Journal of Thermal Analysis and Calorimetry

, Volume 132, Issue 2, pp 1291–1306 | Cite as

Effects of partial slip on entropy generation and MHD combined convection in a lid-driven porous enclosure saturated with a Cu–water nanofluid

  • A. J. Chamkha
  • A. M. Rashad
  • T. Armaghani
  • M. A. Mansour


In this work, the influences of heat generation/absorption and nanofluid volume fraction on the entropy generation and MHD combined convection heat transfer in a porous enclosure filled with a Cu–water nanofluid are studied numerically with of partial slip effect. The finite volume technique is utilized to solve the dimensionless equations governing the problem. A comparison with already published studies is conducted, and the data are found to be in an excellent agreement. The minimization of entropy generation and the local heat transfer according to various values of the controlling parameters are reported in detail. The outcome indicates that an augmentation in the heat generation/absorption parameter decreases the Nusselt number. Also, when the volume fraction is raised, the Nusselt number and entropy generation are reduced. The impact of Hartmann number on heat transfer and the Richardson number on the entropy generation and the thermal rendering criteria are also presented and discussed.


Heat generation/absorption Entropy generation Nanofluid Partial slip Nusselt number 

List of symbols


Dimensionless of heat source/sink length

\( B_{0} \)

Magnetic field strength (T)

\( Be \)

Bejan number

\( b \)

Length of heat source (m)

\( C_{\text{p}} \)

Specific heat at constant pressure \( ( {\text{J}}\;{\text{kg}}\;{\text{K}}^{ - 1} ) \)


Dimensionless heat source position


Darcy number


Location of heat sink and source (m)


Length of cavity (m)


Hartmann number, \( B_{0} L\sqrt {\sigma_{\text{f}} /\rho_{\text{f}} \nu_{\text{f}} } \)


Grashof number, \( g\beta_{\text{f}} H^{3} \Delta T/\upsilon^{2}_{\text{f}} \)


Acceleration due to gravity (m s−2)


Permeability of porous medium (m2)


Thermal conductivity (W m−1 K−1)


Local Nusselt number

\( Nu_{\text{m}} \)

Average Nusselt number of heat source

\( p \)

Fluid pressure (Pa)

\( P \)

Dimensionless pressure, \( pH/\rho_{\text{nf}} \alpha_{\text{f}}^{2} \)


Prandtl number, \( \upsilon_{\text{f}} /\alpha_{\text{f}} \)


Reynolds number, \( V_{0} H/\upsilon_{\text{f}} \)


Entropy generation (W K−1 m−3)


Temperature (K)

\( T_{\text{c}} \)

Cold wall temperature (K)

\( T_{\text{h}} \)

Heated wall temperature (K)


Velocity components in x, y directions (m s−1)

\( U,V \)

Dimensionless velocity components, u/V 0, v/V 0

\( x,y \)

Cartesian coordinates (m)

\( X,Y \)

Dimensionless coordinates, x/L, y/L

Greek symbols

\( \alpha \)

Thermal diffusivity, \( {\text{m}}^{2} \;{\text{s}}^{ - 1} ,{\text{k}}/\rho c_{\text{p}} \)

\( \beta \)

Thermal expansion coefficient, K−1

\( \phi \)

Solid volume fraction


Effective electrical conductivity \( (\upmu {\text{S}}\;{\text{cm}}^{ - 1} ) \)

\( \theta \)

Dimensionless temperature, \( {{(T - T_{\text{c}} )} \mathord{\left/ {\vphantom {{(T - T_{\text{c}} )} {(T_{\text{h}} - T_{\text{c}} }}} \right. \kern-0pt} {(T_{\text{h}} - T_{\text{c}} }}) \)

\( \mu \)

Dynamic viscosity (N s m−2)

\( \nu \)

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

\( \rho \)

Density (kg m−3)







Pure fluid










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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2017

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentPrince Sultan Endowment for Energy and Environment Prince Mohammad Bin Fahd UniversityAl-KhobarKingdom of Saudi Arabia
  2. 2.Rak Research and Innovation CenterAmerican University of Ras Al KhaimahRas Al KhaimahUnited Arab Emirates
  3. 3.Department of Mathematics, Faculty of ScienceAswan UniversityAswânEgypt
  4. 4.Department of Engineering, Mahdishahr BranchIslamic Azad UniversityMahdishahrIran
  5. 5.Department of Mathematics, Faculty of ScienceAssuit UniversityAssuitEgypt

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