Summary: There exists a critical Reynolds number (at which a linear instability first appears) for an incompressible fluid flowing in a channel with compliant walls [F. D. Hains and J. F. Price, Phys. Fluids, 5, 365ff. (1962)]. It is proven that, for fixed nondimensionalized wall parameters, to any unstable disturbance in three dimensions there corresponds an unstable disturbance in two dimensions at a lower Reynolds number. Consequently, the Ginzburg-Landau equation is used to study the weakly nonlinear two-dimensional evolution of a disturbance in a channel with compliant walls for Reynolds number near its critical value. The coefficients of this equation are found by numerically integrating solutions of the Orr-Sommerfeld equation and its adjoint as well as solutions of the perturbation equations.
For rigid walls the finite amplitude two-dimensional plane wave solution that bifurcates from laminar Poiseuille flow at the critical Reynolds number is itself unstable to two-dimensional disturbances. It is found that for sufficiently compliant walls this solution is stable to disturbances of the same type. The formalism developed by M. J. Landman [Stud. Appl. Math., 76, 187-237 (1987; Zbl 0626.76049)] is used to study a class of quasi-steady solutions to the Ginzburg-Landau equation. This class includes soltions describing a transition from the laminar solution to finite amplitude states and nonperiodic, “chaotic” attracting sets. It is shown that for sufficiently compliant walls the transition solutions persist while the “chaotic” ones do not.