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ABSTRACT Recent observations indicate that mm/cm-sized grains may exist in the embedded protostellar discs. How such large grains grow from the micron size (or less) in the earliest phase of star formation remains relatively unexplored. In this study, we take a first step to model the grain growth in the protostellar environment, using 2D (axisymmetric) radiation hydrodynamic and grain growth simulations. We show that the grain growth calculations can be greatly simplified by the ‘terminal velocity approximation’, where the dust drift velocity relative to the gas is proportional to its stopping time, which is proportional to the grain size. We find that the grain–grain collision from size-dependent terminal velocity alone is too slow to convert a significant fraction of the initially micron-sized grains into mm/cm sizes during the deeply embedded Class 0 phase. Substantial grain growth is achieved when the grain–grain collision speed is enhanced by a factor of 4. The dust growth above and below the disc midplane enables the grains to settle faster towards the midplane, which increases the local dust-to-gas ratio, which, in turn, speeds up further growth there. How this needed enhancement can be achieved is unclear, although turbulence is a strong possibility that deserves further exploration.
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The cusp of an apple
Singularities are common in diverse physical systems1 and lead to universal structures2,3. This universality suggests that they should also naturally arise in biological systems, where active growth, autonomous motion, kinesis and taxis focus deformations in spacetime, as exemplified in the morphogenetic processes determining biological size and shape4. A familiar example of a morphogenetic singularity is seen in the humble apple, which forms in the neighbourhood of the stalk as the apple grows. Here we study the geometry and morphogenesis of the cusp of an apple by combining observations of fruit growth with a simple theory, finite element simulations and controlled swelling experiments using a physical gel simulacrum. Our observations show that the axisymmetric cusp develops into a self-similar form, which can be understood in terms of a mechanical theory for the inhomogeneous growth of a soft sphere. Physical experiments using local inhibition in swelling gels corroborate our theoretical predictions. These experiments further show that axisymmetric cusps can lose stability and become lobed. We use simulations to show that the number of cuspidal lobes depends on the ratio of the size of the stalk to the size of the sphere, as well as the amplitude and periodicity of perturbations that mimic the role of fruit anatomy, consistent with observations of multi-cusped fruits.
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- Award ID(s):
- 2011754
- PAR ID:
- 10499811
- Publisher / Repository:
- Nature physics
- Date Published:
- Journal Name:
- Nature physics
- Volume:
- 17
- Issue:
- 10
- ISSN:
- 1745-2473
- Page Range / eLocation ID:
- 1125 to 1129
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1038/s41567-021-01335-8
