| Title:
|
Steady vortex rings with swirl in an ideal fluid: asymptotics for some solutions in exterior domains (English) |
| Author:
|
Tadie |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
44 |
| Issue:
|
1 |
| Year:
|
1999 |
| Pages:
|
1-13 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $d>0$. (English) |
| Keyword:
|
vortex rings |
| Keyword:
|
potential theory |
| Keyword:
|
elliptic equations |
| MSC:
|
31B15 |
| MSC:
|
35Q35 |
| MSC:
|
76B47 |
| MSC:
|
76M25 |
| idZBL:
|
Zbl 1059.76507 |
| idMR:
|
MR1666858 |
| DOI:
|
10.1023/A:1022264002206 |
| . |
| Date available:
|
2009-09-22T17:59:39Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134402 |
| . |
| Reference:
|
[1] M. S. Berger: Mathematical Structures of Nonlinear Sciences, an Introduction.Kluwer Acad. Publ. NTMS 1, 1990. MR 1071172 |
| Reference:
|
[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 |
| Reference:
|
[3] L. E. Fraenkel & M. S. Berger: A global theory of steady vortex rings in an ideal fluid.Acta Math. 132 (1974), 13–51. MR 0422916, 10.1007/BF02392107 |
| Reference:
|
[4] B. Gidas B, WM. Ni & L. Nirenberg: Symmetry and related properties via the maximum principle.Comm. Math. Phys. 68 (1979), 209–243. MR 0544879, 10.1007/BF01221125 |
| Reference:
|
[5] J. Norbury: A family of steady vortex rings.J. Fluid Mech. 57 (1973), 417–431. Zbl 0254.76018, 10.1017/S0022112073001266 |
| Reference:
|
[6] Tadie: On the bifurcation of steady vortex rings from a Green function.Math. Proc. Camb. Philos. Soc. 116 (1994), 555–568. Zbl 0853.35135, MR 1291760, 10.1017/S0305004100072819 |
| Reference:
|
[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. MR 1373832, 10.1007/BF00276697 |
| Reference:
|
[8] Tadie: Radial functions as fixed points of some logarithmic operators.Potential Anal. 9 (1998), 83–89. MR 1644112, 10.1023/A:1008606430233 |
| Reference:
|
[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. . Zbl 0913.76017, MR 1603512 |
| . |
