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Numerical study of singularity formation in a class of Euler and Navier-Stokes flows. (English) Zbl 1184.76402
Summary: We study numerically a class of stretched solutions of the three-dimensional Euler and Navier-Stokes equations identified by J. D. Gibbon, A. S. Fokas and C. R. Doering [Physica D 132, No. 4, 497–510 (1999; Zbl 0956.76018)]. Pseudo-spectral computations of a Euler flow starting from a simple smooth initial condition suggests a breakdown in finite time. Moreover, this singularity apparently persists in the Navier-Stokes case. Independent evidence for the existence of a singularity is given by a Taylor series expansion in time. The mechanism underlying the formation of this singularity is the two-dimensionalization of the vorticity vector under strong compression; that is, the intensification of the azimuthal components associated with the diminishing of the axial component. It is suggested that the hollowing of the vortex accompanying this phenomenon may have some relevance to studies in vortex breakdown.

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
76D17 Viscous vortex flows
35Q30 Navier-Stokes equations
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References:
[1] DOI: 10.1063/1.858849 · Zbl 0800.76083 · doi:10.1063/1.858849
[2] DOI: 10.1007/BF02547354 · JFM 60.0726.05 · doi:10.1007/BF02547354
[3] DOI: 10.1007/BF01212349 · Zbl 0573.76029 · doi:10.1007/BF01212349
[4] Constantin P., Comm. PDEs 21 pp 559– (1996) · Zbl 0853.35091 · doi:10.1080/03605309608821197
[5] DOI: 10.1137/1036004 · Zbl 0803.35106 · doi:10.1137/1036004
[6] Ponce G., Commun. Math. Phys. 98 pp 349– (1985) · Zbl 0589.76040 · doi:10.1007/BF01205787
[7] DOI: 10.1143/JPSJ.54.2132 · doi:10.1143/JPSJ.54.2132
[8] DOI: 10.1103/PhysRevLett.67.3511 · doi:10.1103/PhysRevLett.67.3511
[9] DOI: 10.1063/1.868166 · Zbl 0845.76065 · doi:10.1063/1.868166
[10] Bhattacharjee A., Phys. Rev. E 52 pp 5110– (1995) · doi:10.1103/PhysRevE.52.5110
[11] Bhattacharjee A., Phys. Rev. Lett. 69 pp 2196– (1992) · Zbl 0968.76528 · doi:10.1103/PhysRevLett.69.2196
[12] Pelz R. B., Phys. Rev. Lett. 79 pp 4998– (1997) · doi:10.1103/PhysRevLett.79.4998
[13] DOI: 10.1016/S0065-2156(08)70100-5 · doi:10.1016/S0065-2156(08)70100-5
[14] DOI: 10.1017/S002211209400011X · doi:10.1017/S002211209400011X
[15] Gibbon J. D., Physica D 132 pp 497– (1999) · Zbl 0956.76018 · doi:10.1016/S0167-2789(99)00067-6
[16] Stuart J. T., IMA J. Appl. Math. 46 pp 147– (1991) · Zbl 0724.76019 · doi:10.1093/imamat/46.1-2.147
[17] Cowley S. J., Philos. Trans. R. Soc. Lond., Ser. A 333 pp 343– (1990) · Zbl 0711.76031 · doi:10.1098/rsta.1990.0165
[18] Childress S., J. Fluid Mech. 203 pp 1– (1989) · Zbl 0674.76013 · doi:10.1017/S0022112089001357
[19] Constantin P., Comm. Math. Phys. 104 pp 311– (1986) · Zbl 0655.76041 · doi:10.1007/BF01211598
[20] Moffatt H. K., J. Fluid Mech. 409 pp 51– (2000) · Zbl 0962.76027 · doi:10.1017/S002211209900782X
[21] Oseen C. W., Ark. Mat., Astronom. Fys. 20 pp 1– (1927)
[22] DOI: 10.1063/1.863957 · Zbl 0536.76034 · doi:10.1063/1.863957
[23] DOI: 10.1017/S0022112083001159 · Zbl 0517.76033 · doi:10.1017/S0022112083001159
[24] DOI: 10.1103/PhysRevLett.44.572 · doi:10.1103/PhysRevLett.44.572
[25] Liebovich S., Annu. Rev. Fluid Mech. 10 pp 221– (1978) · doi:10.1146/annurev.fl.10.010178.001253
[26] Lopez J. M., J. Fluid Mech. 221 pp 533– (1990) · Zbl 0715.76096 · doi:10.1017/S0022112090003664
[27] Brown G. L., J. Fluid Mech. 221 pp 553– (1990) · Zbl 0715.76097 · doi:10.1017/S0022112090003676
[28] Stewartson K., J. Fluid Mech. 121 pp 507– (1982) · Zbl 0531.76042 · doi:10.1017/S0022112082002018
[29] Hall P., J. Fluid Mech. 238 pp 297– (1992) · Zbl 0765.76090 · doi:10.1017/S0022112092001721
[30] Okamoto H., Taiwanese J. of Math. 4 pp 65– (2000)
[31] Constantin P., Internat. Math. Res. Notices (IMRN) 9 pp 455– (2000) · Zbl 0970.76017 · doi:10.1155/S1073792800000258
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