Theory and computation of the stability of shear flows over compliant boundaries
File(s)
Author(s)
Chotai, Avni
Type
Thesis
Abstract
Thread annular injection is a minimally invasive technique that entails transporting medical implants into the body via a thread moving through a fluid. It is desirable for this flow to remain laminar so as to ensure that the flow remains predictable and the thread does not suffer any lateral deviations. It is thus of practical interest to determine the range of Reynolds numbers for which this flow is stable. This flow can be modelled by annular Poiseuille-Couette flow (APCF), which is the flow driven by an axial pressure gradient through the annular region between a stationary outer cylinder and a sliding inner cylinder.
In this thesis, the linear stability properties of APCF to infinitesimal, axisymmetric disturbances are studied when the inner cylinder possesses a degree of flexibility. A cylindrical version of the Orr-Sommerfeld equation is derived with appropriate boundary conditions that encompass the compliance of the cylinder. This forms the foundation of our numerical studies at finite Reynolds numbers. It is found that there exist modes of instabilities that are not present in the case of a rigid inner cylinder.
At large Reynolds numbers, an asymptotic approach is used to gain insights into the different physical balances that give rise to neutrally stable modes. Distinguished scalings are found, including those that have no counterpart for a rigid inner cylinder. These asymptotic results are compared to those from our numerical studies.
The inviscid linear stability of this problem is also studied, and analogues to classical inviscid theorems for planar flow over rigid boundaries are provided.
In the final chapter of this thesis, our stability analysis focuses on vortex-wave interaction for planar Couette flow when the lower wall is modelled as compliant. The nonlinear equations governing this interaction are solved numerically, and finite-amplitude solutions are found.
In this thesis, the linear stability properties of APCF to infinitesimal, axisymmetric disturbances are studied when the inner cylinder possesses a degree of flexibility. A cylindrical version of the Orr-Sommerfeld equation is derived with appropriate boundary conditions that encompass the compliance of the cylinder. This forms the foundation of our numerical studies at finite Reynolds numbers. It is found that there exist modes of instabilities that are not present in the case of a rigid inner cylinder.
At large Reynolds numbers, an asymptotic approach is used to gain insights into the different physical balances that give rise to neutrally stable modes. Distinguished scalings are found, including those that have no counterpart for a rigid inner cylinder. These asymptotic results are compared to those from our numerical studies.
The inviscid linear stability of this problem is also studied, and analogues to classical inviscid theorems for planar flow over rigid boundaries are provided.
In the final chapter of this thesis, our stability analysis focuses on vortex-wave interaction for planar Couette flow when the lower wall is modelled as compliant. The nonlinear equations governing this interaction are solved numerically, and finite-amplitude solutions are found.
Version
Open Access
Date Issued
2022-09
Date Awarded
2023-03
Copyright Statement
Creative Commons Attribution NonCommercial Licence
License URL
Advisor
Walton, Andrew
Sponsor
Engineering and Physical Sciences Research Council (EPSRC)
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)