Abstract
The “spatial dynamics” approach is applied to the analysis of bifurcations of the three-dimensional Poiseuille flow between parallel plates. In contrast to the classical studies, we impose time periodicity as well as spatial periodicity with period 2π/α in the streamwise direction. However, we make no assumptions on the behavior in the spanwise direction, except the uniform closeness of the bifurcating solution to the basic flow. In an abstract setting it is shown how the dimension of the critical eigenspace of the spatial dynamics analysis can be uniquely determined from the classical linear stability problem. For the three-dimensional Poiseuille problem we are able to find all relevant coefficients from the analysis of the purely two-dimensional problem. Moreover, we are able to analyze precisely the influence of a spanwise pressure gradient and the associated spanwise mass flux. The study of the reduced problem shows that there are two different kinds of solutions (spirals and ribbons) which are 2αp/β periodic in the spanwise direction, as in the Couette-Taylor problem, and both of them bifurcate in the same direction.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. L. Afendikov, K. I. Babenko & S. P. Yuriev [1982]: On the bifurcation of Couette flow between counter-rotating cylinders in the case of the double eigenvalue. Soviet Phys. Dokl. 27, 706–709.
A. L. Afendikov & K. I. Babenko [1986]: Bifurcation in the presence of the symmetry group and loss of stability of some plane viscous fluid flows. Soviet Math. Dokl. 33, 742–747.
A. L. Afendikov & V. P. Varin [1991]: An analysis of periodic flows in the vicinity of the plane Poiseuille flow. Europ. J. Mech B/Fluids 10, 577–603.
K. I. Babenko [1986]: Foundation of Numerical Analysis. Moscow: Nauka (Russian).
T. J. Bridges [1989]: The Hopf bifurcation with symmetry for the Navier-Stokes equations in (L p(Ω))n, with applications to plane Poiseuille flow. Arch. Rational Mech. Anal. 106, 335–376.
T. J. Bridges [1994]: Erratum to Bridges [1989]. In preparation.
T. S. Chen & D. D. Joseph [1973]: Subcritical bifurcation of the plane Poiseuille flow. J. Fluid Mech. 58, 337–351.
P. Chossat & G. Iooss [1985]: Primary and secondary bifurcations in the Couette-Taylor problem. Japan J. Appl. Math. 2, 37–68.
P. Chossat & G. Iooss [1994]: The Couette-Taylor Problem. New York: Springer-Verlag.
P. Collet & J.-P. Eckmann [1990]: The time dependent amplitude equation for the Swift-Hohenberg problem. Comm. Math. Physics 132, 139–153.
A. Davey, L. M. Hocking & K. Stewartson [1974]: On the nonlinear evolution of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 63, 529–536.
M. Dhanak [1983]: On certain aspects of the 3-D instability of parallel flows. Proc. Royal Soc. London Ser. A 385, 53–84.
U. Ehrenstein & W. Koch [1991]: Three-dimensional wavelike equilibrium states in plane Poiseuille flow. J. Fluid Mech. 288, 111–148.
M. Golubitsky & I. Stewart [1986]: Symmetry and stability in Couette-Taylor flow. SIAM J. Math. Anal. 17, 249–288.
M. Golubitsky, I. Stewart & D. Schaeffer [1988]: Singularities and groups in bifurcation theory. Vol. II, New York: Springer-Verlag.
G. Iooss [1984]: Bifurcation and transition to turbulence in hydrodynamics. In Bifurcation theory and applications, E. Salvatory, ed., Lecture Notes in Math. 1057, Springer-Verlag, 152–201.
G. Iooss & M. Adelmeyer [1992]: Topics in Bifurcation Theory and Applications. Singapore: World Scientific.
G. Iooss, A. Mielke & Y. Demay [1989]: Theory of the steady Ginzburg-Landau equation in hydrodynamic stability problems. Europ. J. Mech. B/Fluids 8, 229–268.
G. Iooss & A. Mielke [1991]: Bifurcating time periodic solutions of Navier-Stokes equations in infinite cylinders. J. Nonlinear Science 1, 106–146.
G. Iooss & A. Mielke [1992]: Time-periodic Ginzburg-Landau equations for one dimensional patterns with large wave length. Z. angew. Math. Phys. 43, 125–138.
V. I. Iudovich [1966]: On the origin of convection. Prikl. Mat. Mek. 30, 1000–1005 (translation pp. 1193–1199).
T. Kato [1966]: Perturbation Theory for Linear Operators. Springer-Verlag, Berlin.
K. Kirchgässner [1982]: Wave solutions of reversible systems and applications. J. Diff. Eqns. 45, 113–127.
O. Ladyzhenskaya [1963]: The Mathematical Theory of Viscous Incompressible Flow. New-York: Gordon and Breach.
A. Mielke [1988]: Reduction of quasilinear elliptic equations in cylindrical domains with applications. Math. Meth. Appl. Sciences 10, 51–66.
L. Ovsiannikov [1982]: Group Analysis of Differential Equations. New York: Academic Press.
G. Schneider [1994A]: A new estimate for the Ginzburg-Landau approximation on the real axis. J. Nonlinear Science 4, 23–34.
G. Schneider [1994B]: Error estimates for the Ginzburg-Landau approximation. Zeits. angew. Math. Physik. 45, 433–457.
H. B. Squire [1933]: On the stability of the three-dimensional disturbances of viscous flow between parallel walls. Proc. Roy. Soc. London Ser. A 142, 621–628.
A. van Harten [1991]: On the validity of Ginzburg-Landau's equation. J. Nonlinear Science 1, 397–422.
Author information
Authors and Affiliations
Additional information
Communicated by M. Golubitsky
Rights and permissions
About this article
Cite this article
Afendikov, A., Mielke, A. Bifurcations of Poiseuille flow between parallel plates: Three-dimensional solutions with large spanwise wavelength. Arch. Rational Mech. Anal. 129, 101–127 (1995). https://doi.org/10.1007/BF00379917
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00379917