Journal of Thermal Analysis and Calorimetry

, Volume 135, Issue 1, pp 729–750 | Cite as

Effects of two-phase nanofluid model on MHD mixed convection in a lid-driven cavity in the presence of conductive inner block and corner heater

  • A. I. AlsaberyEmail author
  • M. A. Ismael
  • A. J. Chamkha
  • I. Hashim


This paper investigates a steady mixed convection in a lid-driven square cavity subjected to an inclined magnetic field and heated by corner heater with an inserted square solid block. Water–Al\(_2\)O\(_3\) nanofluid fills the cavity based on Buongiorno’s two-phase model. A corner heater is configured in the left lower corner of the cavity by maintaining 40% of the bottom and vertical walls at constant hot temperature. The top horizontal wall is moving and maintained at a constant low temperature. The remainder walls are thermally insulated. The governing equations are solved numerically using the finite element method. The governing parameters are the nanoparticles volume fraction (\(0 \le \phi \le 0.04\)), Reynolds number (\(1 \le Re \le 500\)), Richardson number (\(0.01 \le Ri \le 100\)), Hartmann number (\(0 \le Ha \le 50\)) and the size of the inner solid (\(0.1 \le D \le 0.7\)). The other parameters: the Prandtl number, Lewis number, Schmidt number, ratio of Brownian to thermophoretic diffusivity and the normalized temperature parameter, are fixed at \(Pr=4.623\), \(Le=3.5\times 10^{5}\), \(Sc=3.55\times 10^{4}\), \(N_{\mathrm{BT}}=1.1\) and \(\delta =155\), respectively. The inclination of the magnetic field is fixed at \(\gamma =\frac{\pi }{4}\). Results show that at low Reynolds number, the increase in nanoparticles loading more the 2% becomes useless. It is also found that a big size of the solid block can augment heat transfer in the case of high values of both the Reynolds and Richardson numbers.


Lid-driven cavity Magnetic field Thermophoresis Brownian Corner heater Buongiorno’s model 


\(\overrightarrow{\mathbf{B }}\)

Applied magnetic field

\({\mathbf B} \)

Magnitude of magnetic field


Specific heat capacity


Side length of inner block


Diameter of the base fluid molecule


Diameter of the nanoparticle


Dimensionless side length of the inner block, \(D=d/L\)


Brownian diffusion coefficient


Reference Brownian diffusion coefficient


Thermophoretic diffusivity coefficient


Reference thermophoretic diffusion coefficient

\({\mathbf {g}}\)

Gravitational acceleration


Hartmann number


Grashof number


Thermal conductivity


Square wall to nanofluid thermal conductivity ratio, \(K_{\mathrm{r}}=k_{\mathrm{w}}/k_{\mathrm{nf}}\)


Width and height of enclosure


Lewis number


Ratio of Brownian to thermophoretic diffusivity


Average Nusselt number


Prandtl number


Reynolds number


Brownian motion Reynolds number


Richardson number, \(Ri=Gr/{Re}^2\)


Schmidt number




Reference temperature (310 K)


Freezing point of the base fluid (273.15 K)

\({\mathbf {v}} \), \({\mathbf {V}} \)

Velocity and dimensionless velocity vector, respectively


Brownian velocity of the nanoparticle

x, y and X, Y

Space coordinates and dimensionless space coordinates

Greek symbols

\(\alpha \)

Thermal diffusivity

\(\gamma \)

Inclination angle of magnetic field

\(\beta \)

Thermal expansion coefficient

\(\delta \)

Normalized temperature parameter

\(\theta \)

Dimensionless temperature

\(\mu \)

Dynamic viscosity

\(\nu \)

Kinematic viscosity

\(\rho \)


\(\sigma \)

Electrical conductivity

\(\varphi \)

Solid volume fraction

\(\varphi ^*\)

Normalized solid volume fraction

\(\phi \)

Average solid volume fraction



Bottom wall




Base fluid






Solid nanoparticles


Top wall


Solid wall



The work was supported by the Universiti Kebangsaan Malaysia (UKM) research Grant DIP-2017-010.


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Refrigeration & Air-conditioning Technical Engineering Department, College of Technical EngineeringThe Islamic UniversityNajafIraq
  2. 2.School of Mathematical Sciences, Faculty of Science & TechnologyUniversiti Kebangsaan MalaysiaBangiMalaysia
  3. 3.Mechanical Engineering Department, Engineering CollegeUniversity of BasrahBasrahIraq
  4. 4.Department of Mechanical Engineering, Prince Sultan Endowment for Energy and the EnvironmentPrince Mohammad Bin Fahd UniversityAl-KhobarSaudi Arabia
  5. 5.RAK Research and Innovation CenterAmerican University of Ras Al KhaimahRas Al KhaimahUnited Arab Emirates

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