Journal of Thermal Analysis and Calorimetry

, Volume 138, Issue 5, pp 3183–3203 | Cite as

Mixed convective magnetonanofluid flow over a backward facing step and entropy generation using extended Darcy–Brinkman–Forchheimer model

  • Shafqat HussainEmail author
  • Khalid Mehmood
  • Muhammad Sagheer
  • Asifa Ashraf


This article focuses on a numerical study for the effect of porosity and internal heat generation/absorption parameters on the mixed convective flow of nanofluid over a backward facing step along with the entropy generation. The channel downstream bottom wall is isothermally heated while the remaining walls of the channel are thermally insulated. The dimensionless governing equations are discretized utilizing the Galerkin-based finite element method. The discrete systems of nonlinear algebraic equations are computed using the Newton method and the corresponding linearized systems are treated using the monolithic geometric multigrid solver. Effect of different physical parameters in the specified ranges such as \((10 \le Re \le 200)\), \((0.01 \le Ri \le 20)\), \((0\le Ha \le 100)\), \((-10 \le q \le 10)\), \((10^{-6} \le Da \le 10^{-3})\), \((0.2 \le \epsilon \le 0.8)\) and \((0^\circ \le \gamma \le 180^\circ )\) on the fluid flow is analyzed with the help of the streamlines, isotherm patterns, and various graphs. It is noticed that the average heat transfer increases with the porosity parameter and reduces with the internal heat generation parameter. Furthermore, the entropy generation produced by the heat transfer and fluid friction is found to amplify with the porosity parameter, whereas a decline is recorded in it due to the magnetic field.


Magnetohydrodynamics Mixed convection Backward facing step Porous media Newton-multigrid Galerkin finite element method 

List of symbols


Specific heat (\(\hbox {J kg}^{-1}\hbox {K}^{-1}\))


Bejan number


Gravitational acceleration (\(\hbox {m s}^{-2}\))


Hartmann number


Grashof number


Effective thermal conductivity of porous medium (\(\hbox {W m}^{-1}\hbox {K}^{-1}\))


Permeability parameter (\(\hbox {m}^2\))


Thermal conductivity (\(\hbox {W m}^{-1}\hbox {K}^{-1}\))


Step size (m)

\(Nu_{\text {avg}}\)

Average Nusselt number


Dimensionless pressure


Pressure (\(\hbox {N m}^{-2}\))


Prandtl number


Richardson number


Dimensionless heat generation/absorption parameter


Reynolds number


Temperature (K)


Dimensionless entropy generation due to heat transfer


Dimensionless entropy generation due to fluid friction


Dimensionless entropy generation due to magnetic field


Dimensional velocity components (\(\hbox {m s}^{-1}\))


Dimensional space coordinates (m)


Velocity at the inlet


Dimensionless space coordinates


Dimensionless velocity components

Greek symbols


Thermal diffusivity (\(\hbox {m}^2\hbox { s}^{-1}\))


Magnetic field inclination angle


Density (\(\hbox {kg m}^{-3}\))


Expansion coefficient (\(\hbox {K}^{-1}\))


Dynamic viscosity (\(\hbox {kg m}^{-1}\hbox {s}^{-1}\))


Electrical conductivity (\(\hbox {S m}^{-1}\))


Dimensionless temperature


Solid volume fraction


Kinematic viscosity (\(\hbox {m}^2\hbox { s}^{-1}\))


\(\text {avg}\)














Calculations have been carried out on the LiDOng cluster at Technische Universität, Dortmund, Germany. The support by the LiDOng team at the ITMC at TU Dortmund is gratefully acknowledged. We would like to thank the LiDOng cluster team for their help and support. We also used FeatFlow ( solver package and would like to acknowledge the support by the FeatFlow team.


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Department of MathematicsCapital University of Science and TechnologyIslamabadPakistan
  2. 2.Institut für Angewandte Mathematik (LS III)Technische UniversitätDortmundGermany

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