buoyant forces-I. Transverse and longitudinal disturbances, Chem. Eng. Sci. 47 (5), 1992, 1247-1264.
[2]  L.B. Wang, N.I. Wakayama, Control of natural convection in non- and low-conducting diamagnetic fluids in a cubical enclosure using inhomogeneous magnetic fields with different directions, Chem. Eng. Sci. 57 (11), 2002, 1867-1876.
[3]  I.E. Sarris, I. Lekakis, N.S. Vlachos, Natural convection in a 2D enclosure with sinusoidal upper wall temperature, Num Heat Transfer A 42 (2002) 513-530.
[4]  A. Ousegui, A. Le Bail, M. Havet, Numerical and experimental study of a natural convection thawing process, AIChE J. 52 (12), 2006, 4240-4247.
[5]  O.G. Martynenko, P.P. Khramtsov, Free-Convective Heat Transfer, Springer, Berlin, 2005.
[6]  J. Patterson, J. Imberger, Unsteady natural convection in a rectangular cavity, J.Fluid Mech., 100, 1980, 65-86.
[7]  J.D. Hall, A. Bejan, J.B. Chaddock, Transient natural convection in a rectangular enclosure with one heated side wall, Int. J. Heat Fluid Flow, 9, 1988, 396-404.
[8]  J.M. Hyun, J.W. Lee, Numerical solutions of transient natural convection in a square cavity with different sidewall temperature, Int. J. Heat Fluid Flow, 10, 1989, 146-151.
[9]  A.M. Al-Amiri, K.M. Khanafer, Pop I, Numerical simulation of combined thermal and mass transport in a square lid-driven cavity, Int. J. Thermal Sci., 46(7), 2007, 662-671.
[10]  Y.E. Karyakin, Y.A. Sokovishin, O.G. Martynenko, Transient natural convection in triangular enclosures, Int. J. Heat Mass Transfer, 31, 1988, 1759-1766.
[11]  A. Bejan, D. Poulikakos, Natural convection in an attic shaped space filled with porous material, J. Heat Transfer – Trans. ASME, 104, 1982, 241-247.
[12]  D. Poulikakos, A. Bejan, Numerical study of transient high Rayleigh number convection in an attic-shaped porous layer, J. Heat Transfer – Trans. ASME, 105, 1983, 476-484.
[13]  T. Basak, R. Anandalakshmi, Kumar Pushpendra, S. Roy, “Entropy generation vs energy flow due to natural convection in a trapezoidal cavity with isothermal and non-isothermal hot bottom wall”, Energy, 37(1), January 2012, 514.
[14]  T. Basak, D. Ramakrishna, S. Roy, A. Matta, I. Pop, “A comprehensive heatline based approach for natural convection flows in trapezoidal enclosures: Effect of various walls heating”, International Journal of Thermal Sciences, 50 (8), August 2011, 1385-1404.
[15]  R. Anandalakshmi, T. Basak, “Heat flow visualization for natural convection in rhombic enclosures due to isothermal and non-isothermal heating at the bottom wall”, International Journal of Heat and Mass Transfer, 55(4), 31 January 2012, 1325-1342.
[16]  M. Hasanuzzaman, H.F. Öztop, M. M. Rahman, N.A. Rahim, R. Saidur and Y. Varol,“Magnetohydrodynamic natural convection in trapezoidal cavities”,  International Communications in Heat and Mass Transfer, 39(9), November 2012, 1384-1394.
[17]  F. Selimefendigil, H. F. Öztop and N. Abu-Hamdeh “Natural convection and entropy generation in nanofluid filled entrapped trapezoidal cavities under the influence of magnetic field”, Entropy 2016, 18(2), 43.
[18]  B. J. Gireesha, C.S. Bagewadi, B.C. Prasannakumar, “Study of unsteady dusty fluid Flow through rectangular channel in Frenet Frame Field system”, International Journal of Pure And Applied Mathematics, 34 (4), January 2007, 525-535.
[19]  M. S. Hossain, M. A. Alim, “MHD free convection within trapezoidal cavity with non-uniformly heated bottom wall”, International Journal of Heat and Mass Transfer, 69, February 2014, 327-336.
[20]  M. S. Hossain, M. A. Alim, “MHD free convection within trapezoidal cavity with uniformly heated bottom wall”, Annals of Pure and Applied Mathematics, 3(1), 2013, 41-55.
[21]  M. S. Hossain, M. N. Uddin, M. A. Alim, M. Shobahani, “Finite Element Analysis of MHD Free Convection within trapezoidal Eenclosures with uniformly heated side walls.”, IOSR Journal of Mathematics, 6(3), May-Jun 2013, 64-73.
[22]  S. K. Natarajan, K.S. Reddy, and T. K. Mallick, “Heat loss characteristics of trapezoidal cavity receiver for solar linear concentrating system”, Applied Energy, 93, May 2012, 523-531.
[23]  T. Basak, S. Roy, A. Matta and I. Pop, “Analysis of heatlines for natural convection within porous trapezoidal enclosures: Effect of uniform and non-uniform heating of bottom wall” International Journal of Heat and Mass Transfer, Vol. 53, Issues 25-26, December 2010, 5947-5961.
[24]  T. Basak, S. Roy, and I. Pop, Heat flow analysis for natural convection within trapezoidal enclosures based on heatline concept, Int.J. Heat Mass Transfer, 52, March 2009, 2471-2483.
[25]  C. Taylor and P. Hood, “A Numerical Solution of the Navier-Stokes Equations Using Finite Element echnique”, Computer and Fluids, 1( 1), 1973, 73-100.
[26]  P. Dechaumphai, Finite Element Method in Engineering, 2nd ed., Chulalongkorn University Press, Bangkok, 1999.