MHD Mixed Convection of Developing Slip Flow in a Vertical Porous Microchannel Under Local Thermal Non–Equilibrium Conditions

ABSTRACTMHD mixed convection heat transfer of an ionized gas in a vertical microchannel filled with a porous medium is simulated and discussed in this study. The considered flow is hydrodynamically and thermally developing with Local Thermal Non – Equilibrium (LTNE) between the gas and the solid matrix. The Darcy – Brinkman – Forchheimer model is utilized to describe the flow filed in the porous medium. Moreover, both velocity – slip and temperature – jump boundary conditions are applied to the gas at the walls. The governing equations are solved by the finite – volume method. Results are presented and discussed in terms of the developed profiles of velocity and temperature of the constituents as well as the variations of the Nusselt number through the microchannel, the numerical values of the hydrodynamic and thermal entry lengths, and the fully – developed Nusselt number for different conditions. It is found that direct relations exist between the fully – developed Nusselt number and the Richardson number, the Reynolds number, the Hartmann number, the Biot number, the thermal conductivity ratio, and the Forchheimer number. With rise in the Knudsen number or the Darcy number, however, the Nusselt number deteriorates. The results indicate that the Knudsen number, the Hartmann number, the Biot number, and the thermal conductivity ratio are the most influential parameters on the fully – developed Nusselt number. It is envisaged that a tenfold increase in the Hartmann number and a hundredfold elevation in the Knudsen number are accompanied by 14% rise and 42% reduction in the fully – developed Nusselt number, respectively.KEYWORDS: magnetohydrodynamicsmixed convectiondeveloping flowporous mediamicrochannelslip flow Disclosure statementNo potential conflict of interest was reported by the author(s).Nomenclature Bi=Biot numberB0=magnetic field strength (T)cf=coefficient in the Forchheimer termDa=Darcy numberGr=Grashof numberh=local heat transfer coefficient (W/m2.K)hsf=fluid to solid heat transfer coefficient (W/m2.K)H=channel width (m)Ha=Hartmann numberk=thermal conductivity (W/m.K)K=permeability of the porous medium (m2)Kn=Knudsen numberKr=conductivity ratioL=channel length (m)Lh=non – dimensional value of the hydrodynamic entry lengthLt=non – dimensional value of the thermal entry lengthNu=local Nusselt numberp=pressure (Pa)Pr=Prandtl numberRe=Reynolds numberRi=Richardson numberrT=defined in EquationEquation 23(23) θf,m=∫01UθfdY∫01UdY(23) T=temperature (K)u=vertical component of the gas velocity (m/s)u0=reference velocity (m/s)U=dimensionless value of the vertical velocityv=horizontal component of the gas velocity (m/s)V=dimensionless value of the horizontal velocityx=vertical coordinate (m)X=dimensionless vertical coordinatey=horizontal coordinate (m)Y=dimensionless horizontal coordinateGreek symbols=α=thermal diffusivity (m2/s)γ=specific heat ratioΓ=Forchheimer numberλ=mean – free–path (m)ε=medium porosityμ=dynamic viscosity (kg/m.s)ρ=density (kg/m3)σ=electrical conductivity (1/Ω.m)σT=thermal – accommodation coefficientσV=tangential – momentum–accommodation coefficientθ=dimensionless temperatureSubscripts=f=fluidfd=fully – developedm=mean values=solid matrixw=wallr=rightl=leftAbbreviations LTE=local thermal equilibriumMHD=MagnetohydrodynamicsLTNE=local thermal non – equilibrium