# MHD Convection with Heat Generation in a Porous Cavity

## Abstract

The present study deals with the magnetic-field-affected heat generation–absorption undergoing natural convection in a differentially heated cavity packed with porous media. A two-dimensional porous cavity with adiabatic top and bottom is investigated numerically considering its left wall heated isothermally and right wall maintained at ambient temperature. The solution of the governing equations and subsequent post-processing is conducted using finite volume-based in-house CFD code. The flow through the porous medium has been modeled using Brinkman–Forchheimer–Darcy model (BFDM). The results obtained from the wide range of parameters are examined graphically using streamlines, isotherms, and average Nusselt number (*Nu*) plots and discussed to know the effects of different flow parameters like modified Rayleigh number (*Ra*_{m} = 1–1000), Darcy number (*Da* = 10^{−3} − 10^{−6}), porosity (*ε *= 0.1 − 1.0), Hartmann number (*Ha* = 10 − 100) along with its inclination angle (*γ *= 0 − 180°), in the presence of heat generation and absorption. It is found that as the magnetic field strength increases, heat transfer rate decreases substantially, and it is further affected by heat generation–absorption parameter.

## Keywords

Magneto-hydrodynamics Porous cavity Heat generation–absorption Thermal convection## Nomenclature

*B*Uniform magnetic field (tesla)

*Da*Darcy number

*G*Ratio of heat generation–absorption

*H*Height of the cavity/length scale, m

*Ha*Hartmann number

- \(K\)
Permeability of porous medium, m

^{2}- \(L\)
Length of the cavity, m

- \(Nu\)
Average Nusselt number

- \(p\)
Pressure, Pa

- \(P\)
Dimensionless pressure

- \(Pr\)
Prandtl number

- \(Ra\)
External Rayleigh number

- \(Ra_{\text{I}}\)
Internal Rayleigh number

- \(Ra_{\text{m}}\)
Modified Darcy–Rayleigh number

- \(T\)
Temperature, K

- \(g\)
Velocity components, m/s

- \(U,V\)
Dimensionless velocity components

- \(x,y\)
Cartesian coordinates, m

- \(X,Y\)
Dimensionless coordinates

## Greek Symbols

- \(\alpha\)
Thermal diffusivity, m

^{2}/s- \(\beta\)
Thermal expansion coefficient, K

^{−1}- \(\gamma\)
Inclination angle of the magnetic field

- \(\theta\)
Dimensionless temperature

*ε*Porosity

- \(\upsilon\)
Kinematic viscosity, m

^{2}/s- \(\rho\)
Density, kg/m

^{3}- \(\kappa\)
Electrical conductivity (μS cm

^{−1})- \(\psi\)
Dimensionless stream function

## Subscripts

- \({\text{c}},{\text{h}}\)
Cooling, heating

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