# Mixed convection in a trapezoidal enclosure filled with two layers of nanofluid and porous media with a rotating circular cylinder and a sinusoidal bottom wall

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## Abstract

The laminar two-dimensional mixed convection in a trapezoidal enclosure with a rotating inner circular cylinder and a sinusoidal bottom wall is studied numerically. The fluid inside the enclosure is a CuO–water nanofluid layer in the top space of it, while the bottom space includes a CuO–water nanofluid saturated with a porous medium. Both the right and left sidewalls are assumed adiabatic, while the bottom and the top walls of the enclosure are maintained, respectively, at the hot and cold temperatures. The dimensionless governing equations are expressed for velocity and temperature formulation and modeled by using COMSOL code based on the Galerkin finite element method. Parametric studies on the effects of various significant parameters such as Rayleigh number, Darcy number, the inner cylinder radius, the porous layer thickness, the angular rotational velocity, the solid volume fraction and the number of undulations on the flow and thermal fields together with the heat transfer rate have been performed. The highest value of the stream function for (Ra = 10^{3} and Ra = 10^{5}) is seen at (*R* = 0.2 and *S* = 0.2). The same thing is observed, when the bottom wall is considered wavy. For (Ra = 10^{3} and *N* = 0) and (0.5 ≤ *S* ≤ 0.8), it can be seen that as the inner cylinder radius increases from (*R* = 0.1) to (*R* = 0.3), the stream function values increase continuously. It is found that the average Nusselt number increases as the Rayleigh and Darcy numbers, the solid volume fraction, inner cylinder radius and the angular rotational velocity of the cylinder increase, while it decreases as the porous layer thickness and the number of undulations increase. Comparisons with previously published numerical works are performed, and good agreements between the results are observed.

## Keywords

Mixed convection Nanofluid Trapezoidal enclosure Porous media Rotating cylinder Sinusoidal wall## List of symbols

- \(c_{\text{p}}\)
Heat capacitance (J kg

^{−1}°C^{−1})- Da
Darcy number

*g*Gravitational acceleration (m s

^{−2})- \(k_{\text{f}}\)
Thermal conductivity of the fluid (W m

^{−1}°C^{−1})*L*Height and width of the trapezoidal enclosure (m)

- Nu
Nusselt number

*N*Number of undulations

*P*Dimensionless pressure

*p*Pressure (N m

^{−2})- Pr
Prandtl number

*R*Dimensionless radius of the inner cylinder

*r*Radius of the inner cylinder (m)

- Ra
Rayleigh number

- Re
Reynolds number

- Ri
Richardson number

*S*Porous layer thickness (m)

*T*Temperature (°C)

*U*Dimensionless velocity component in

*x*-direction*u*Dimensional velocity component in

*x*-direction (m s^{−1})*V*Dimensionless velocity component in

*y*-direction*v*Dimensional velocity component in

*y*-direction (m s^{−1})*w*Width of the trapezoidal enclosure (m)

*X*Dimensionless coordinate in horizontal direction

*x*Cartesian coordinate in horizontal direction (m)

*Y*Dimensionless coordinate in vertical direction

*y*Cartesian coordinate in vertical direction (m)

## Greek symbols

*α*Thermal diffusivity (m

^{2}s^{−1})*β*Volumetric thermal expansion coefficient (K

^{−1})*θ*Dimensionless temperature distribution

*φ*Solid volume fraction

*ε*Porosity

- \(\zeta\)
Amplitude of the wavy wall

*Ω*Dimensionless angular rotational velocity

- \(\omega\)
Angular rotational velocity (rad s

^{−1})- \(\nu\)
Kinematic viscosity (m

^{2}s^{−1})*ρ*Density (kg m

^{−3})- \(\mu\)
Dynamic viscosity (kg m

^{−1}s^{−1})*Ψ*Dimensionless stream function

*λ*Permeability of the porous medium (m

^{2})

## Subscripts

- ave
Average

- c
Cold

- eff
Effective

- fl
Base fluid

- h
Hot

- loc
Local

- Max
Maximum

- Min
Minimum

- na
Nanofluid particle

- o
Center of enclosure

- po
Porous medium

- so
Solid particles

## Notes

### Acknowledgements

The fourth author would like to acknowledge the research deanship of University of Ha’il, KSA for funding the project “191225”.

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