Journal of Thermal Analysis and Calorimetry

, Volume 136, Issue 4, pp 1489–1514 | Cite as

Effects of nonhomogeneous nanofluid model on convective heat transfer in partially heated square cavity with conducting solid block

  • A. I. AlsaberyEmail author
  • M. H. Yazdi
  • A. A. Altawallbeh
  • I. Hashim


In this study, the conjugate natural convection in a square cavity filled with \(\hbox {Al}_2\hbox {O}_3\)–water nanofluid with an inner conducting solid block is studied numerically using nonhomogeneous Buongiorno’s two-phase model. The left wall of the cavity is partially heated and the remaining parts of the wall are adiabatic, while the right wall is fully cooled. The top and bottom horizontal walls are adiabatic. The numerical simulations are based on the finite difference method. The results are simulated for various values of the nanoparticle volume fraction \((0\le \phi \le 0.04)\), Rayleigh number \((10^2\le Ra\le 10^6)\), thermal conductivity of the conjugate square \((k_{\mathrm{w}}=0.28, 0.76, 1.95, 7.0)\) and 16.0 (epoxy: 0.28, brickwork: 0.76, granite: 1.95, solid rock: 7, stainless steel: 16), the size of the inner solid \((0\le D\le 0.7)\), and the length of the heater (\(0.1\le H\le 1.0\)). The numerical results for the average and local Nusselt numbers, isotherms, distribution of nanoparticles, and streamlines are presented graphically. The findings indicate that increasing the average solid volume fraction and the size of the solid block as well as the thermal conductivity will enhance the rate of the heat transfer at low values of Rayleigh number \(Ra=10^3\). On the other hand, increasing these parameters at high values of Rayleigh number (\(Ra>10^5\)) decreases the average Nusselt number.


Conjugate natural convection Square cavity Brownian motion Thermophoresis effect Nanoparticle distribution Partially heating 

List of symbols


Specific heat capacity


Width and height of inner square


Diameter of the base fluid molecule


Diameter of the nanoparticle


Dimensionless length of the conductive solid block (\(D=d/L\))


Brownian diffusion coefficient


Reference Brownian diffusion coefficient


Thermophoretic diffusivity coefficient


Reference thermophoretic diffusion coefficient

\({\mathbf {g}}\)

Gravitational acceleration


Dimensionless length of the heat source (\(H=h/L\))


Thermal conductivity


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


Width and height of cavity


Lewis number


Ratio of Brownian to thermophoretic diffusivity


Average Nusselt number


Prandtl number


Rayleigh number


Brownian motion Reynolds number




Reference temperature (310 K)


Freezing point of the base fluid (273.15 K)

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

Velocity and dimensionless velocity vector


Brownian velocity of the nanoparticle

x, y and X, Y

Space coordinates and dimensionless space coordinates

Greek symbols


Thermal diffusivity


Thermal expansion coefficient


Normalized temperature parameter


Dimensionless temperature


Dynamic viscosity


Kinematic viscosity




Solid volume fraction

\(\varphi ^*\)

Normalized solid volume fraction


Average solid volume fraction

\(\psi\) and \(\varPsi\)

Stream function and dimensionless stream function

\(\omega\) and \(\varOmega\)

Vorticity and dimensionless vorticity





Base fluid






Solid nanoparticles


Solid wall



The work was supported by the Universiti Kebangsaan Malaysia (UKM) research Grant DIP-2017-010. We thank the respected reviewers for their constructive comments which clearly enhanced the quality of the manuscript.


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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 Malaysia (UKM)BangiMalaysia
  3. 3.Department of Mechanical Engineering, Neyshabur BranchIslamic Azad UniversityNeyshaburIran
  4. 4.Department of Basic Sciences and Mathematics, Faculty of SciencePhiladelphia UniversityAmmanJordan

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