Numerical Solutions of Serrin’s Equations by Double Exponential Transformation

Hamada, Shinsuke

doi:10.2977/prims/1201012042

Cited by 4 publications

(1 citation statement)

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“…Lack of analytic solutions for b = 1 and constant nonzero viscosity led to numerical approaches presented, e.g., in [17,24,30]. We have not been able to find analytic expressions for any nontrivial solutions in the case 0 < b < 1 either, but we used a numerical approach to generate their approximations for various values of b between 0 and 1.…”

Section: Case 0 < B < 1 (Numerical Solutions)mentioning

confidence: 99%

Fractal powers in Serrin's swirling vortex solutions

Bělík

Dokken

Scholz

et al. 2014

Asymptotic Analysis

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We consider a modification of the fluid flow model for a tornado-like swirling vortex developed by Serrin [Phil. Trans. Roy. Soc. London, Series A, Math & Phys. Sci. 271(1214) (1972), 325-360], where velocity decreases as the reciprocal of the distance from the vortex axis. Recent studies, based on radar data of selected severe weather events [Mon. Wea. Rev. 133(9) (2005), 2535-2551; Mon. Wea. Rev. 128(7) (2000), 2135-2164; Mon. Wea. Rev. 133(1) (2005), 97-119], indicate that the angular momentum in a tornado may not be constant with the radius, and thus suggest a different scaling of the velocity/radial distance dependence.Motivated by this suggestion, we consider Serrin's approach with the assumption that the velocity decreases as the reciprocal of the distance from the vortex axis to the power b with a general b > 0. This leads to a boundary-value problem for a system of nonlinear differential equations. We analyze this problem for particular cases, both with nonzero and zero viscosity, discuss the question of existence of solutions, and use numerical techniques to describe those solutions that we cannot obtain analytically.

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Section: Case 0 < B < 1 (Numerical Solutions)mentioning

confidence: 99%