An unsteady two-cell vortex solution of the Navier—Stokes equations

Abstract

The steady two-cell viscous vortex solution of Sullivan (1959) is extended to yield unsteady two-cell viscous vortex solutions which behave asymptotically as certain analogous unsteady one-cell solutions of Rott (1958). The radial flux is a parameter of the solution, and the effect of the radial flow on the circumferential velocity, is analyzed. The work suggests an explanation for the eventual dissipation of meteorological flow systems such as tornadoes.

Exact solutions of the Navier-Stokes equations via Leray’s scheme

“…Other equations ave satisfied if the following three equations ave satisfied: (a-1)r r where a and d are constants, exists. This is closely related with the solution obtained by Bellamy-Knights [1]. His solution will be discussed in Section 14 below.…”

“…As in the blow-up case, we have the following solutions: (U~,Uo, U~) = This is a solution obtained in [1]. The streamlines of this vector fields are depicted in Figure 3.…”

“…One of the purposes of the present paper is to note that there are many unbounded solutions to (1) and (2). Because of (3), the domain in whieh ~ runs must be a cone with the originas its vertex.…”

Dynamically stretched vortices as solutions of the 3D Navier–Stokes equations

“…To see how ω 3 and W evolve individually we must restrict the flow to that of the strain field of Eq. (23). In this case, Eqs.…”

“…(1) but with W = 0, Sullivan [21] extended the steady solution for a viscous vortex embedded in a radially inward asymmetric stagnation point flow over a plane boundary [1,22]. This vortex naturally divides into two cells and the solution has been extended by Bellamy-Knights to take account of a moving separation surface between them [23].…”