Mechanics and Instabilities of Fluid-Conveying Elastic-Walled Tubes

Dr. Robert Whittaker (University of Norwich)

Thursday 29th October, 2015 14:00-15:00 Maths 417


Fluid flows through elastic-walled tubes are common in biological 
and industrial systems, and have received much attention through 
experimental, numerical and analytical studies. Experiments show 
that steady flow along an elastic-walled tube can become unstable 
to large-amplitude oscillations involving both the tube wall and 
the fluid.

In this talk, I shall describe theoretical work examining a 
particular instability that leads to high-frequency oscillations 
in a long finite length of thin-walled tube. The fluid and wall 
mechanics are considered as two separate problems, coupled by the 
kinematic and dynamic boundary conditions. A relatively simple 
asymptotic model is derived, whose solution gives neutrally 
stable normal modes of oscillation at leading order. The slow 
growth or decay of these modes is determined by examining the 
energy budget of the system, which is then used to determine the 
stability of the system to these modes. A critical Reynolds 
number is found in terms of the problem parameters, which gives 
the minimum axial flow rate at which instability is observed.

In an extension to the basic problem, we examine how the addition 
of wall inertia affects the stability of the system. Wall inertia 
decreases the frequency of the normal modes, leading to a 
lowering of the critical Reynolds number. Also, the increased 
kinetic energy penalty in the wall reduces the growth rate of any 
growing modes. So wall inertial will enhance the instability at 
some points in parameter space, while reducing it at others.

We also look in more detail at the solid-mechanics modelling of 
the tube wall. The initial asymptotic model of the wall is a PDE 
that is only second-order in the axial coordinate. This does not 
allow enough degrees of freedom to satisfy the full set of 
`clamped' boundary conditions at the tube ends, though in some 
regimes this doesn't matter. Further investigation reveals a 
variety of different bending and shearing tube-end boundary 
layers in different regimes, some of which can have significant 
effects on the interior solutions for the tube wall.

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