Hydrodynamic Oscillations

The hydrodynamic oscillations of a liquid drop, where the restoring force is the surface tension of the liquid, have been treated by Chandrasekhar (1961). For a drop with surface tension $\gamma$ , density $\rho$ and radius a , the period of oscillation is given by

$$\pi \sqrt{ \rho a^2 \over l (l-1)(l+2) \gamma}$$ (11)

where l is an integer greater than unity. We may gain some estimate of the surface tension from March and Tosi (1984), who show that for molten alkali halides a typical value is of order 0.1 J m ^{- 2}. Thus for a sphere of radius 10 $\mbox{\AA}$ , the lowest-order oscillation has a period of approximately 17 ps . This may be compared with the periods of the oscillations of the liquid drop in the simulations, shown in region B of figure 1, which are between 5 and 10 ps.

The shape assumed by the surface of the sphere is given by

$$\theta \phi \theta \phi \omega$$ (12)

where B is the amplitude of the surface tension oscillation and Y_l,m is a spherical harmonic. The radial and tangential velocities in this oscillation are

$$\omega {r^{l-1} \over a^{l-1}} \theta \phi \omega$$ (13)

$$\omega {r^{l-1} \over a^{l-1}} {1 \over l \sin \theta} {\partial Y_{l,m} (\theta , \phi ) \over \partial ( \cos \theta )} \omega$$ (14)

from which we may evaluate the kinetic energy. In the fundamental (l = 2) mode this is

$$\left \langle KE \right\rangle= \gamma$$ (15)

If we equate this energy to ${\frac{1}{2}}$kT for T = 2000 K , we find the amplitude to be about 3 $\mbox{\AA}$ , and independent of the radius of the drop. For a drop of radius 10 $\mbox{\AA}$ this amplitude is about one third of the drop radius. This, of course, invalidates the perturbative approach by which the surface tension oscillations were described, but it is consistent with the large distortions of shape seen in the simulations of the molten LaF₃ drops, as shown in Figure 1.