Page 207 - Percolation Models for Transport in Porous Media With

12.3 CAVITATION 205

On the other hand [85), the mass velocity is

-lc-1
- Ot
"-PaP! 1 {12.12)
where C1 is the velocity of sound in the fluid. In this case, as in {12.9), after
taking into account {12.11) we obtain

vf < v >"' (< r > /r) 2

and using {12.12) and the fact that < v >"' ii,
"!:!! _1!2.._ (< r >)2

P!Cf r
After substituting this value of v in {12.10), we finally obtain the formula for the
pressure difference when the fluid passes from a thick capillary to a thin one:
p=Po-! ~ (<r>)4 {12.13)
2 P!C} r

For a fluid, P!C} "' 1 GPa; Pa "'0.01- 0.1 MPa for the considered phenomena.
As it was demonstrated above, one can take < r >"' 10- 5 m and the minimal value
of radius r "' 10- 6 m. In this case < r > fr "' 10, ( < r > /r) 4 "' 10 4 , and if the
background pressure of the order 10 MPa is taken as Po, we obtain from {12.13)
that p "' 10 7 - {1/2) · 10- 4 • 10 4 "' 10 7 - {1 + 0.1) "' 10 Pa. This means that
7
in a reasonable range of radii of the joining capillaries (rfr 0 < 10), the pressure
difference, as fluid passes from a thick capillary to a thin one, is of the order of
the wave amplitude Pa, and therefore no negative pressure develops with respect
to the high reservoir pressure Po.
If we take an "exotic" case < r > fr "' 10 2 , i.e., r "' w- 7 m, then ( < r >
fr ) 4 "' 10 8 and p "' -1 GPa. At such values of negative pressure one can expect the
effect of cavitation to work. For the mentioned rarefaction pressure, the existing
hydrodynamic pressure, and the given frequency of action, estimate the range of
radii of the cavitating gas bubbles.
Suppose that due to some reasons, the so-called germs, i.e., bubbles of gas,
exist inside the fluid. Formed inside the cone of depression during the evolution
of gas because of the local temperature, concentration and other fluctuations,
these germs can oscillate in the pressure wave and collapse with the formation of
cumulative jets directed towards the interface of the fluid and the solid capillary
surface [89, 90). As in the approach [91), consider the variation of the initial
pressure Po in the fluid in the vicinity of the oscillating bubble with initial radius
ao. If we neglect the elasticity of vapor, the pressure of the gas inside the bubble
is
Pn =Po+ 2xfao