×

zbMATH — the first resource for mathematics

On the spectral instability of parallel shear flows. (English) Zbl 1353.76022
Summary: This short note is to announce our recent results [the authors, Adv. Math. 292, 52–110 (2016; Zbl 1382.76079); Duke Math. J. 165, No. 16, 3085–3146 (2016; Zbl 1359.35129)] which provide a complete mathematical proof of the viscous destabilization phenomenon, pointed out by W. Heisenberg [“Über Stabilität und Turbulenz von Flüssigkeitsströmen”, Ann. Phys. 379, No. 15, 577–627 (1924; doi:10.1002/andp.19243791502)], C. C. Lin [The theory of hydrodynamic stability. London, New York: Cambridge University Press (1955; Zbl 0068.39202)] and Tollmien (1940s), among other prominent physicists. Precisely, we construct growing modes of the linearized Navier-Stokes equations about general stationary shear flows in a bounded channel (channel flows) or on a half-space (boundary layers), for sufficiently large Reynolds number \(R \rightarrow \infty\). Such an instability is linked to the emergence of Tollmien-Schlichting waves in describing the early stage of the transition from laminar to turbulent flows.
MSC:
76E05 Parallel shear flows in hydrodynamic stability
35B35 Stability in context of PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
76D05 Navier-Stokes equations for incompressible viscous fluids
76F10 Shear flows and turbulence
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] P. G. Drazin, W. H. Reid, Hydrodynamic stability. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University, Cambridge-New York, 1981.
[2] E. Grenier, Y. Guo, and T. Nguyen, Spectral instability of symmetric shear flows in a two-dimensional channel, . · Zbl 1382.76079
[3] E. Grenier, Y. Guo, and T. Nguyen, Spectral instability of characteristic boundary layer flows, . · Zbl 1359.35129
[4] W. Heisenberg, Über Stabilität und Turbulenz von Flüssigkeitsströmen. Ann. Phys. 74, 577-627 (1924)
[5] W. Heisenberg, On the stability of laminar flow. Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, pp. 292-296. Amer. Math. Soc., Providence, R. I., 1952.
[6] C. C. Lin, The theory of hydrodynamic stability. Cambridge, at the University Press, 1955.
[7] W. Orr, Stability and instability of steady motions of a perfect liquid and of a viscous fluid, Parts I and II, Proc. Ir. Acad. Sect. A, Math Astron. Phys. Sci., 27 (1907), pp. 9-68, 69-138.
[8] Lord Rayleigh, On the stability, or instability, of certain fluid motions. Proc. London Math. Soc. 11 (1880), 57-70. · JFM 12.0711.02
[9] H. Schlichting, Boundary layer theory, Translated by J. Kestin. 4th ed. McGraw-Hill Series in Mechanical Engineering. McGraw-Hill Book Co., Inc., New York, 1960. · Zbl 0096.20105
[10] A. Sommerfeld, Ein Beitrag zur hydrodynamischen Erklärung der turbulent Flussigkeitsbewe-gung, Atti IV Congr. Internat. Math. Roma, 3 (1908), pp. 116-124.
[11] W. Wasow, The complex asymptotic theory of a fourth order differential equation of hydrodynamics. Ann. of Math. (2) 49, (1948). 852-871. · Zbl 0031.40202
[12] W. Wasow, Asymptotic solution of the differential equation of hydrodynamic stability in a domain containing a transition point. Ann. of Math. (2) 58, (1953). 222-252. · Zbl 0051.06602
[13] W. Wasow, Linear turning point theory. Applied Mathematical Sciences, 54. Springer-Verlag, New York, 1985. ix+246 pp. · Zbl 0558.34049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.