English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We completely determine all the unstable modes and neutral modes for the Kolmogorov flow (sin(2x2), 0) on the two-dimensional torus [0, 2n\ x [0, 2n] for both the Navier-Stokes equations and the Euler equations. We also give comments on "vorticity generation". 1. Main Results. It is a difficult problem to determine every unstable and neutral direction for the linearized Navier-Stokes equations around a steady flow for any large Reynolds numbers, unless the linear operator is self-adjoint or its unstable and neutral spectra are trivial. In this paper, for the special basic flow (sin(2x2), 0) on the two-dimensional torus [0, 2n] x [0, 2n], we give all the unstable modes and neutral modes for the Navier-Stokes equations for any Reynolds numbers. We also determine all the unstable modes and neutral modes for the corresponding Euler equations (that is, the case when the Reynolds number is infinity). We show that the unstable eigenvalues for the linearized Navier-Stokes equations approach the unstable eigenvalues for the linearized Euler equations when the viscosity approaches zero. We believe this is the first nontrivial example in which the linearized operators are not self-adjoint but all the unstable and neutral modes can be determined theoretically. We also comment on Vishik and Friedlander's result [18] on hydrodynamic stability. The result here is a starting point for our study on stationary and Hopf bifurcations of the Navier-Stokes equations on the 2D torus (cf. [9]), and also for our future theoretical investigations on more complex dynamical behavior of the Navier-Stokes equations on the 2D torus. Let £2 = [0, 2n] x [0, 2n] be the 2D torus, and let H be the subspace of (L2(Q))2 that consists of sinusoidal periodic flows on Q with space average zero. An orthonorReceived December 11, 1992. 1991 Mathematics Subject Classification. Primary 76D05, 76E05, 76N10, 35Q30.

On unstable and neutral spectra of incompressible inviscid and viscid fluids on the 2D torus