Two-layer shear flows in the presence of slippage at the liquid-liquid interface
File(s)
Author(s)
Katsiavria, Anna
Type
Thesis
Abstract
Two immiscible and Newtonian superposed fluid layers are sheared in a plane Couette
flow configuration in two dimensions. Slippage is allowed to occur at the liquid-liquid
interface and it is described through a Navier-slip type boundary condition, whilst absence
of slip and impermeability are imposed at the solid walls.
After the mathematical model is constructed, two approaches are followed. First, the
linear stability is addressed asymptotically for the limits of long-waves, short-waves and
large slip. Slip is found capable of destabilising otherwise stable long-waves and make
short-waves more unstable in the absence of surface tension. Infinite slip gives rise to
a novel instability, viscous analogue of the Kelvin-Helmholtz but with bounded growth
rate. To account for arbitrary values of the parameters, analytical calculations are also
performed and used for numerical purposes. Through them it is found that in addition to
the above effects, slip also induces a Turing type instability by destabilisation of a band of
finite wavenumbers. An extensive parametric study is performed to identify the role of the
different parameters, i.e. viscosity ratio, surface tension, Reynolds number and interfacial
depth.
For the second approach, one of the layers is considered asymptotically thin and a
weakly non-linear equation for the evolution of the interface is derived. The novel form of
the non-local term coupling the two layers creates a novel type of PDEs, where dispersion
is also introduced in the system. A similar combination of analytical and numerical
calculations as before enables a parametric study, where slip is found to both stabilise
systems that would exhibit instability in its absence and induce a Turing-type instability
for small surface tension.
For appropriate matching of the parameters the two approaches are linked exhibiting
remarkable agreement in describing the behaviour of the system in the case where one of
the layers is very thin.
flow configuration in two dimensions. Slippage is allowed to occur at the liquid-liquid
interface and it is described through a Navier-slip type boundary condition, whilst absence
of slip and impermeability are imposed at the solid walls.
After the mathematical model is constructed, two approaches are followed. First, the
linear stability is addressed asymptotically for the limits of long-waves, short-waves and
large slip. Slip is found capable of destabilising otherwise stable long-waves and make
short-waves more unstable in the absence of surface tension. Infinite slip gives rise to
a novel instability, viscous analogue of the Kelvin-Helmholtz but with bounded growth
rate. To account for arbitrary values of the parameters, analytical calculations are also
performed and used for numerical purposes. Through them it is found that in addition to
the above effects, slip also induces a Turing type instability by destabilisation of a band of
finite wavenumbers. An extensive parametric study is performed to identify the role of the
different parameters, i.e. viscosity ratio, surface tension, Reynolds number and interfacial
depth.
For the second approach, one of the layers is considered asymptotically thin and a
weakly non-linear equation for the evolution of the interface is derived. The novel form of
the non-local term coupling the two layers creates a novel type of PDEs, where dispersion
is also introduced in the system. A similar combination of analytical and numerical
calculations as before enables a parametric study, where slip is found to both stabilise
systems that would exhibit instability in its absence and induce a Turing-type instability
for small surface tension.
For appropriate matching of the parameters the two approaches are linked exhibiting
remarkable agreement in describing the behaviour of the system in the case where one of
the layers is very thin.
Version
Open Access
Date Issued
2023-06
Date Awarded
2023-09
Copyright Statement
Creative Commons Attribution NonCommercial Licence
License URL
Advisor
Papageorgiou, Demetrios
Sponsor
Engineering and Physical Sciences Research Council
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)