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On the Stability of Shear Flows in Bounded Channels, II: Non-monotonic Shear Flows
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Tobias Ekholm (Ed.)We prove nonlinear asymptotic stability of a large class of monotonic shear flows among solutions of the 2D Euler equations in the channel $$\mathbb{T}\times[0,1]$$. More precisely, we consider shear flows $(b(y),0)$ given by a function $$b$$ which is Gevrey smooth, strictly increasing, and linear outside a compact subset of the interval $(0,1)$ (to avoid boundary contributions which are incompatible with inviscid damping). We also assume that the associated linearized operator satisfies a suitable spectral condition, which is needed to prove linear inviscid damping. Under these assumptions, we show that if $$u$$ is a solution which is a small and Gevrey smooth perturbation of such a shear flow $(b(y),0)$ at time $t=0$, then the velocity field $$u$$ converges strongly to a nearby shear flow as the time goes to infinity. This is the first nonlinear asymptotic stability result for Euler equations around general steady solutions for which the linearized flow cannot be explicitly solved.more » « less
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Aguirre, M.A.; Luding, S.; Pugnaloni, L.A.; Soto, R. (Ed.)Particle segregation in geophysical and industrial granular flows is typically driven by gravity and shear. While gravity-induced segregation is relatively well understood, shear-induced segregation is not. In particular, what controls segregation in the absence of gravity and the interplay between shearand gravity-driven segregation remain unclear. Here, we explore the shear-induced segregation force on an intruder particle in controlled-velocity granular flows where the shear profile is systematically varied. The shear-induced segregation force is found to be proportional to the shear rate gradient, which effectively pushes the large intruder from lower to higher shear rate regions. A scaling law is developed for the segregation force that is accurate over a wide range of overburden pressures and shear rates, and hence inertial numbers.more » « less
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s10013-023-00661-z
