Determining the shapes of a rotating liquid droplet bound by surface tension is an archetypal problem in the study of the equilibrium shapes of a spinning and charged droplet, a problem that unites models of the stability of the atomic nucleus with the shapes of astronomical-scale, gravitationally-bound masses. The shapes of highly deformed droplets and their stability must be calculated numerically. Although the accuracy of such models has increased with the use of progressively more sophisticated computational techniques and increases in computing power, direct experimental verification is still lacking. Here we present an experimental technique for making wax models of these shapes using diamagnetic levitation. The wax models resemble splash-form tektites, glassy stones formed from molten rock ejected from asteroid impacts. Many tektites have elongated or ‘dumb-bell' shapes due to their rotation mid-flight before solidification, just as we observe here. Measurements of the dimensions of our wax ‘artificial tektites' show good agreement with equilibrium shapes calculated by our numerical model, and with previous models. These wax models provide the first direct experimental validation for numerical models of the equilibrium shapes of spinning droplets, of importance to fundamental physics and also to studies of tektite formation.

Splash-form tektites are glassy rocks ranging in size from roughly 1 to 100 mm that are believed to have formed from the splash of silicate liquid after a large terrestrial impact from which they are strewn over thousands of kilometres. They are found in an array of shapes including spheres, oblate ellipsoids, dumbbells, rods and possibly fragments of tori. It has recently become appreciated that surface tension and centrifugal forces associated with the rotation of fluid droplets are the main factors determining the shapes of these tektites. In this contribution, we compare the shape distribution of 1163 measured splash-form tektites with the results of the time evolution of a 3D numerical model of a rotating fluid drop with surface tension. We demonstrate that many aspects of the measured shape distribution can be explained by the results of the dynamical model.

Rotating superfluid He droplets of approximately 1 μm in diameter were obtained in a free nozzle beam expansion of liquid He in vacuum and were studied by single-shot coherent diffractive imaging using an x-ray free electron laser. The formation of strongly deformed droplets is evidenced by large anisotropies and intensity anomalies (streaks) in the obtained diffraction images. The analysis of the images shows that, in addition to previously described axially symmetric oblate shapes, some droplets exhibit prolate shapes. Forward modeling of the diffraction images indicates that the shapes of rotating superfluid droplets are very similar to their classical counterparts, giving direct access to the droplet angular momenta and angular velocities. The analyses of the radial intensity distribution and appearance statistics of the anisotropic images confirm the existence of oblate metastable superfluid droplets with large angular momenta beyond the classical bifurcation threshold.3

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Abstract. We present a series of simulations of the mantle convection process based upon an axisymmetric numerical model and highlight a wide range of results in which scaling emerges. For the more challenging simulations it was found necessary to employ a finite difference mesh with uneven grid spacing in the radial coordinate, and we present the appropriate transformed field equations that are required to implement a model of this kind. The statistics of mass flux events transiting the 660-km phase transition are calculated for a large number of high-resolution calculations, and some of these are shown to display scale invariance properties in the high Rayleigh number regime. We also present a new parameterized model of convection and demonstrate its success in predicting the manner in which many of the bulk properties of the convection process scale with convection control parameters. Results are also presented which demonstrate that quantities such as heat flow, characteristic velocity, and thermal boundary layer thickness scale with the mean viscosity even in time-dependent simulations in which the effects of phase transitions, depth-varying viscosity, and internal heating are active. The heat flow scaling exponent is seen to decrease in magnitude with increasing internal heating rate and Clapeyron slope of the 660-km phase transition, but it is shown to be insensitive to depth variation of viscosity. Heat flow is seen to be reduced only modestly as the degree of layering increases unless layering is extreme. These calculations clearly demonstrate that in order for the surface heat flow predicted by the model to equal that characteristic of Earth, the mean viscosity of the mantle that controls the convection process must be considerably higher than the viscosity inferred on the basis of postglacial rebound and/or the flow must be significantly layered by the endothermic phase transition at 660 km depth. If mantle viscosity may be assumed to be Newtonian, in which case the creep resistance that controls rebound and convection must be the same, this constitutes a strong argument for the importance of layering. The force of this argument depends upon the existence of an accurate estimate of the temperature at the core-mantle boundary which has only recently become available.

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