Page 206 - Percolation Models for Transport in Porous Media With

204 CHAPTER 12. ACOUSTIC WAVES AND PERMEABILITY

Figure 73: A sketch of a capillary junction element for capillaries with different
radii

in the absolute value develop at the relief stage. Consider two possible mecha-
nisms for the development of negative pressure during ultrasonic action. This can
happen at the junctions of capillaries with different radii during the flow from the
thicker to the thinner one and at the relief stage of the traveling pressure wave in
the fluid.
1. Suppose two capillaries in a chain form a junction of the type presented in
fig. 73. As was shown by the estimates presented above, the displacement of the
fluid during a half-period of the wave in a capillary chain is rather small ("' 10- 7
m); however, in the immediate vicinity of the junction, as the fluid passes from
the thick capillary to the thin one, the increase of the flow rate and the pressure
drop take place. We will estimate this pressure drop using Bernoulli's integral
along a horizontal flow tube, when the z-coordinate is constant. In this case the
gz term can be included in the general constant, so that the following equality can
be written
(12.8)
The quantities marked with index "0" correspond to the thick capillary, and
those without an index, to the thin one. From the mass conservation equation we
have pvSk = canst, where Sk is the area of the cross-section of the capillary, and
we obtain
(12.9)
where r and r 0 are the radii of the corresponding capillaries. After substituting
(12.9) in (12,8), we obtain

(12.10)

since we are discussing the case when (rfr 0 ) 4 « 1.
The average velocity < v > of Poiseuille flow along a chain in the high pressure
phase can be estimated using the average radius < r > of a capillary chain for a
given function f(r). In this case

(12.11)