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β μ β
2
0 = −∇ p β + ρ β g − v ~
β + μ β ∇ v β
( 2-20)
K
β
the third term on the RHS is a shear stress term such as would be required by no-slip
condition. The coefficient μ β
~ is an effective viscosity, which in general is not equal to the
fluid viscosity, μ β , as discussed by Nield and Bejan (1999) [70]. For many situations, the
use of the boundary shear term is not necessary. Without discussing the validity of
Brinkman’s equation near a wall or in areas of rapid porosity variations, the effect is only
significant in a region close to the boundary whose thickness is of order of the square root
0.
5
of the gas permeability, K β , (assuming ~ μ β = μβ
), so for most applications and also in
this study the effect can be ignored.
The Brinkman equation is also often employed at the interface between a porous medium
and a free fluid (fluid with no porous medium), in order to obtain continuity of shear stress
(more detail in [70] and [47])
2.4.2 No-linear case
At low pore velocities, Darcy’s law works quite well. However, as the pore velocities
increase, the inertial effect becomes very important, the flow resistance becomes nonlinear,
and the Forchheimer equation is more appropriate as
β μ
0 = −∇ p + ρ − v v
( 2-21)
β
β
β g − v β ρ β β F
K β
β
The third term on the RHS is a nonlinear flow resistance term. According to Nield and
Bejan (1999), the above equation is based on the work of Dupuit (1863) and Forchheimer
(1901) as modified by Ward (1964). β F is a factor to be experimentally deduced.
Whitaker, (1996) derived Darcy's law with the Forchheimer correction for homogeneous
porous media using the method of volume averaging. Beginning with the Navier-Stokes
equations, they found that the volume averaged momentum equation to be given by
β ( ∇ β − ρ β g)
F v β
K
= p ( 2-22)
β
v β − . − .
μβ
32
β