On the existence of conically self-similar free-vortex solutions to the Navier-Stokes equations

摘要

In this paper a proof of existence and non-existence of the conically self-similar free-vortex solutions to the Navier-Stokes equations, originally found by Yih et al. (1982, Phys. Fluids. 25, 2147-2158), is presented. This proof clearly establishes that these solutions do not have any kind of singularity at the symmetry axis. This analysis gives considerably improved existence and non-existence bounds and it is shown that these bounds are close to optimal in the low-swirling limit. This approach links the questions of existence and non-existence for the swirling case and for the non-swirling case. The proof, which is an extension of techniques developed by Serrin (1972, Phil. Trans. R. Soc. Lend. 271, 325-360), is based on Schauder's Fixed Point Theorem and is, therefore, non-constructive, Therefore, the paper ends with a brief discussion of the question of how to compute the conically selfsimilar free-vortex solutions to the Navier-Stokes equations. [References: 15]