We consider N oscillators coupled by a mean field as in the Winfree model. The model is governed by two parameters: the coupling strength κ and the spectrum width γ of the frequencies of each oscillator centered at 1. In the uncoupled regime, κ = 0, each oscillator possesses its own natural frequency, and the difference between the phases of any two oscillators grows linearly in time. In the zero-width regime for the spectrum, the oscillators are simultaneously in the death state if and only if κ is above some positive value κ * . We say that N oscillators are synchronized if the difference between any two phases is uniformly bounded in time. We identify a new hypothesis for the existence of synchronization. The domain in (γ, κ) of synchronization contains {0} × [0, κ * ] in its closure. Moreover the domain is independent of the number of oscillators and the distribution of the frequencies. We show numerically that the above hypothesis is necessary for the existence of synchronization.
In this article we prove the stability of some mean field systems similar to the Winfree model in the synchronized state. The model is governed by the coupling strength parameter κ and the natural frequency of each oscillator. The stability is proved independently of the number of oscillators and the distribution of the natural frequencies.The main result is proved using the positive invariant cone method for the linearized system. This method can be applied to other mean field models as in the Kuramoto model.
This paper presents a quantitative visualization study and a theoretical analysis of two-phase flow relevant to polymer electrolyte membrane fuel cells (PEMFCs) in which liquid water management is critical to performance. Experiments were conducted in an air-flow microchannel with a hydrophobic surface and a side pore through which water was injected to mimic the cathode of a PEMFC. Four distinct flow patterns were identified: liquid bridge (plug), slug/plug, film flow, and water droplet flow under small Weber number conditions. Liquid bridges first evolve with quasi-static properties while remaining pinned; after reaching a critical volume, bridges depart from axisymmetry, block the flow channel, and exhibit lateral oscillations. A model that accounts for capillarity at low Bond number is proposed and shown to successfully predict the morphology, critical liquid volume and evolution of the liquid bridge, including deformation and complete blockage under specific conditions. The generality of the model is also illustrated for flow conditions encountered in the manipulation of polymeric materials and formation of liquid bridges between patterned surfaces. The experiments provide a database for validation of theoretical and computational methods.
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