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Application of nanofluid and optimization of pore size arrangement of heterogeneous porous media to enhance mixed convection inside a two-sided lid-driven cavity

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Mixed convection of Cu–water nanofluid inside a two-sided lid-driven cavity filled with heterogeneous porous media is optimized. The horizontal walls are adiabatic and movable, and the vertical walls are exposed to constant hot and cold temperatures. Two-phase mixture model and Darcy–Brinkman–Forchheimer relation are implemented, respectively, for simulation of nanofluid and fluid flow through porous media. Pores size diameters of the porous medium in different regions are considered as decision variables for optimization process. In this regard, the cavity is divided into 25 parts, and the pore size of each part is found through the pattern search optimization algorithm. The optimization is performed in order to maximize Nuavg of the flow for various Rayleigh (Ra = 103–106) and Richardson (Ri = 0.01, 0.1, 1, 10 and 100) numbers. Gaining the optimized heterogeneous structure of the porous medium in which Nuavg is greater than that of the homogeneous medium with the highest Nusselt (\({\text{Nu}}_{{{\text{dp}}_{ \hbox{max} } }}\)) is the main goal of optimization. Results indicate that for more convection dominated flows (lower Ri and higher Ra numbers), the optimized heterogeneous porous medium could enhance heat transfer up to 8.3%. But the optimal porous medium for natural convection dominated flows (high Ri and low Ra values) is the homogeneous porous case with maximum pore size diameter. Furthermore, drag force on the driven lid increased up to 0.34% for the optimal cases which is very low and can be disregarded.

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C p :

Specific heat (J kg−1 K−1)

C d :

Drag coefficient


Darcy number

d p :

Pore size diameter of porous zone (m)

d np :

Nanoparticles diameter (m)

f drag :

Drag function


Grashof number

g :

Gravity acceleration (m s−2)

h :

Heat transfer coefficient (W m−2 K−1)

k b :

Boltzmann constant (J K−1)


Nusselt number


Prandtl number

p :

Pressure (Pa)


Heat flux (W m−2)


Rayleigh number


Richarson number

T :

Temperature (K)

\(\vec{V}\) :

Velocity vector (m s−1)

\(\alpha\) :

Thermal diffusivity (m2 s−1)

\(\beta\) :

Thermal expansion coefficient (K−1)

ε :


\(\varphi\) :

Volume fraction

\(\kappa\) :

Permeability (m2)

\(\mu\) :

Dynamic viscosity (kg m−1 s−1)

\(\nu\) :

Kinematic viscosity (m2 s−1)

\(\rho\) :

Density (kg m−3)




Cold wall






Hot wall


Mixture (nanofluid)




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Correspondence to Majid Siavashi.

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Maghsoudi, P., Siavashi, M. Application of nanofluid and optimization of pore size arrangement of heterogeneous porous media to enhance mixed convection inside a two-sided lid-driven cavity. J Therm Anal Calorim 135, 947–961 (2019).

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