Modeling Material Microstructure - Continuum Mechanics of Anisotropic Materials - page 187

more general, more rigorous, and more elegant proofs of this result (Ene and
Sanchez-Palencia 1975 ; Sanchez-Palencia 1980 ; Burridge and Keller 1981 ) but
the one presented below suffices to make the point.
Consider the bar with axially aligned cylindrical cavities illustrated in Fig. 7.3 as
a porous medium, the pores being the axially aligned cylindrical cavities. Each pore
is identical and can be treated as a pipe for the purpose of determining the fluid flow
through it. In the case of pipe flow under a steady pressure gradient

x 3 , the
velocity distribution predicated by the Navier-Stokes equations is a parabolic
profile (compare Example 6.4.3),

@ p =@

r o
4

;

r 2
r o

v 3 ¼ @

p

1 (7.28)

@ x 3 m

where is the viscosity, r o is the radius of the pipe and r and x 3 are two of the three

cylindrical coordinates. The volume flow rate through the pipe is given by

m

p

r o

ð

¼ @

p r o

8

Q ¼ 2 p rv 3 dr

m ;

(7.29)

@ x 3

0

which, multiplied by the total number of axially aligned cylindrical cavities per unit
area, n c , gives the average volume flow rate per unit area along the bar,

þ
@V ð

n c p

r o
8

1
V RVE

¼ @

p

h q i¼ n q Þ x d s

m :

(7.30)

@ x 3

x 3 is a constant in the bar, hence the average of the
pressure gradient over the bar is given by the constant value;

The pressure gradient @ p =@

þ
V r

1
V RVE

¼ @

p

hr p i¼ p d v

x 3 ;

(7.31)

@

a result that, combined with ( 7.28 ), yields

r o
8

n c p

h q i¼

m hr

p i: (7.32)

A comparison of this representation for h q i with that of ( 7.19 ) yields the

representation for the hydraulic permeability H eff

33 in the x 3 direction,

r o

n c p H eff

33

¼

8 m :

(7.33)