Linear stability analysis of a multi-layered system with a throughflow
File(s)
Author(s)
Yu, Jian
Type
Thesis
Abstract
This thesis, consisting of two parts, investigates stability of a multilayered system with a thorughflow. The first part concerns the combined effects of viscous dissipation and a horizotnal throughflow on the linear stability behaviour of two homogeneous porous layers interbedded with a solid partition. The stability of one porous layer can effectively affect the stability of another porous layer.
The second problem investigates the two-dimensional linear stability of a pressure-
driven laminar flow in a fluid layer between two homogeneous and identical porous layers with a transverse throughflow. The flow stability is characterised by performing a parametric study where the Reynolds number associated with the transverse throughflow velocity, Rev, and the width of the porous layers, H1, are varied. When
Rev is beyond a critical value, Rev*, a new mode occurs with lower Reynolds number, Renew, and wavenumber, a. The new mode is at the core of this thesis.
For the new mode, its linear and weakly nonlinear stability is analysed in the limit
a ≪ 1 and Renew ≫ 1 as Rev → Rev*. The asymptotic neutral curve is obtained by solving the asymptotic problem numerically, which disappears as Rev is below Rev*. Weakly nonlinear solutions are found to bifurcate supercritically from the neutral curve when Rev is just above Rev*, but to ‘bifurcate from infinity’ just below Rev*.
We consider the large-Rev asymptotic properties of the neutral curve of the new mode. From the asymptotic analysis, we obtain the dispersion relations of the upper and lower branches of the asymptotic neutral curve respectively.
Finally, we consider the asymptote of the lower-branch of the original mode in the limit of Renew ≫ 1. Two regimes are considered in detail: Rev = O(1) and Rev = O(Renew^(1/6)), for which a = O(1). The dispersion relations are derived for these two regimes.
The second problem investigates the two-dimensional linear stability of a pressure-
driven laminar flow in a fluid layer between two homogeneous and identical porous layers with a transverse throughflow. The flow stability is characterised by performing a parametric study where the Reynolds number associated with the transverse throughflow velocity, Rev, and the width of the porous layers, H1, are varied. When
Rev is beyond a critical value, Rev*, a new mode occurs with lower Reynolds number, Renew, and wavenumber, a. The new mode is at the core of this thesis.
For the new mode, its linear and weakly nonlinear stability is analysed in the limit
a ≪ 1 and Renew ≫ 1 as Rev → Rev*. The asymptotic neutral curve is obtained by solving the asymptotic problem numerically, which disappears as Rev is below Rev*. Weakly nonlinear solutions are found to bifurcate supercritically from the neutral curve when Rev is just above Rev*, but to ‘bifurcate from infinity’ just below Rev*.
We consider the large-Rev asymptotic properties of the neutral curve of the new mode. From the asymptotic analysis, we obtain the dispersion relations of the upper and lower branches of the asymptotic neutral curve respectively.
Finally, we consider the asymptote of the lower-branch of the original mode in the limit of Renew ≫ 1. Two regimes are considered in detail: Rev = O(1) and Rev = O(Renew^(1/6)), for which a = O(1). The dispersion relations are derived for these two regimes.
Version
Open Access
Date Issued
2023-08-31
Date Awarded
2024-06-01
Copyright Statement
Attribution-NonCommercial 4.0 International Licence (CC BY-NC)
License URL
Advisor
Wu, Xuesong
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)