# zbMATH — the first resource for mathematics

The energetics of flow through a rapidly oscillating tube. II: Application to an elliptical tube. (English) Zbl 1189.76132
Summary: In Part 1 of this work [J. Fluid Mech. 648, 83–121 (2010; Zbl 1189.76133)], we derived general asymptotic results for the three-dimensional flow field and energy fluxes for flow within a tube whose walls perform prescribed small-amplitude periodic oscillations of high frequency and large axial wavelength. In the current paper, we illustrate how these results can be applied to the case of flow through a finite-length axially non-uniform tube of elliptical cross-section - a model of flow in a Starling resistor. The results of numerical simulations for three model problems (an axially uniform tube under pressure-flux and pressure-pressure boundary conditions and an axially non-uniform tube with prescribed flux) with prescribed wall motion are compared with the theoretical predictions made in Part 1, each showing excellent agreement. When upstream and downstream pressures are prescribed, we show how the mean flux adjusts slowly under the action of Reynolds stresses using a multiple-scale analysis. We test the asymptotic expressions obtained for the mean energy transfer $$E$$ from the flow to the wall over a period of the motion. In particular, the critical point at which $$E = 0$$ is predicted accurately: this point corresponds to energetically neutral oscillations, the condition which is relevant to the onset of global instability in the Starling resistor.

##### 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.)
Full Text:
##### References:
 [1] Bertram, Flow Past Highly Compliant Boundaries and in Collapsible Tubes pp 51– (2003) [2] DOI: 10.2307/2005509 · Zbl 0271.35017 [3] DOI: 10.1016/j.euromechflu.2009.03.002 · Zbl 1167.76329 [4] DOI: 10.1103/PhysRevE.74.017301 [5] DOI: 10.1017/S0022112008000293 · Zbl 1151.76455 [6] DOI: 10.1017/S002211200300394X · Zbl 1049.76015 [7] Whittaker, J. Fluid Mech. 648 pp 83– (2010) [8] DOI: 10.1017/S0022112008000463 · Zbl 1151.76418 [9] DOI: 10.1017/S0022112005007214 · Zbl 1082.74015 [10] DOI: 10.1007/3-540-34596-5_2 · Zbl 1323.74085 [11] DOI: 10.1146/annurev.fluid.36.050802.121918 · Zbl 1081.76063 [12] DOI: 10.1017/S0022112097007313 · Zbl 0903.76029 [13] DOI: 10.1016/j.jfluidstructs.2006.07.005 [14] DOI: 10.1017/S0022112090003408 · Zbl 0708.76056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.