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Finite Element Solution for MHD Flow of Nanofluids with Heat and Mass Transfer through a Porous Media with Thermal Radiation, Viscous Dissipation and Chemical Reaction Effects

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Incompressible viscous fluids Partial differential equations

Published online by Cambridge University Press:  18 January 2017

Shafqat Hussain
Shafqat Hussain*
Affiliation:
Department of Mathematics, Capital University of Science & Technology, Islamabad, Pakistan
*
*Corresponding author. Email:[email protected] (S. Hussain)
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Abstract

In this paper, the problem of magnetohydrodynamics (MHD) boundary layer flow of nanofluid with heat and mass transfer through a porous media in the presence of thermal radiation, viscous dissipation and chemical reaction is studied. Three types of nanofluids, namely Copper (Cu)-water, Alumina (Al 2 O 3)-water and Titanium Oxide (TiO 2)-water are considered. The governing set of partial differential equations of the problem is reduced into the coupled nonlinear system of ordinary differential equations (ODEs) by means of similarity transformations. Finite element solution of the resulting system of nonlinear differential equations is obtained using continuous Galerkin-Petrov discretization together with the well-known shooting technique. The obtained results are validated using MATLAB “bvp4c” function and with the existing results in the literature. Numerical results for the dimensionless velocity, temperature and concentration profiles are obtained and the impact of various physical parameters such as the magnetic parameter M, solid volume fraction of nanoparticles 𝜙 and type of nanofluid on the flow is discussed. The results obtained in this study confirm the idea that the finite element method (FEM) is a powerful mathematical technique which can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering.

Keywords

Boundary layer flownanofluidmagnetohydrodynamicsstretching sheetporous mediumheat and mass transferfinite element method (FEM)continuous Galerkin-Petrov (cGP) discretizationnumerical comparison

