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addition wall  flow  frequency  high frequency  high  inertia  instability  interface  mechanics  model  rate  system  tube  wall inertia  wall 
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Preview: The Quarterly Journal of Mechanics and Applied Mathematics - Advance Access

The Quarterly Journal of Mechanics and Applied Mathematics Advance Access

Published: Wed, 15 Nov 2017 00:00:00 GMT

Last Build Date: Wed, 15 Nov 2017 03:49:49 GMT


The Effect of Wall Inertia on High-Frequency Instabilities of Flow Through an Elastic-Walled Tube


We examine the effect of wall inertia on the onset of high-frequency self-excited oscillations in flow through an elastic-walled tube. The previous asymptotic model of Whittaker et al. (Proc. Roy. Soc. A466, 2010), for a long-wavelength high-frequency instability in a Starling-resistor set-up, neglected inertia in the tube wall. Here, we extend this model by modifying the ‘tube-law’ for the wall mechanics to include inertial effects. The resulting coupled model for the fluid and solid mechanics is solved to find the normal modes of oscillation for the system, together with their frequencies and growth rates. In the system and parameter regime considered, the addition of wall inertia reduces the oscillation frequency of each mode, however its effect on the stability of the system is not as straightforward. Increasing wall inertia lowers the mean flow rate required for the onset of instability, and is therefore destabilising. However, at higher flow rates the instability growth rate is decreased, and so wall inertia is stabilising here. Overall, the addition of wall inertia decreases the sensitivity of the system to the mean axial flow rate. The theoretical results show good qualitative and reasonable quantitative agreement with direct numerical simulations performed using the oomph-lib framework.

Interfacial behaviour in two-fluid Taylor–Couette flow


The flow of a system of two viscous fluids between two concentric counter-rotating cylinders is discussed. A simple theory is presented that describes the evolution of shape of the interface between the fluids when they have near equal densities and identical viscosities. This suggests that the interface is neutrally stable, but that after sufficient time there are nevertheless points on the profile at which the curvature becomes very large. As a consequence, the interface develops cusp-like portions in its profile. A novel spectral method is developed for this problem in which the interface is represented as a region of finite width and over which the density changes rapidly but smoothly. The results confirm the general predictions of the asymptotic theory for rotation in a horizontal plane but when the rotation occurs vertically additional features develop in the flow.