TITLE:
Hopf bifurcations to quasi-periodic solutions for the two-dimensional
Poiseuille flow
AUTHORS:
Pablo S. Casas^(1), Angel Jorba^(2)
(1) pablo@casas.upc.es
Departament de Matematica Aplicada I,
Universitat Politecnica de Catalunya,
Diagonal 647, 08028 Barcelona (Spain).
(2) angel@maia.ub.es
Departament de Matematica Aplicada i Analisi,
Universitat de Barcelona,
Gran Via 585, 08007 Barcelona (Spain).
ABSTRACT:
This paper studies various Hopf bifurcations in the two-dimensional
Poiseuille problem. For several values of the wavenumber $\alpha$, we
obtain the branch of periodic flows which are born at the Hopf
bifurcation of the laminar flow. It is known that, taking
$\alpha\approx1$, the branch of periodic solutions has several Hopf
bifurcations to quasi-periodic orbits. For the first of them, previous
calculations seem to indicate that the bifurcating quasi-periodic
flows are stable and go backwards with respect to the Reynolds number
$\Rey$. By improving the precision of those previous works we find
that the bifurcating flows are unstable and go forward with respect to
$\Rey$. We have also analysed the second Hopf bifurcation of periodic
orbits for $\alpha\approx1$ to find again quasi-periodic solutions
with increasing $\Rey$. In this case they are stable to superharmonic
disturbances.
The proposed numerical scheme is based on a full numerical integration
of the Navier-Stokes equations and the use of suitable Poincar\'e
sections. The most intensive part of the computations has been done in
parallel. We believe that this methodology can also be applied to
similar problems.