Instabilities in flexible channel flow with large external pressure

Abstract

We examine the stability of laminar high-Reynolds-number flow through an asymmetric flexible-walled channel driven by a fixed upstream flux and subject to a (large) uniform external pressure. We construct a long wavelength, spatially one-dimensional model using a flow profile assumption, modelling the flexible wall as a thin tensioned membrane subject to a large axial pre-stress. We numerically construct the non-uniform static shape of the flexible wall and consider its stability using both a global eigensolver and numerical simulation of the nonlinear governing equations. The system admits multiple static solutions, including a highly collapsed steady state where the membrane has a single constriction which increases with increasing external pressure. We demonstrate that the non-uniform static state is unstable to two distinct (infinite) families of normal modes which we characterise in the limit of large external pressure. In particular, there is a family of low-frequency oscillatory modes which each persist to low membrane tensions, where the most unstable mode has an oscillating membrane profile which is outwardly bulged at the centre of the domain with a narrow constriction at the downstream end. In addition, there is a family of high-frequency oscillatory modes which are each unstable beyond a critical value of the tension within a two-branch neutral curve. Unstable modes along the lower branch of the neutral curve are sustained by a leading-order balance between unsteady inertia and the restoring force of membrane tension along the channel. In addition, we elucidate the mechanism of energy transfer to sustain the self-excited oscillations: oscillations decrease the mean maximal constriction of the channel over a period, which reduces the overall dissipation of the mean flow and releases energy to sustain the instability. Fully nonlinear simulations indicate that as the Reynolds number increases these unstable normal modes can grow supercritically into sustained large amplitude 'slamming' oscillations, where the membrane is periodically drawn very close to the opposite rigid wall before recovering.

Self-Excited Oscillations in a Collapsible Channel With Applications to Retinal Venous pulsation

“…Comprehensive literature reviews are available elsewhere [13,18]. Of most pertinent interest to the present study are long-wavelength models for flow through collapsible channels with prescribed inlet flux, where the wall elasticity takes the form of an axial membrane tension [40,[49][50][51]. These models are similar in spirit to our previous model for the onset of RVp [42], albeit with only one compliant compartment rather than two.…”

“…The flow of blood was modelled using the incompressible Navier-Stokes equations, while the elastic portions of the channel wall were modelled as a tensioned membrane with an elastic pre-stress. Similar to many previous studies of flow in collapsible vessels [40,49], the flow was simplified using a long-wavelength approximation with a flow profile assumption. The ICP and IOP were chosen to be prescribed functions of time with simple oscillatory profiles.…”

“…In particular, we look for the onset of self-excited oscillations by recasting the governing equations as a linear stability problem about a nonuniform static state. In a similar manner to [40], we construct a global (linear) stability eigensolver which can be solved to isolate the critical conditions for instability with much reduced computational cost, compared to full nonlinear simulations of the same system. Such an approach provides explicit access to the corresponding oscillation frequency at neutral stability and elucidates all the normal modes of the system.…”

“…These models are similar in spirit to our previous model for the onset of RVp [42], albeit with only one compliant compartment rather than two. For a constant external pressure on the collapsible segment, we previously showed that this collapsible channel system with one flexible compartment admits two distinct families of self-excited oscillation [40]: for low tensions the system admits a family of low-frequency oscillatory modes where the flexible portion of the channel is highly inflated along most of its length; for large tensions the system admits a family of high-frequency oscillatory modes where the channel is highly collapsed. We now revisit these predictions in a model with two compliant compartments rather than one.…”

Three-dimensional flows in a hyperelastic vessel under external pressure

“…The next level of modelling is the onedimensional approach (Cancelli and Pedley 1985;Jensen 1990Jensen , 1992Shapiro 1977), which enabled wave propagation to be included in the system, leading to the "choking" mechanism. Indeed, some of these one-dimensional models are extremely useful in explaining some system mechanisms (Stewart et al 2010;Stewart 2017). A more rational two-dimensional fluid-membrane model was developed by Pedley and coworkers, in which the collapsible tube is represented by a channel with part of the upper wall replaced by a thin and inextensible membrane (Pedley and Luo 1998;Lowe and Pedley 1995;Pedley 1996, 2000).…”