A viscous solution of the spherical vortex to the Navier-Stokes equations

TL;DR

This paper extends the classical Hill's spherical vortex solution from Euler equations to viscous Navier-Stokes equations, providing exact solutions that enable analysis of flows with vorticity discontinuities and including a time-dependent rotating vortex.

Contribution

It presents the first exact viscous solutions to the Navier-Stokes equations based on Hill's spherical vortex, incorporating viscosity and time evolution with rotation.

Findings

Derived a viscous solution satisfying Navier-Stokes equations

Provided a time-dependent vortex solution with rotation

Enabled analysis of vorticity discontinuities in viscous flows

Abstract

We deal with the Hill's spherical vortex, which is an exact solution to the Euler equation, and manage the solution to satisfy the incompressible Navier-Stokes(INS) equations with a viscous term. Once we get a viscous solution to the INS equations, we will be able to analyze the flows with discontinuities in vorticity. In the same procedure, we also present a time developing exact solution to the INS equations, which has a rotation on the axis besides the Hill's vortex.