Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
vortex rings; potential theory; elliptic equations
vortex rings; potential theory; elliptic equations
Summary:
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 $d>0$.
References:
[1] M. S. Berger: Mathematical Structures of Nonlinear Sciences, an Introduction. Kluwer Acad. Publ. NTMS 1, 1990. MR 1071172
[2] L. E. Fraenkel: On Steady Vortex Rings with Swirl and a Sobolev Inequality. Progress in PDE: Calculus of Variations, Applications, C. Bandle et al. (eds.), Longman Sc. & Tech., 1992, pp. 13–26. MR 1194186
[3] L. E. Fraenkel & M. S. Berger: A global theory of steady vortex rings in an ideal fluid. Acta Math. 132 (1974), 13–51. DOI 10.1007/BF02392107 | MR 0422916
[4] B. Gidas B, WM. Ni & L. Nirenberg: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), 209–243. DOI 10.1007/BF01221125 | MR 0544879
[5] J. Norbury: A family of steady vortex rings. J. Fluid Mech. 57 (1973), 417–431. DOI 10.1017/S0022112073001266 | Zbl 0254.76018
[6] Tadie: On the bifurcation of steady vortex rings from a Green function. Math. Proc. Camb. Philos. Soc. 116 (1994), 555–568. DOI 10.1017/S0305004100072819 | MR 1291760 | Zbl 0853.35135
[7] Tadie: Problèmes elliptiques à frontière libre axisymetrique: estimation du diamètre de la section au moyen de la capacité. Potential Anal. 5 (1996), 61–72. DOI 10.1007/BF00276697 | MR 1373832
[8] Tadie: Radial functions as fixed points of some logarithmic operators. Potential Anal. 9 (1998), 83–89. DOI 10.1023/A:1008606430233 | MR 1644112
[9] Tadie: Steady vortex rings in an ideal fluid: asymptotics for variational solutions. Integral methods in sciences and engineering Vol 1 (Oulu 1996), Pitman Res. Notes Math. 374, Longman, Harlow, 1997, pp. . MR 1603512 | Zbl 0913.76017
Search
Partner of
