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2 results

  • Cavitation of a spherical body under mechanical and self-gravitational forces

  • Part of
  • Pablo V. Negrón–Marrero, Jeyabal Sivaloganathan
  • Journal:
    Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
    Published online by Cambridge University Press:
    08 January 2024, pp. 1-31
    • Article
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  • In this paper, we look for minimizers of the energy functional for isotropic compressible elasticity taking into consideration the effect of a gravitational field induced by the body itself. We consider two types of problems: the displacement problem in which the outer boundary of the body is subjected to a Dirichlet-type boundary condition, and the one with zero traction on the boundary but with an internal pressure function. For a spherically symmetric body occupying the unit ball $\mathcal {B}\in \mathbb {R}^3$, the minimization is done within the class of radially symmetric deformations. We give conditions for the existence of such minimizers, for satisfaction of the Euler–Lagrange equations, and show that for large displacements or large internal pressures, the minimizer must develop a cavity at the centre. We discuss a numerical scheme for approximating the minimizers for the displacement problem, together with some simulations that show the dependence of the cavity radius and minimum energy on the displacement and mass density of the body.

  • 4 - Planetesimal Formation

  • Philip J. Armitage, Stony Brook University, State University of New York
  • Book:
    Astrophysics of Planet Formation
    Published online:
    27 January 2020
    Print publication:
    30 January 2020, pp 141-180

  • Summary

    Chapter 4 covers the evolution of the solid component of protoplanetary disks, from dust through to the formation of km-scale planetesimals. The physics of how dust particles interact with the gas through aerodynamic drag is described, together with the consequences - vertical settling, radial drift, particle trapping, and particle pile-up. The outcome of particle collisions, and their theoretical description using the coagulation equation, are reviewed. Collective mechanisms for planetesimal formation via gravitational collapse are discussed, starting with the classical Goldreich-Ward theory, and concluding with the streaming instability of aerodynamically coupled mixtures of gas and dust.