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

ABSTRACT

MHD 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:

• magnetohydrodynamics
• mixed convection
• developing flow
• porous media
• microchannel
• slip flow

Disclosure statement

No potential conflict of interest was reported by the author(s).

Nomenclature

$Bi$	=	Biot number
$B_0$	=	magnetic field strength (T)
$c_f$	=	coefficient in the Forchheimer term
$Da$	=	Darcy number
$Gr$	=	Grashof number
$h$	=	local heat transfer coefficient (W/m2.K)
$h_{sf}$	=	fluid to solid heat transfer coefficient (W/m2.K)
$H$	=	channel width (m)
$Ha$	=	Hartmann number
$k$	=	thermal conductivity (W/m.K)
$K$	=	permeability of the porous medium (m2)
$Kn$	=	Knudsen number
$Kr$	=	conductivity ratio
$L$	=	channel length (m)
$L_h$	=	non – dimensional value of the hydrodynamic entry length
$L_t$	=	non – dimensional value of the thermal entry length
$Nu$	=	local Nusselt number
$p$	=	pressure (Pa)
$Pr$	=	Prandtl number
$Re$	=	Reynolds number
$Ri$	=	Richardson number
$r_T$	=	defined in Equation 23(23) ${\theta }_{f,m}=\frac{{\int }_0^{1}U{\theta }_{f}dY}{{\int }_0^{1}UdY}$(23)
$T$	=	temperature (K)
$u$	=	vertical component of the gas velocity (m/s)
$u_0$	=	reference velocity (m/s)
$U$	=	dimensionless value of the vertical velocity
$v$	=	horizontal component of the gas velocity (m/s)
$V$	=	dimensionless value of the horizontal velocity
$x$	=	vertical coordinate (m)
$X$	=	dimensionless vertical coordinate
$y$	=	horizontal coordinate (m)
$Y$	=	dimensionless horizontal coordinate
Greek symbols	=
$\alpha$	=	thermal diffusivity (m2/s)
$\gamma$	=	specific heat ratio
$\mathrm{\Gamma }$	=	Forchheimer number
$\lambda$	=	mean – free–path (m)
$\epsilon$	=	medium porosity
$\mu$	=	dynamic viscosity (kg/m.s)
$\rho$	=	density (kg/m3)
$\sigma$	=	electrical conductivity (1/Ω.m)
${\sigma }_T$	=	thermal – accommodation coefficient
${\sigma }_V$	=	tangential – momentum–accommodation coefficient
$\theta$	=	dimensionless temperature
Subscripts	=
$f$	=	fluid
$fd$	=	fully – developed
$m$	=	mean value
$s$	=	solid matrix
$w$	=	wall
$r$	=	right
$l$	=	left

Abbreviations

LTE	=	local thermal equilibrium
MHD	=	Magnetohydrodynamics
LTNE	=	local thermal non – equilibrium