A paradoxical solution of the Navier-Stokes equations

Gol'dshtik, M. A.

doi:10.1016/0021-8928(60)90070-8

Cited by 53 publications

(59 citation statements)

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“…, which also satisfies the boundary conditions (5)- (9). According to Theorem 3.1 we have that such a solution is real analytic in (x c , 1) and analytic in some disc of the complex plane, B (1, r) , r > 0 .…”

Section: Existence Of Solutions To (109)mentioning

confidence: 96%

“…First we will prove the real analyticity of a conically self-similar solution outside the singularity at x = 1 . 1)) and which satisfies the boundary conditions (5)- (9) is real analytic in [x c , 1) . If x c = −1 the same statement holds, but we must exclude the point x = −1 everywhere.…”

Section: Proof Of Theorem 31mentioning

confidence: 99%

“…For these solutions the velocity components decay as R −1 , where R is the distance to the apex of the cone, and the quotient of any two velocity components depends only on the polar angle of a spherical coordinate system. The swirling conically self-similar solutions, which were originally discovered by Long [13,14] and Goldshtik [9] fall into two categories. In the first one, the solutions satisfy the no-slip condition at the conical boundary, but are singular on the symmetry axis.…”

Section: Introductionmentioning

confidence: 99%

“…In the first one, the solutions satisfy the no-slip condition at the conical boundary, but are singular on the symmetry axis. This class of solutions was first studied by Goldshtik [9] and more generally by Serrin [16]. The second class of swirling conically self-similar solutions to the Navier-Stokes equations is the class of free-vortex solutions, where there are no forces acting along the symmetry axis.…”

Section: Introductionmentioning

confidence: 99%

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On the regularity and uniqueness of conically self-similar free-vortex solutions to the Navier-Stokes equations

Stein

2001

Z. angew. Math. Phys.

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In this paper the real analyticity of all conically self-similar free-vortex solutions to the Navier-Stokes equations is proven. Furthermore, it is mathematically established that such solutions are uniquely determined by the values of three derivatives on the symmetry axis, and hence a numerical method, invented and successfully used by Shtern & Hussain (1993,1996, is justified mathematically. In addition, it is proven that these results imply that for any conically self-similar free-vortex solution to the Navier-Stokes equations there exists a second order nonswirling correction term. For this term it is also shown that the second order contribution to the total axial flow force vanishes in the cases of the entire space and a half-space, but that it need not vanish for general conical domains. In doing so an old claim by Burggraf & Foster (1977) is established mathematically, however not for Long's problem but for Shtern & Hussain's (1996) extension of this problem to the full Navier-Stokes equations and the entire space. Mathematics Subject Classification (2000). 34A12, 34B15, 76D25, 76U05.

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Section: Existence Of Solutions To (109)mentioning

confidence: 96%

Section: Proof Of Theorem 31mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the regularity and uniqueness of conically self-similar free-vortex solutions to the Navier-Stokes equations

Stein

2001

Z. angew. Math. Phys.

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“…The various ways, mentioned in Idso's article, in which tornadoes can form seem to indicate the possibility of tornadoes with different structures, and to encourage theoretical constructions of tornado-like flows. For previous theoretical treatments of the tornado problem, see the papers by Goldstik, 7 Serrin, 8 and Head, Prahlad, and Phillips. 9 Reference to earlier works can be found in the powerful paper of Serrin.…”

Section: Introductionmentioning

confidence: 98%