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Construction of circle bifurcations of a two-dimensional spatially periodic flow

Authors
Journal
Journal of Mathematical Analysis and Applications
0022-247X
Publisher
Elsevier
Publication Date
Volume
324
Issue
1
Identifiers
DOI: 10.1016/j.jmaa.2005.11.060
Keywords
  • Steady-State Bifurcation
  • Pitchfork Bifurcation
  • Navier–Stokes Equation
  • Kolmogorov Flow

Abstract

Abstract The study by Yudovich [V.I. Yudovich, Example of the generation of a secondary stationary or periodic flow when there is loss of stability of the laminar flow of a viscous incompressible fluid, J. Math. Mech. 29 (1965) 587–603] on spatially periodic flows forced by a single Fourier mode proved the existence of two-dimensional spectral spaces and each space gives rise to a bifurcating steady-state solution. The investigation discussed herein provides a structure of secondary steady-state flows. It is constructed explicitly by an expansion that when the Reynolds number increases across each of its critical values, a unique steady-state solution bifurcates from the basic flow along each normal vector of the two-dimensional spectral space. Thus, at a single Reynolds number supercritical value, the bifurcating steady-state solutions arising from the basic solution form a circle.

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