Page 203 - Percolation Models for Transport in Porous Media With

12.1 DISSIPATION. POISEUILLE FLOW 201

ultrasound, i.e., the oscillations with frequency no less than v0 EO!! (1.5 + 2.0) · 10 4
Hz, is going to be used, the obtained estimate does not allow to decide whether
the considered mechanism is or is not going to work at this frequency.
Therefore we will analyze the possibility of its realization more closely. The
velocity profile in a capillary for Poiseuille flow is described by the following rela-
tionship (89),
(12.3)
where r is the capillary radius, r0 is the distance from the axis of the capillary, and
Vp is the local pressure gradient in the given capillary. In this case the average
velocity in the capillary is

(12.4)

and the flow is

Since for each marked chain we have q = const, it follows that

Therefore
v(ro) ""r~fr , ovforo ""2rofr 4
4
and the release of energy per unit time per unit length of a capillary is

r
2
E' ""j<ovf8r0 ) rodr0 ""r- 4
0
i.e., when there exist non-vanishing total flows under the action of the acoustic
wave one could expect localization of the dissipated energy release in the thinnest
capillaries. However it is clear that in the considered case of the acoustic wave
propagation, no integral displacement of the fluid in the capillary chain is observed
when a fixed cross-section of the fluid passes through several capillaries of the chain.
The average velocity of Poiseuille flow in a capillary with a variable cross-
section can be estimated using the relationship {12.4), with the average capillary
radius substituted for radius.
After substituting (12.2} in {12.4), we obtain the characteristic velocity< v >EO!!
2.5 . w- 2 mfs.
At the frequency v0 = 20 kHz, the period of the directed motion is T /2 =
v0 /2 = (1/4}10- 4 s and the displacement during this period is ill
1
=< v > T /2 "" w- 6 + w- 7 m. This value is substantially less than the char-
acteristic length of a capillary in the chain, since it is assumed that the lengths of