FIG. 2.
(color online). Results of the continuum hydrodynamic model in the overdamped limit using c(z) = A + (z/Lz)2, Lz = 1, α = 1. (a) Three regimes of narrow-channel flows: symmetric (1), asymmetric (2), and periodic (3) flows; (b) Bifurcation diagram of the narrow-channel flows in the (A, γ) plane. Symbol S denotes symmetric flow, A, asymmetric, and O, oscillatory flow regimes, double symbols (A/S, O/S) indicate domains of bistability according to local bifurcation analysis. Points 1,2,3 correspond to time series shown in (a). (c) Streaming instability domain in the parameter plane (A, γ) for different values of viscosity coefficient μ. Streaming for a given μ occurs for A below the corresponding curve; (d) Velocity field for the eigenfunction near the onset of a streaming instability, with wave vector k = 2, γ = 1, A = 0.045, μ = 0.001. Since the flow is inverted for π/4 ≤ x ≤ 3π/4, only half the x period is displayed.