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vortex rings; potential theory; elliptic equations
In this paper, the axisymmetric flow in an ideal fluid outside the infinite cylinder ($r \le d$) where $ (r,\theta ,z)$ denotes the cylindrical co-ordinates in ${\mathbb{R}}^3$ is considered. The motion is with swirl (i.e. the $\theta $-component of the velocity of the flow is non constant). The (non-dimensional) equation governing the phenomenon is (Pd) displayed below. It is known from e.g. that for the problem without swirl ($f_q=0$ in (f)) in the whole space, as the flux constant $k$ tends to $\infty $, 1) $\mathrm{dist}(0z,\partial A)=O(k^{1/2})$; $\mathrm{diam}A = O(\exp (-c_0k^{3/2}))$; 2) $(k^{1/2} \Psi )_{k \in \mathbb{N}}$ converges to a vortex cylinder $U_m$ (see (1.2)). We show that for the problem with swirl, as $k\nearrow \infty $, 1) holds; if $m \le q+2$ 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 $d>0$.
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