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∞). If, however, h˃δ (i.e., βh ˃ 3), then this difference is small. In such a case, Equations (2.24) and (2.26) provide close results. For this reason, liquid velocity in a δ-wide fracture may be determined with an accuracy sufficient for practical calculations from a simple Stokes formula Equation (2.26) considering the liquid outside of this layer immobile. If, however, h ˂ δ, (βh ˂ 0.5), the calculations should be conducted using a more complex formula Equation (2.24). Liquid layers for which practically move together with the fracture surface (as a solid body).

Figure 2.3 Vibration velocity distribution in a fracture (solid curves) and over a free surface (dashed line).

      In a finite thickness layer above the harmonically vibrating fracture surface, for which βh ≥ c ≈ 3.69 (where c is the root of equation sh²c + ch²c = 400), and the penetration depth determined as previously is somewhat greater than the value δ = 3/β, but does not exceed the value c/β which it assumes at βh = c. In layers where βh = c, the 20-fold decline in the velocity amplitude is not reached.

      Due to linearity of the problem, the above results are easily generalized for cases of rectilinear periodic arbitrary vibrations and periodic vibrations in two mutually perpendicular directions on the fracture surface.

At periodic rectilinear motion of the fracture surface (represented by a sum of harmonics with the frequency kω, where k 1, 2, …), the penetration depth for some m^th harmonic, according to Equation (2.27), will be times lower than for the first one. For this reason, at sufficient distance from the fracture surface, liquid vibration is determined by the first harmonic; in other words, it approaches harmonic character regardless of the pattern of periodic vibrations (of course, on condition that h ≥ δ and the harmonic amplitudes do not grow with the increase of their numerical order).

If the fracture surface is making random periodic vibrations in two mutually perpendicular directions in two mutually perpendicular directions lying on its surface, the liquid velocity in each of these directions are determined independent from one another. But, the aforementioned qualitative patterns remain valid. In particular, if periods of mutually perpendicular vibrations coincide, then the liquid particles’ trajectories at a sufficient distance from the fracture surface approach ellipses, and the size of these ellipses is exponentially declining. In a case when periods are substantially different, the liquid motion at distancing from the fracture surface approaches rectilinear harmonic vibrations coinciding in the direction and frequency with longer period vibrations. In a manner of speaking, the liquid layer has properties of a high frequency filter.

      The quoted results are comparable with the solution of a problem of fading the agitations caused by a spherical fractured surface in the process of its harmonic vibrations along some direction in an incompressible liquid.

For instance, we will review for a case of Reynolds’ small numbers a spherical surface within which 95% of the total energy loss in a viscous incompressible liquid is dispersed (Figure 2.4a). Assume that R₀ is radius of vibration penetration and r₀ is radius of the spherical surface. Then, the value δ = R₀ − r₀ may be considered the appropriate depth of vibrations’ penetration into the liquid. Then, correlation δ = R₀ − r₀ may be treated as the appropriate depth of vibrations’ penetration into the liquid. For the correlation δ/r₀ vs. the parameter we obtain a solid line diagram in Figure 2.4b. At that, is the same value with the dimension of length as in Equation (2.27). The same diagram within the considered limits of β variation may be approximated as follows:

(2.28)

      (dashed line in Figure 2.3). Upon reducing by r₀, this equality is exactly equal to (2.27).

Thus, there is practically total coincidence of the results for a case of the flat and spherical fracture surfaces’ vibrations. The study result is also close if the agitation area from the spherical surface is defined not by the energy dissipation, but by leveling of the pressure field or by fading of the velocity field as it has been done for a case of a flat fracture. The agitation zone in the case of a spherical surface’s incremental advance in a viscous liquid under the problem’s conditions is by an order of magnitude greater in size than in the case of its vibration. This conclusion is in the qualitative agreement with correlations of Equations (2.27) and (2.28), which indicate an expansion of the agitation zone with the decline in the vibration frequency.

      In conclusion, one may note that under the assumption of incompressibility of liquids, a case of lateral fracture surface vibrations is trivial as the liquid vibrates in this direction together with the fracture surface. If compressibility is present, then the process is described by a wave equation which includes an addend accounting for the dissipation of the vibration energy.

Figure 2.4 Vibration in a liquid volume at progressive vibrations of the fracture’s spherical surface.

Figure 2.5 Schematic advance of a mixture of a wetting and nonwetting liquids in a unit pore volume in a field of elastic vibrations. A_r and A_a are the relative and absolute displacement amplitudes of the nonwetting liquid; ε_r and ε_a are the relative and absolutes dislocations.

In the case of pore volume filled up with a wetting liquid with floating droplets of nonwetting liquid, the vibration penetration depth may be estimated treating the mixture equivalent to some liquid with viscosity greater than that of the wetting liquid with droplets of a nonwetting liquid. In order to estimate the absolute and relative amplitudes of displacements, force, and energy expense needed for providing

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