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The energetics of flow through a rapidly oscillating tube. I: General theory. (English) Zbl 1189.76133
Summary: We examine the effect of prescribed wall-driven oscillations of a flexible tube of arbitrary cross-section, through which a flow is driven by prescribing either a steady flux at the downstream end or a steady pressure difference between the ends. A large-Womersley-number large-Strouhal-number regime is considered, in which the oscillations of the wall are small in amplitude, but sufficiently rapid to ensure viscous effects are confined to a thin boundary layer. We derive asymptotic expressions for the flow fields and evaluate the energy budget. A general result for the conditions under which there is zero net energy transfer from the flow to the wall is provided. This is presented as a critical inverse Strouhal number (a dimensionless measure of the background flow rate) which is expressed only in terms of the tube geometry, the fluid properties and the profile of the prescribed wall oscillations. Our results identify an essential component of a fundamental mechanism for self-excited oscillations in three-dimensional collapsible tube flows, and enable us to assess how geometric and flow properties affect the stability of the system.

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
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[1] DOI: 10.1017/S0022112008000293 · Zbl 1151.76455 · doi:10.1017/S0022112008000293
[2] DOI: 10.1017/S0022112003003987 · Zbl 1034.76020 · doi:10.1017/S0022112003003987
[3] DOI: 10.1017/S0022112005004295 · Zbl 1074.76016 · doi:10.1017/S0022112005004295
[4] DOI: 10.1063/1.2824401 · Zbl 1182.76359 · doi:10.1063/1.2824401
[5] DOI: 10.1017/S002211200300394X · Zbl 1049.76015 · doi:10.1017/S002211200300394X
[6] Heil, Flow Past Highly Compliant Boundaries and in Collapsible Tubes pp 15– (2003) · doi:10.1007/978-94-017-0415-1_2
[7] Hinch, Perturbation Methods (1991) · doi:10.1017/CBO9781139172189
[8] DOI: 10.1146/annurev.fluid.36.050802.121918 · Zbl 1081.76063 · doi:10.1146/annurev.fluid.36.050802.121918
[9] DOI: 10.1017/S0022112006009220 · Zbl 1156.76320 · doi:10.1017/S0022112006009220
[10] DOI: 10.1063/1.2186673 · Zbl 1185.76576 · doi:10.1063/1.2186673
[11] DOI: 10.1007/s00348-005-0029-1 · doi:10.1007/s00348-005-0029-1
[12] DOI: 10.1063/1.1511545 · Zbl 1185.76127 · doi:10.1063/1.1511545
[13] DOI: 10.1017/S0022112008000463 · Zbl 1151.76418 · doi:10.1017/S0022112008000463
[14] DOI: 10.1017/S0022112005007214 · Zbl 1082.74015 · doi:10.1017/S0022112005007214
[15] DOI: 10.1063/1.2890790 · Zbl 1182.76087 · doi:10.1063/1.2890790
[16] Bertram, Flow Past Highly Compliant Boundaries and in Collapsible Tubes pp 51– (2003) · doi:10.1007/978-94-017-0415-1_3
[17] DOI: 10.1017/S0022112008002012 · Zbl 1145.76355 · doi:10.1017/S0022112008002012
[18] Aris, Vectors, Tensors, and the Basic Equations of Fluid Mechanics (1962)
[19] Whittaker, J. Fluid Mech. 648 pp 123– (2010)
[20] DOI: 10.1017/S0022112006002023 · Zbl 1106.76035 · doi:10.1017/S0022112006002023
[21] DOI: 10.1016/j.euromechflu.2009.03.002 · Zbl 1167.76329 · doi:10.1016/j.euromechflu.2009.03.002
[22] DOI: 10.1093/qjmam/29.3.365 · Zbl 0359.76027 · doi:10.1093/qjmam/29.3.365
[23] DOI: 10.1093/qjmam/29.3.343 · Zbl 0359.76026 · doi:10.1093/qjmam/29.3.343
[24] DOI: 10.1017/S0022112088001223 · doi:10.1017/S0022112088001223
[25] DOI: 10.1017/S0022112085003512 · doi:10.1017/S0022112085003512
[26] DOI: 10.1017/S0022112096000286 · Zbl 0875.76264 · doi:10.1017/S0022112096000286
[27] DOI: 10.1007/BF00644490 · doi:10.1007/BF00644490
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