MSC classification

Secondary: 35A25: Other special methods 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 76D05: Navier-Stokes equations
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 4 , August 2017 , pp. 904 - 923
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Sakiadis, B. C., Boundary-layer behavior on continuous solid surfaces: I. boundary-layer equations for two-dimensional and axisymmetric flow, AIChE J., 7(1) (1961), pp. 26–28.CrossRefGoogle Scholar
[2] Sakiadis, B. C., Boundary-layer behavior on continuous solid surfaces: II. the boundary layer on a continuous flat surface, AIChE J., 7(2) (1961), pp. 221–225.CrossRefGoogle Scholar
[3] Sakiadis, B. C., Boundary-layer behavior on continuous solid surfaces: III. the boundary layer on a continuous cylindrical surface, AIChE J., 7(3) (1961), pp. 467–472.CrossRefGoogle Scholar
[4] Tsou, F., Sparrow, E. and Goldstein, R., Flow and heat transfer in the boundary layer on a continuous moving surface, Int. J. Heat Mass Transfer, 10(2) (1967), pp. 219–235.CrossRefGoogle Scholar
[5] Crane, L., Flow past a stretching plate, Zeitschrift für angewandte Mathematik und Physik ZAMP, 21(4) (1970) pp. 645–647.CrossRefGoogle Scholar
[6] Chakrabarti, A. and Gupta, A. S., Hydromagnetic flow and heat transfer over a stretching sheet, Quarterly Appl. Math., 37 (1979), pp. 73–78.CrossRefGoogle Scholar
[7] Grubka, L. J. and Bobba, K. M., Heat transfer characteristics of a continuous, stretching surface with variable temperature, J. Heat Transfer, 107 (1985), pp. 248–250.Google Scholar
[8] Andersson, H., Bech, K. and Dandapat, B., Magnetohydrodynamic flow of a power-law fluid over a stretching sheet, Int. J. Nonlinear Mech., 27(6) (1992), pp. 929–936.CrossRefGoogle Scholar
[9] Chen, C. H., Laminar mixed convection adjacent to vertical, continuously stretching sheets, Heat Mass Transfer, 33(5-6) (1998), pp. 471–476.CrossRefGoogle Scholar
[10] Vajravelu, K. and Rollins, D., Hydromagnetic flow of a second grade fluid over a stretching sheet, Appl. Math. Comput., 148(3) (2004), pp. 783–791.Google Scholar
[11] Cortell, R., A note on magnetohydrodynamic flow of a power-law fluid over a stretching sheet, Appl. Math. Comput., 168(1) (2005), pp. 557–566.Google Scholar
[12] Abel, M. S. and Mahesha, N., Heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivity, non-uniform heat source and radiation, Appl. Math. Model., 32(10) (2008), pp. 1965–1983.CrossRefGoogle Scholar
[13] Aziz, A., A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition, Commun. Nonlinear Sci. Numer. Simulation, 14(4) (2009), pp. 1064–1068.CrossRefGoogle Scholar
[14] Yohannes, K. and Shankar, B., Heat and mass transfer in mhd flow of nanofluids through a porous media due to a stretching sheet with viscous dissipation and chemical reaction effects, Caribbean J. Sci. Tech., 1 (2013),, pp. 01–17.Google Scholar
[15] Choi, S., Enhancing thermal conductivity of fluids with nanoparticles, developments and applications of non-newtonian flows, Argonne National Laboratory, United States, 66 (1993), pp. 99–105.Google Scholar
[16] Kang, H. U., Kim, S. H. and Oh, J. M., Estimation of thermal conductivity of nanofluid using experimental effective particle volume, Exp. Heat Transfer, 19(3) (2006), pp. 181–191.CrossRefGoogle Scholar
[17] Vasu, V., Rama, K. K. and Chandra, A. K. S., Empirical correlations to predict thermophysical and heat transfer characteristics of nanofluids, Thermal Sci., 12 (2008), pp. 27–37.Google Scholar
[18] Rudyak, V., Belkin, A. and Tomilina, E., On the thermal conductivity of nanofluids, Tech. Phys. Lett., 36(7) (2010), pp. 660–662.CrossRefGoogle Scholar
[19] Aziz, A., Khan, W. and Pop, I., Free convection boundary layer flow past a horizontal flat plate embedded in porous medium filled by nanofluid containing gyrotactic microorganisms, Int. J. Thermal Sci., 56(0) (2012), pp. 48–57.CrossRefGoogle Scholar
[20] Wang, X.-Q. and Mujumdar, A. S., Heat transfer characteristics of nanofluids: a review, Int. J. Thermal Sci., 46(1) (2007), pp. 1–19.Google Scholar
[21] Wen, D. and Ding, Y., Experimental investigation into convective heat transfer of nanofluids at the entrance region under laminar flow conditions, Int. J. Heat Mass Transfer, 47(24) (2004), pp. 5181–5188.Google Scholar
[22] Xuan, Y. and Li, Q., Heat transfer enhancement of nanofluids, Int. J. Heat Fluid Flow, 21(1) (2000), pp. 58–64.Google Scholar
[23] Albadr, J., Tayal, S. and Alasadi, M., Heat transfer through heat exchanger using al2o3 nanofluid at different concentrations, Case Studies Thermal Eng., 1(1) (2013), pp. 38–44.Google Scholar
