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On steady solutions of symmetry-preserving perturbations of the two-dimensional Couette flow problem
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10.1063/1.2059571
/content/aip/journal/pof2/17/9/10.1063/1.2059571
http://aip.metastore.ingenta.com/content/aip/journal/pof2/17/9/10.1063/1.2059571

Figures

Image of FIG. 1.
FIG. 1.

Numerically computed spectrum of planar Couette flow (, ) together with the continuous spectrum in the Euler limit .

Image of FIG. 2.
FIG. 2.

Eigenvalue and bifurcation diagrams for the degenerate-Hopf normal form: (a) is the bifurcating case, (b) is the degenerate case where the eigenvalue momentarily touches the imaginary axis before retreating, and (c) is the nonbifurcating case.

Image of FIG. 3.
FIG. 3.

Bifurcation diagram for the traveling-wave solutions of the Poiseuille-Couette homotopy, where the “One” case applies.

Image of FIG. 4.
FIG. 4.

Plot of the critical Reynolds number : -perturbed Couette flow bifurcates at .

Image of FIG. 5.
FIG. 5.

Bifurcation diagram for the traveling-wave solutions of the perturbed Couette flow .

Image of FIG. 6.
FIG. 6.

Wave speed as the traveling-wave solution is continued in the homotopy parameter (for ): denote the forward (backward) traveling waves and (o) denote the frozen waves.

Image of FIG. 7.
FIG. 7.

Streamlines for the traveling-wave solutions of the perturbed Couette flow with (a) , and (b) , .

Image of FIG. 8.
FIG. 8.

Velocities for the traveling-wave solutions whose streamlines appear in Fig. 7.

Image of FIG. 9.
FIG. 9.

Boxes denote the smallest Reynolds number Re for which a bifurcating (from Reynolds number ) solution of the perturbed Couette flow was detected.

Image of FIG. 10.
FIG. 10.

Primary eigenvalue of Couette flow: the eigenvalue is real for Reynolds number and comes close to imaginary axis before temporarily retreating.

Image of FIG. 11.
FIG. 11.

Local bifurcation diagram for steady solutions of the perturbed Couette flow .

Image of FIG. 12.
FIG. 12.

Global bifurcation diagram for steady solutions of the perturbed Couette flow .

Image of FIG. 13.
FIG. 13.

Streamlines for the steady solutions of the perturbed Couette flow with (a) , , (b) , , and (c) , .

Image of FIG. 14.
FIG. 14.

Velocities for the steady solutions whose streamlines appear in Fig. 13.

Image of FIG. 15.
FIG. 15.

Linear spectral near-degeneracy for (a) real double eigenvalue closing in before retreating, (b) two pair of real eigenvalues colliding, and (c) resulting double complex eigenvalue pair closing in to imaginary axis and corresponding local nonlinear solution branches for (a) steady (o) solutions, (b) forward and backward traveling-wave solutions colliding to yield a frozen wave steady solution, and (c) distinct forward and backward traveling waves.

Image of FIG. 16.
FIG. 16.

Bifurcation diagram for steady solutions obtained using continuation in the wave-number parameter (, ).

Image of FIG. 17.
FIG. 17.

Bifurcation diagram for steady solutions obtained using continuation in the homotopy parameter for four distinct wave numbers .

Image of FIG. 18.
FIG. 18.

Bifurcation diagram for the long-wavelength steady solutions obtained using continuation in the wave-number parameter .

Image of FIG. 19.
FIG. 19.

Streamlines for the low-energy long-wavelength steady solutions with (a) , (b) , and (c) (, ).

Image of FIG. 20.
FIG. 20.

Velocities for the low-energy long-wavelength steady solutions whose streamlines appear in Fig. 19.

Image of FIG. 21.
FIG. 21.

Streamlines for the high-energy long-wavelength steady solutions with (a) , (b) , and (c) (, ).

Image of FIG. 22.
FIG. 22.

Velocities for the high-energy long-wavelength steady solutions whose streamlines appear in Fig. 21.

Image of FIG. 23.
FIG. 23.

Bifurcation diagram for the homoclinic solution obtained using continuation in the parameter (, ).

Tables

Generic image for table
Table I.

Summary of past computational work using homotopy methods for obtaining solutions of planar Couette flow.

Generic image for table
Table II.

Summary of bifurcation characteristics of the perturbed Couette flow: denotes the wave number, denotes the size of identity perturbation to induce bifurcation at Reynolds number is the Landau constant (subcritical for ).

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/content/aip/journal/pof2/17/9/10.1063/1.2059571
2005-09-26
2014-04-25
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: On steady solutions of symmetry-preserving perturbations of the two-dimensional Couette flow problem
http://aip.metastore.ingenta.com/content/aip/journal/pof2/17/9/10.1063/1.2059571
10.1063/1.2059571
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