Page 85 - gas transport in porous media

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and a little thought will indicate that this result takes the form
A=N A=N Whitaker

ρ A v A v A = ρvv + ρ A v A u A (6.40)
A=1 A=1
With the aid of Eq. (6.37) one can express the last term in this expression as

A=N A=N A=N

ρ A v A u A = ρ A vu A + ρ A u A u A (6.41)
A=1 A=1 A=1
and the constraint on the mass diffusion velocities given by Eq. (6.38) leads to the
simplification

A=N A=N

ρ A v A u A = ρ A u A u A (6.42)
A=1 A=1
Use of this result in Eq. (6.40) provides
A=N A=N

ρ A v A v A = ρvv + ρ A u A u A (6.43)
A=1 A=1

and this allows us to write Eq. (6.36) as
A=N A=N
∂
(ρv) +∇ · (ρvv) +∇ · ρ A u A u A = ρb +∇ · T A (6.44)
∂t
A=1 A=1
At this point it is convenient to define a total stress tensor for multicomponent
systems as
A=N

T = T A − ρ A u A u A (6.45)
A=1
in which the terms represented by ρ A u A u A are referred to as the diffusive stresses
(Truesdell and Toupin, 1960). Use of this result in Eq. (6.44) leads to
∂
(ρv) +∇ · (ρvv) = ρb +∇ ·T (6.46)
∂t
This result is identical in form to Cauchy’s first equation for single component
systems; however, in this case each term has a greater physical significance because
of the definitions given by Eqs. (6.32), (6.34), (6.35), and (6.45). It is easy to see that
one can use Eq. (6.30) along with Eq. (6.45) to produce the symmetry condition

T = T T (6.47)