Magnetohydrodynamic flow and heat transfer of ferrofluid in a channel with non-symmetric cavities

  • Shafqat HussainEmail author
  • Hakan F. Öztop
  • Muhammad Amer Qureshi
  • Nidal Abu-Hamdeh


This paper explores the heat transfer characteristics and fluid flow of ferrofluid in a channel having non-symmetric cavities under the applied magnetic field. Bottom surface of the cavity is uniformly heated, whereas ceiling of the top cavity is cooled isothermally. The dimensionless governing equations for various physical parameters are computed via a higher-order and stable Galerkin-based finite element technique. Effective governing parameters are nanoparticle volume fraction; \((0 \le \phi \le 0.15\)), aspect ratio of the cavities; (\(0.2\le h/H \le 1.0\) ), Richardson number; (\(0.01\le \mathrm{Ri} \le 10\)), Hartmann number; (\(0 \le \mathrm{Ha} \le 100\) ); and Reynolds number; (\(1\le \mathrm{Re} \le 200\)). It is found that the most important parameter is the geometry such that there is an optimal value to maximize the heat transfer. Moreover, it is also noticed that the heat transfer is reduced with strong magnetic field, namely Hartmann number.


Mixed convection Ferrofluid Magnetohydrodynamics Galerkin finite element method Channel with non-symmetric cavities 

List of symbols


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


Channel height (m)


Cavity height (m)


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


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


Nusselt number (local)


Temperature (K)


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


Dimensionless pressure


Hartmann number \(B_{0}H\sqrt{\dfrac{\sigma _\mathrm{f}}{\mu _\mathrm{f}}}\),


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


Prandtl number \(\nu _\mathrm{f}/\alpha _\mathrm{f}\)


Reynolds number \({\bar{u}} H/\nu _\mathrm{f}\)


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


Velocity components (dimensionless)


Dimensionless space coordinates

\(\mathrm{Nu}_{\text {avg}}\)

Average Nusselt number


Dimensional space coordinates (m)

\(\text {div}\)

Divergence operator

\(\mathbf e\)

Unit vector (0,1)

Greek symbols


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


Volume fraction of the nanoparticles


Thermal expansion coefficient (\(\hbox {K}^{-1}\))


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


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


Dimensionless temperature


\(\text {avg}\)














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. Second and last authors extend their appreciation to the International Scientific Partnership Program (ISPP) at King Saud University for funding this research work through ISPP#131.


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Department of MathematicsCapital University of Science & TechnologyIslamabadPakistan
  2. 2.Institut für Angewandte Mathematik (LS III)Technische UniversitätDortmundGermany
  3. 3.Department of Mechanical EngineeringFirat UniversityElazigTurkey
  4. 4.PYP-Math, College of General StudiesKFUPMDhahranSaudi Arabia
  5. 5.Department of Mechanical EngineeringKAUJeddahSaudi Arabia

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