Periodic waves in rotating plane Couette flow

Abstract

A class of nonlinear equations of Navier-Stokes type of the form

$$\frac{{dw}}{{dt}} + L_0 w + (\lambda - \lambda _0 )L_1 w + \gamma \lambda (M_1 w + M_2 w) + \gamma ^2 L_3 (\lambda ,\gamma )w + B(w) = 0$$

((**))

is investigated, where λ is a “load” parameter (i.e., a Reynolds number), γ is a “structure” parameter,L _o L ₁,M ₁,M ₂ andL ₃(λ, γ) are linear operators, andB is a quadratic operator. An equation of the form (**) describes a variety of spiral flow problems including rotating plane Couette flow which is studied here in detail. Under suitable hypotheses on the operators in (**), it is shown that Hopf bifurcation occurs for γ sufficiently small. In the problem of rotating plane Couette flow, by determining the sign of the real part of a certain “cubic” coefficient, it is shown, in addition, that the bifurcating periodic orbits are supercritical and asymptotically stable, and correspond to periodic waves.