# Natural convection of a non-Newtonian ferrofluid in a porous elliptical enclosure in the presence of a non-uniform magnetic field

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## Abstract

In the present study, laminar natural convection of a non-Newtonian ferrofluid inside an elliptical porous cavity was numerically simulated in the presence of a non-uniform external magnetic field. This natural convection problem was relevant to the cooling of micro-sized electronic devices. The well-known finite volume method was employed to discretize the governing equations for ferrofluid flow under the effect of an external magnetic field. The effects of pertinent non-dimensional numbers including the Rayleigh number, the magnetic number, the power-law index, and the Darcy number were studied on the flow pattern and the heat transfer rate of the non-Newtonian ferrofluid. The results showed that by applying the magnetic field by a wire, the overall heat transfer rate increased significantly. Moreover, to achieve the maximum heat transfer enhancement, the wire should have been placed at the center of the elliptical walls of the enclosure. It was also shown that the impact of the power-law index on the heat transfer rate was considerable, and using a shear-thinning liquid increased the average Nusselt number in the porous elliptical enclosure.

## Keywords

Ferrofluid Porous media Non-Newtonian fluid Magnetic field Natural convection## List of symbols

*a*Large inner ellipse radius

*b*Small inner ellipse radius

- \(\vec{B}\)
Magnetic induction

*C*Consistency index (Ns

^{n}m^{−2})*C*_{P}Specific heat capacity (J kg

^{−1}K^{−1)}*C*_{d}Inertia coefficient of porous media

*d*Outer ellipse radius

- Da
Darcy number

*D*_{ij}Rate of deformation tensor

*g*Gravitational acceleration (ms

^{−2})- \(\vec{H}\)
Magnetic field vector (A m

^{−1})*I*Electrical intensity (A)

*L*Reference length (m)

*m*Consistency index

- Mn
Magnetic non-dimensional number

*n*Power-law index

- Nu
Nusselt number

*P*Pressure (Pa)

- Pr
Prandtl number

- Ra
Rayleigh number

- \(\vec{u},\,\vec{v}\)
Velocity vector components (m s

^{−1})*x*,*y*Cartesian coordinates (m)

## Greek symbols

*α*Thermal diffusivity (m

^{2}s^{−1})*θ*Non-dimensional temperature

*ν*Kinematic viscosity (m

^{2}s^{−1})*μ*Dynamic viscosity (kg m

^{−1}s^{−1})*μ*_{0}Magnetic permeability in a vacuum (= 4

*π*× 10^{−7}T m A^{−1})*χ*Magnetic susceptibility

*β*Thermal expansion coefficient (1 K

^{−1})*ρ*Density (kg m

^{−3})*φ*Solid volume fraction

*κ*Permeability of porous medium (m

^{2})*τ*Shear stress (Pa)

*ε*Porosity

*λ*Thermal conductivity (W m

^{−1}K^{−1})

## Subscript

- avg
Average

- c
Cold

- eff
Effective (porous media)

- f
Base fluid

- h
Hot

- nf
Mixture (nanofluid)

- p
Particle

- w
Wall

## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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