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Local and global instabilities of flow in a flexible-walled channel
Authors:Peter S Stewart  Sarah L Waters  Oliver E Jensen
Institution:1. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK;2. Oxford Centre for Industrial and Applied Mathematics, University of Oxford, 24–29 St. Giles'', Oxford, OX1 3LB, UK;1. Dipartimento di Matematica, Politecnico di Milano, P.za L. da Vinci 32, I-20133 Milano, Italy;2. Dipartimento di Matematica, Università di Milano, Via C. Saldini 50, I-20133 Milano, Italy;3. Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Italy;1. Vascular and Endovascular Surgery Unit, University Hospital IRCCS San Martino-IST, University of Genova, Genova, Italy;2. Department of Civil, Environmental and Architectural Engineering, University of Genova, Genova, Italy;1. Molecular & Cell Biophysics Laboratory;2. Graduate Group in Pharmacological Science, University of Pennsylvania, Philadelphia, PA;3. Divison of Hematology, The Children''s Hospital of Philadelphia, Philadelphia, PA;4. Department of Pediatrics, Perelman School of Medicine at the University of Pennsylvania, Philadelphia, PA
Abstract:We consider laminar high-Reynolds-number flow through a long finite-length planar channel, where a segment of one wall is replaced by a massless membrane held under longitudinal tension. The flow is driven by a fixed pressure difference across the channel and is described using an integral form of the unsteady boundary-layer equations. The basic flow state, for which the channel has uniform width, exhibits static and oscillatory global instabilities, having distinct modal forms. In contrast, the corresponding local problem (neglecting boundary conditions associated with the rigid parts of the system) is found to be convectively, but not absolutely, unstable to small-amplitude disturbances in the absence of wall damping. We show how amplification of the primary global oscillatory instability can arise entirely from wave reflections with the rigid parts of the system, involving interacting travelling-wave flutter and static-divergence modes that are convectively stable; alteration of the mean flow by oscillations makes the onset of this primary instability subcritical. We also show how distinct mechanisms of energy transfer differentiate the primary global mode from other modes of oscillatory instability.
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