content_image/430911.jpg content_image/430920.jpg content_image/430921.jpg content_image/430922.jpg content_image/430906.jpg content_image/430908.jpg content_image/430909.jpg content_image/430907.jpg content_image/430910.jpg
perfluid v s = ∇ φ$_{s}$ motions of the form 

Using the same boundary conditions as above, we find the velocities of the unperturbed flow of the liquid phase 

and coefficients A$_{l}$ ( t ) and B$_{l}$ ( t ) describing the perturbed motion of the interface 

Employing the same boundary relation for the pressures in the phases and the same dependence between growth rate and chemical potential difference, we obtain for the undisturbed growth of the solid phase after some algebra 

<img src='content_image/430911.jpg'>

The growth equation looks exactly like the motion of a particle with the effective mass M ( R ) = 4 πρ$_{ef}$R $^{3}$, drag force 4 πR $^{2}$ρ $^{′}$K − 1 ˙ R , and potential energy U ( R ) = 4 παR 2 − (( ρ ′ − ρ ) /ρ )Δ P (4 πR $^{3}$/ 3), Δ P = P$_{∞}$ − P$_{c}$ being overpressure. 

The small amplitude perturbations for the spherical surface of the solid phase are described by the relation 

<img src='content_image/430920.jpg'>

For large radius R and l → ∞ so that q = l/R is fixed, equation (11) goes over to the result for the plane geom- etry. Under steady conditions R ( t ) ≡ R the spectrum of crystallization waves on the spherical interface is deter- mined from 

<img src='content_image/430921.jpg'>

Equation (11) can have unstable solutions giving rise to the generation of crystallization waves at the interface. The answer whether the spherical interface will be stable or unstable depends strongly on the history of the time behavior R ( t ). To illustrate, we consider the Richtmyer- Meshkov situation when a solid with radius R and inter- face rate ˙ R is subjected to a spherical shock acceleration ¨ R = V (0) δ ( t ). In the linear regime the perturbation am- plitude can be estimated according to 

<img src='content_image/430922.jpg'>

<img src='content_image/430906.jpg'>

<img src='content_image/430908.jpg'>

<img src='content_image/430909.jpg'>

Here ζ$_{l}$ (0) is the initial perturbation amplitude of l - th harmonic at the interface immediately after the shock acceleration and V (0) is an additional velocity acquired by the interface. The mode with l = 1 is obviously not involved because this harmonic describes the displace- ment of a sphere as a whole. Like the case of the planar geometry, the initial growth of the interface perturbation is a linear function of time 

The long-time behavior of the interface perturbation is strongly governed by the magnitude and the sign of the growth rate ˙ R . In fact, provided that a solid $^{4}$He globe either grows at any rate or melts not so fast at the mo- ment of shock acceleration, i.e., 

the growth of the perturbation amplitude saturates. For larger harmonics, the stabilizing role of the finite damp- ing of crystallization waves increases. In the opposite regime when a $^{4}$He solid melts sufficiently fast, the lin- ear growth of the interface amplitude in time will cross over into an exponential increment of the perturbation amplitude. 

The effects in the spherical geometry can be studied by focusing a high-intensity sound wave in the middle of an experimental cell. The experiments [31] on nucleation of solid $^{4}$He with two hemispherical piezoelectric trans- ducers glued together to make a spherical geometry have shown that it is possible to achieve very high pressure amplitudes, more than 100 bar in the bulk liquid $^{4}$He. 

We have investigated here the analogue of the Ray- leigh-Taylor and Richtmyer-Meshkov instabilities of crys- tallization waves at the accelerated superfluid-solid $^{4}$He interface. Our analysis, made within the linear the- ory in perturbation, shows that the Rayleigh-Taylor and Richtmyer-Meshkov instabilities can occur as well as the parametric Faraday instability [18] which we have treated as a periodically driven version of the Taylor instability. The plane and spherical interfaces are considered, and the first-order linearized equations are found for the per- turbation amplitudes. 

Unfortunately, it is difficult to make a well-founded conclusion in favor of destructing the crystal faceting as a result of the impulsively accelerated interface. For a quantitative analysis, it is necessary to have the images of crystal evolution taken successively in time. How- ever, such aspects as the Taylor-like instability cannot be rejected for the highly accelerated superfluid-solid $^{4}$He boundaries. 

<img src='content_image/430907.jpg'>

<img src='content_image/430910.jpg'>

## SUMMARY 

