Natural convection of a non-Newtonian ferrofluid in a porous elliptical enclosure in the presence of a non-uniform magne
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Natural convection of a non‑Newtonian ferrofluid in a porous elliptical enclosure in the presence of a non‑uniform magnetic field M. R. Daneshvar Garmroodi1 · A. Ahmadpour1 · M. R. Hajmohammadi1 · S. Gholamrezaie1 Received: 18 September 2019 / Accepted: 9 November 2019 © Akadémiai Kiadó, Budapest, Hungary 2019
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 ⃗ Magnetic induction B C Consistency index (Nsn m−2) CP Specific heat capacity (J kg−1 K−1) Cd Inertia coefficient of porous media d Outer ellipse radius Da Darcy number Dij Rate of deformation tensor g Gravitational acceleration (ms−2) ⃗ Magnetic field vector (A m−1) H 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) * A. Ahmadpour [email protected] 1
Department of Mechanical Engineering, Amirkabir University of Technology, P.O. Box 15878‑4413, Vali‑Asr Sq., Hafez Ave., Tehran, Iran
Pr Prandtl number Ra Rayleigh number u⃗ , v⃗ Velocity vector components (m s−1) x, y Cartesian coordinates (m) Greek symbols α Thermal diffusivity (m2 s−1) θ Non-dimensional temperature ν Kinematic viscosity (m2 s−1) μ Dynamic viscosity (kg m−1 s−1) μ0 Magnetic permeability in a vacuum (= 4π × 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 (m2) 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
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nf Mixture (nanofluid) p Particle w Wall
Introduction Natural convection heat transfer has been continuously investigated because o
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