Flow-induced oscillations of fluid-conveying elastic vessels arise in many engineering and biomechanical systems. Examples include pipe flutter, wheezing during forced expiration from the pulmonary airways, and the development of Korotkoff sounds during blood pressure measurement by sphygmomanometry (Heil & Jensen 2003; Grotberg & Jensen 2004; Heil & Hazel 2011).

Experimental studies of flow in elastic-walled tubes are typically performed with a Starling resistor. A finite-length elastic tube is mounted on two rigid tubes and flow is driven through the system either by imposing the flow rate (using a volumetric pump) or by applying a fixed pressure drop between the far-upstream and far-downstream ends of the two rigid tubes. The collapsible segment is contained inside a pressure chamber which allows the external pressure acting on the elastic tube to be controlled independently of the fluid pressure. If the transmural (external minus internal) pressure becomes sufficiently large, the elastic tube buckles non-axisymmetrically. Once buckled, the tube is very flexible and, as a result, small changes in transmural pressure suffice to induce large changes in the tube shape, resulting in strong fluid-structure interaction. Experiments show that the elastic tube segment has a propensity to develop large-amplitude self-excited oscillations of great complexity when the flow rate is increased beyond a certain value (Bertram 2003).

I am currently studying a particular class of high-frequency instability of flow through a collapsible tube. The basic setup comprises an elastic-walled tube along which a steady axial flow is driven either by a prescribed pressure drop between the ends, or by a flux condition at one end. The aim is then to consider self-excited oscillations of this system involving bother the fluid and the tube walls. The mechanism I am looking at was first proposed by Jensen & Heil (2003) for 2D channel flows, and I have shown that it persists in 3D tubes. (The mechanism involves the extraction of kinetic energy from the mean flow via a larger oscillatory velocity perturbation at the upstream end than at the downstream end.)

I have been carrying out theoretical work in collaboration with Sarah Waters at Oxford and Oliver Jensen at Nottingham. Another team in Manchester, lead by Matthias Heil have been carrying out numerical simulations with which we compare our results.

As a first step in analysing the instability, I have used asymptotic analysis to derive leading order solutions for the fluid flow in response to prescribed oscillations of the tube wall. Changes in cross-sectional area result in an axial sloshing flow in addition to the mean flow.

Asymptotic analysis of the fluid problem reveals the energy budget for the system and shows that the instability mechanism seen in 2D channel flows persists in 3D. A general stability criterion is derived in terms of the oscillation mode shape, original cross-section, and the physical parameters of the system (Whittaker et al. 2010a). The theoretical predictions are found to be in good agreement with numerical simulations (Whittaker et al. 2010b).

The next stage in the work is to develop a simple model to capture
the wall mechanics. I have derived a so-called *tube law*
linking the cross-sectional area changes to the transmural pressure
for small-amplitude long-wavelength deformations. We start by deriving
a rational approximation to a full shell-theory description, which
consists of an infinite set of coupled ODEs in an infinite set of
axially varying amplitude parameters
`b`_{n}(`z`). Truncating after
the first term yields a simple tube law which is a good approximation
to the full solution (Whittaker et al. 2010b).

Finally, we combine the solid and fluid mechanics descriptions to
consider the full fluid–structure interaction problem. We
determine the shapes of the normal modes of oscillation, and the
growth or decay rate of each mode as a function of the problem
parameters. Our theoretical predictions are in good agreement with
numerical simulations conducted using
the `oomph-lib`
C++ library (Whittaker et
al. 2010d; Whittaker et al.2011).

A PhD student, Martin Walters, examined the effect of inertia in the tube wall on the stability and growth of self-excited oscillations. Since the wall inertia is associated with the pressure in each cross-section, rather than the axial pressure gradient, the additional terms from wall inertia appear in a different place in the equations from the terms describing the dominant (axial) fluid inertia.

The tube law described above is only second-order in the axial
coordinate `z`, and hence solutions are incapable of
satisfying the full set of 'clamped' boundary conditions that should
be applied where an elastic tube is joined to a rigid
section. Higher axial derivatives were neglected during the
derivation as they were asymptotically small on the axial
length-scale of the whole system. However, we would expect these
terms to re-enter on shorter axial length-scales, and so give rise
to boundary layers close to the ends of the elastic tube. Work by
myself and my PhD Student Martin Walters has revealed a rich variety
of such boundary layers, involving in-plane shearing, axial bending,
and transverse shearing. Which boundary layers are needed, and their
thicknesses depend on the key ratios of the wall thickness to tube
diameter `δ`, and the axial tension to bending
stiffness `F`.

