Systems of nonlinear PDEs arising in multilayer channel flows
File(s)
Author(s)
Papaefthymiou, Evangelos
Type
Thesis
Abstract
This thesis presents analysis and computations of systems of nonlinear partial differential
equations (PDEs) modelling the dynamics of three stratified immiscible viscous layers
flowing inside a channel with parallel walls inclined to the horizontal. The three layers are
separated by two fluid-fluid interfaces that are free to evolve spatiotemporally and nonlinearly when the flow becomes unstable. The determination of the flow involves solution
of the Navier-Stokes in domains that are changing due to the evolution of the interfaces
whose position must be determined as part of the solution, providing a hard nonlinear
moving boundary problem. Long-wave approximation and a weakly nonlinear analysis
of the Navier-stokes equations along with the associated boundary conditions, leads to
reduced systems of nonlinear PDEs that in general form are systems of coupled Kuramoto-
Sivashinsky equations. These physically derived coupled systems are mathematically rich
due to the rather generic presence of coupled nonlinearities that undergo hyperbolic-elliptic
transitions, along with high order dissipation. Analysis and numerical computations of the
resulting coupled PDEs is presented in order to understand the stability of multilayer channel
flows and explore and quantify the different types of underlying nonlinear phenomena
that are crucial in applications. Importantly, it is found that multilayer flows can be unstable
even at zero Reynolds numbers, in contrast to single interface problems. Furthermore,
the thesis investigates the dynamical behaviour of the zero viscosity limits of the derived
systems in order to verify their physical relevance as reduced models. Strong evidence of
the existence of the zero viscosity limit is provided for mixed hyperbolic-elliptic type systems
whose global existence is an open and challenging mathematical problem. Finally, a
novel sufficient condition is derived for the occurrence of hyperbolic-elliptic transitions in
general conservation laws of mixed type; the condition is demonstrated for several physical systems that have been studied in the literature.
equations (PDEs) modelling the dynamics of three stratified immiscible viscous layers
flowing inside a channel with parallel walls inclined to the horizontal. The three layers are
separated by two fluid-fluid interfaces that are free to evolve spatiotemporally and nonlinearly when the flow becomes unstable. The determination of the flow involves solution
of the Navier-Stokes in domains that are changing due to the evolution of the interfaces
whose position must be determined as part of the solution, providing a hard nonlinear
moving boundary problem. Long-wave approximation and a weakly nonlinear analysis
of the Navier-stokes equations along with the associated boundary conditions, leads to
reduced systems of nonlinear PDEs that in general form are systems of coupled Kuramoto-
Sivashinsky equations. These physically derived coupled systems are mathematically rich
due to the rather generic presence of coupled nonlinearities that undergo hyperbolic-elliptic
transitions, along with high order dissipation. Analysis and numerical computations of the
resulting coupled PDEs is presented in order to understand the stability of multilayer channel
flows and explore and quantify the different types of underlying nonlinear phenomena
that are crucial in applications. Importantly, it is found that multilayer flows can be unstable
even at zero Reynolds numbers, in contrast to single interface problems. Furthermore,
the thesis investigates the dynamical behaviour of the zero viscosity limits of the derived
systems in order to verify their physical relevance as reduced models. Strong evidence of
the existence of the zero viscosity limit is provided for mixed hyperbolic-elliptic type systems
whose global existence is an open and challenging mathematical problem. Finally, a
novel sufficient condition is derived for the occurrence of hyperbolic-elliptic transitions in
general conservation laws of mixed type; the condition is demonstrated for several physical systems that have been studied in the literature.
Version
Open Access
Date Issued
2014-07
Date Awarded
2014-10
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
Papageorgiou, Demetrious
Pavliotis, Grigoris
Sponsor
Engineering and Physical Sciences Research Council
Imperial College London
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)