Steady vortex rings with swirl in an ideal fluid: asymptotics for some solutions in exterior domains

Tadie

Abstract: In this paper, the axisymmetric flow in an ideal fluid outside the infinite cylinder ($r \leq d$) where $ (r,\theta,z)$ denotes the cylindrical co-ordinates in ${\Bbb R}^3$ is considered. The motion is with swirl (i.e. the $\theta$-component of the velocity of the flow is non constant). The (non-dimensional) equation governing the phenomenon is (Pd) displayed below. It is known from e.g. \cite9 that for the problem without swirl ($f_q=0$ in (f)) in the whole space, as the flux constant $k$ tends to $\infty$, \endgraf1) $\dist(0z,\partial A)=O(k^{1/2})$; $\diam A = O(\exp(-c_0k^{3/2}))$; \endgraf2) $(k^{1/2} \Psi)_{k \in\Bbb N}$ converges to a vortex cylinder $U_m$ (see (1.2)). \endgraf We show that for the problem with swirl, as $k\nearrow\infty$, 1) holds; if $m \leq q+2$ then 2) holds and if $m> q+2$ it holds with $U_{q+2}$ instead of $U_m$. Moreover, these results are independent of $f_0$, $f_q$ and $d>0$.

Keywords: vortex rings, potential theory, elliptic equations

Classification (MSC 1991): 76M25 31B15 35J

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