The initial tube-law did not even allow the canonical 'pinned'
boundary conditions to be satisfied at the tube ends. This was found
to cause discrepancies between the model and numerical simulations
when `δ`^{2}`F` was small. The
discrepancies were found to be caused by the neglect of in-plane
shear forces, rather than axial bending. A shear-relaxation boundary
layer was found and solved asymptotically to allow the pinned
boundary conditions to be
satisfied. (Whittaker 2015).

Work by Martin Walters has examined the boundary layers required to allow the full set of clamped conditions to be satisfied at the tube ends. Various regimes have been discovered with either axial bending or normal-shear effects included.

The work above has used a tube law to describe the elastic response of the tube wall, which has been linearised for small-amplitude deformations about an axially uniform initial state. Even if the oscillations take place about a slightly non-uniform mean state, the linearisation means that certain effects are neglected. I am currently working with a postdoctoral research assistant, Dr Thomas Ward, on an EPSRC-funded project to consider the inclusion of these effects.

Specifically, we are investigating what happens in the case of
O(`ε`) amplitude oscillations about an
O(`δ`) steady deformation, in the case
where `ε` ≪ `δ` ≪ 1. Here
additional terms appear in the tube law due to axial stretching
arising from the curvature of the mean state.

For a 2D channel, the extra term in the tube law is a dimensionless
parameter `K`, times an integral `I` times the
dimensionless axial curvature
perturbation `η`_{zz}. The
integral `I` is the product of the dimensionless base-state
axial curvature and the dimensionless normal displacements, which
represents the increase in axial tension due changes in axial
extension. The dimensionless parameter `K` gives the size
of this non-linear effect relative to the other restoring forces in
the tube law, and involves the mean-deformation
amplitude `δ`, the extensional-to-bending-stiffness
ratio `K`/`D`=`h`^{2}/12 of the
wall, and the aspect
ratio `ℓ`=`L`/`a` of the channel.

For sufficiently large `K`, the effect of the additional
term in the tube law is to force the oscillatory mode to alter its
shape so that `I`=O(`K`^{-1}). For
larger `K`, this forces the oscillatory mode to be almost
orthogonal to the base-state curvature. In the case where the
base-state curvature has an axial mode-1 shape, this forces the
primary (lowest frequency) oscillatory mode towards an axial mode-2
shape, and the secondary oscillatory mode towards an axial mode-3
shape. This in turn affects the stability properties of the modes,
and we have found a region of parameter space in which the secondary
oscillatory mode is more unstable than the primary oscillatory
mode.

- The Energetics of Flow through a Rapidly Oscillating Tube. Part 1. General Theory
- , 2010.
- Journal
of Fluid Mechanics
**648**, 83–121. - The Energetics of Flow through a Rapidly Oscillating Tube. Part 2. Application to an Elliptical Tube
- , 2010.
- Journal
of Fluid Mechanics
**648**, 123–153. - A Rational Derivation of a Tube Law from Shell Theory
- , 2010.
- Quarterly Journal of Mechanics
and Applied Mathematics
**63**(4), 465–496. - Predicting the Onset of High-Frequency Self-Excited Oscillations in Elastic-Walled Tubes
- , 2010.
- Proceedings of the
Royal Society A
**466**(2124), 3635–3657. - The Energetics of Flow through a Rapidly Oscillating Tube with Slowly Varying Amplitude
- , 2011.
- Phil.
Trans. Roy. Soc. A
**369**(1947), 2989–3006. - A Shear-Induced Boundary Layer near the Pinned Ends of a Buckled Elastic-Walled Tube
- , 2015.
- IMA
Journal of Applied Mathematics,
**80**(6) 1932–1967. - The Effect of Wall Inertia on High-Frequency Oscillations in an Elastic-Walled Tube (free access link)
- , 2017.
- Q. J. Mech. Appl. Math.,
**71**(1), 47–77.

- Fluid–Structure Interaction: Shell Theory and 3D Flow Instability in Collapsible Tubes
- , 2007.
- Conference Poster for the Fourth Physiological Flow Network Meeting, University of Manchester.
- High-Frequency Self-Excited Oscillations in Collapsible Tube Flows
- , 2009.
- Poster at the Cardiac Physiome Project Meeting, Isaac Newton Institute, University of Cambridge.

- Bending boundary layers at the Clamped Ends of a Buckled Elastic-Walled Tube
- In preparation.
- The Non-Linear Effect of Base-State Curvature on Self-Excited Oscillations in an Elastic-Walled Channel
- Submitted to Phys. Rev. Fluids.