[24] Abbasbandy, S., A numerical solution of blasius equation by adomian's decomposition method and comparison with homotopy perturbation method, Chaos, Solitons & Fractals, 31(1) (2007), pp. 257–260.CrossRefGoogle Scholar
[25] Elgazery, N. S., Numerical solution for the falknerskan equation, Chaos, Solitons & Fractals, 35(4) (2008), pp. 738–746.CrossRefGoogle Scholar
[26] Kuo, B. L., Heat transfer analysis for the Falkner–Skan wedge flow by the differential transformation method, Int. J. Heat Mass Transfer, 48(2324) (2005), pp. 5036–5046.Google Scholar
[27] Wazwaz, A. M., The variational iteration method for solving two forms of blasius equation on a half-infinite domain, Appl. Math. Comput., 188(1) (2007), pp. 485–491.Google Scholar
[28] Liao, S., Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC Series–Modern Mechanics and Mathematics, Chapman & Hall/CRC Press, 2004.Google Scholar
[29] Yao, B. and Chen, J., A new analytical solution branch for the blasius equation with a shrinking sheet, Appl. Math. Comput., 215(3) (2009), pp. 1146–1153.Google Scholar
[30] Yao, B. and Chen, J., Series solution to the falknerskan equation with stretching boundary, Appl. Math. Comput., 208(1) (2009), pp. 156–164.Google Scholar
[31] Noiey, A. R., Haghparast, N., Miansari, M. and Ganji, D. D., Application of homotopy perturbation method to the mhd pipe flow of a fourth grade fluid, J. Physics: Conference Series, 96(1) (2008), 012079.Google Scholar
[32] Motsa, S. S. and Shateyi, S., A new approach for the solution of three-dimensional magnetohydrodynamic rotating flow over a shrinking sheet, Math. Problems Eng., 2010 (2010), 15 pages.Google Scholar
[33] Makukula, Z. G., Sibanda, P., Motsa, S. S. and Shateyi, S., On new numerical techniques for the mhd flow past a shrinking sheet with heat and mass transfer in the presence of a chemical reaction, Math. Problems Eng., 2011 (2011), 19 pages.Google Scholar
[34] Schieweck, F., A-stable discontinuous Galerkin-Petrov time discretization of higher order, J. Numer. Math., 18 (2010), pp. 25–57.CrossRefGoogle Scholar
[35] Matthies, G. and Schieweck, F., Higher order variational time discretizations for nonlinear systems of ordinary differential equations, Preprint 23/2011, Otto-von-Guericke Universität Magdeburg, Fakultät für Mathematik (2011).Google Scholar
[36] Hussain, S., Schieweck, F. and Turek, S., Higher order Galerkin time discretizations and fast multigrid solvers for the heat equation, J. Numer. Math., 19(1) (2011), pp. 41–61.Google Scholar
[37] Haile, E. and Shankar, B., Heat and mass transfer through a porous media of MHD flow of nanofluids with thermal radiation, viscous dissipation and chemical reaction effects, American Chemical Science J., 4 (2014), pp. 828–846.Google Scholar
[38] Hamad, M., Analytical solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field, Int. Commun. Heat Mass Transfer, 38(4) (2011), pp. 487–492.Google Scholar
[39] Kameswaran, P., Narayana, M., Sibanda, P. and Murthy, P., Hydromagnetic nanofluid flow due to a stretching or shrinking sheet with viscous dissipation and chemical reaction effects, Int. J. Heat Mass Transfer, 55 (2526) (2012), pp. 7587–7595.CrossRefGoogle Scholar
[40] Rosseland, S., Theoretical Astrophysics: Atomic Theory and the Analysis of Stellar Atmosphere and Envelopes, The International Series of Monographs on Nuclear Energy: Reactor Design Physics, At the Clarendon Press, 1936.Google Scholar
[41] Cess, R., The interaction of thermal radiation with free convection heat transfer, Int. J. Heat Mass Transfer, 9(11) (1966), pp. 1269–1277.Google Scholar
[42] Arpaci, V. S., Effect of thermal radiation on the laminar free convection from a heated vertical plate, Int. J. Heat Mass Transfer, 11(5) (1968), pp. 871–881.CrossRefGoogle Scholar
[43] Cheng, E. and Özişik, M., Radiation with free convection in an absorbing, emitting and scattering medium, Int. J. Heat Mass Transfer, 15(6) (1972), pp. 1243–1252.Google Scholar
[44] Hossain, M. and Takhar, H., Radiation effect on mixed convection along a vertical plate with uniform surface temperature, Heat Mass Transfer, 31(4) (1996), pp. 243–248.CrossRefGoogle Scholar
[45] Siddiqa, S., Hossain, M. and Saha, S. C., The effect of thermal radiation on the natural convection boundary layer flow over a wavy horizontal surface, Int. J. Thermal Sci., 84 (2014), pp. 143–150.Google Scholar
[46] Srivastava, A. C. and Hazarika, G. C., Shooting method for third order simultaneous ordinary differential equations with application to magnetohydrodynamic boundary layer, Math. Problems Eng., 40 (1990), pp. 263–273.Google Scholar