Abstract
We investigate the influence of a horizontal plane Poiseuille shear flow transversal to the convective roll chain of the Rayleigh-B\'enard problem. Using a one-dimensional (1D) amplitude equation and a 2D numerical simulation of the basic field equations, we study how different boundary conditions at the inlet and outlet of the channel affect nonlinear convection. If convection is suppressed near the cell apertures, spatially localized traveling-wave states appear with a uniquely selected bulk wavelength. For convectively unstable parameters this pattern is pushed out of the channel; however, the system becomes very sensitive to perturbations, and noise-driven structures occur. Phase-pinning boundary conditions lead for very small flows to stationary roll patterns with a space-dependent wavelength decreasing downstream. Strengthening the throughflow causes local Eckhaus instabilities, which finally generate a transition to propagating rolls.
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