# MHD natural convection from two heating modes in fined triangular enclosures filled with porous media using nanofluids

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## Abstract

In this paper, numerical investigations for magnetohydrodynamic natural convection from two heating systems inside fined triangular enclosures filled with an isotropic porous medium using the nanofluids are performed. The two heating modes are represented by two cases, namely, case 1 a triangular enclosure with a heated part at the left wall and including a cold fin at the bottom wall and case 2 in which a cold part at the left wall and a heated fin located at the bottom wall. The copper is considered as nanoparticles and the Darcy model is applied to the porous medium. The triangular physical model is transformed to a rectangular computational model using suitable grid transformations and then the finite-volume method is applied to solve the resulting system. The key parameters in this study are the height, width and locations of the fin, different lengths and locations of the active part, nanoparticles volume fraction, heat generation/absorption parameter, and the Hartmann number. The results revealed that the increase in height of the fins decays the nanofluid flow in case 1, but in case 2, it accelerates the fluid motion. In addition, the increase in width and height of the fin enhances the rate of the heat transfer regardless the heating mode.

## Keywords

MHD Active part Heated/cold fin Nanofluids Heating mode## List of symbols

*b*Heat source/sink length (m)

*B*Dimensionless heat source/sink length

*B*_{0}External magnetic field (Tesla)

*C*_{p}Specific heat at constant pressure (kg m

^{2}K s^{−2})*d*Heat source/sink position (m)

*D*Dimensionless heat source/sink position

*Da*Darcy number

*d*1Location of the fin

*D*1Dimensionless location of the fin

*g*Gravitational acceleration (m s

^{−2})*h*Height of fin

*H*Dimensionless height of fin

*L*Length of the bottom and height wall of triangle (m)

*Ha*Hartmann number

*k*Thermal conductivity (W mK

^{−1})*K*Permeability of the porous medium (m

^{2})*Nu*_{s}Local Nusselt number

*Nu*_{m}Average Nusselt number

*P*Pressure (N/m

^{2})*Pr*Prandtl number \(\Pr = \nu_{\text{f}} /\alpha_{\text{f}}\)

*Q*Dimensionless heat generation/absorption parameter

*Q*_{0}Dimensional heat generation/absorption parameter

*Ra*Rayleigh number \({\text{Ra}} = g\,\beta \,\Delta T\,H^{3} /\nu_{\text{f}} \alpha_{\text{f}}\)

*T*Temperature (K)

*t*Dimensional temperature

*U*Dimensionless velocity component along

*x*-direction*V*Dimensionless velocity component along

*y*-direction*x, y*Cartesian coordinates (m)

*w*Width of fin

*W*Dimensionless fin width

*X, Y*Dimensionless Cartesian coordinates

## Greek symbols

- \(\alpha\)
Thermal diffusivity (m

^{2}s^{−1})- \(\beta\)
Thermal expansion coefficient (K

^{−1})- \(\phi\)
Nanoparticles volume fraction

- \(\mu\)
Dynamic viscosity (Pa s)

- \(\omega\)
Dimensionless vorticity

*ε*Porosity

*θ*Dimensionless temperature

*ρ*Density (kg m

^{−3})*σ*Heat capacity ratio

*σ*_{f}Electrical conductivity of fluid (S m

^{−1})*σ*_{nf}Electrical conductivity of nanofluid (S m

^{−1})*τ*Dimensionless time parameter

*ψ*Stream function (m

^{2}s^{−1})- Ψ
Dimensionless stream function

## Subscripts

*C*Cold

*f*Fluid

*h*Hot

*m*Average

- nf
Nanofluid

## Notes

### Acknowledgements

The authors would like to extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through General Research Project under Grant Number (R.G.P1./91/40).

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