# Nonlinear instabilities in multilayer shear flows

Tuesday, March 13, 2018 - 10:00am - 10:50am

Lind 305

Demetrios Papageorgiou (Imperial College London)

This talk will be in two parts. The first will describe some recent results

on the stability of multilayer shear flows of three or more immiscible viscous

fluids caused to flow by gravity and/or a pressure gradient. There are now at

least two free boundaries and asymptotic solutions will be described that

yield a system of coupled partial differential equations for the interfacial positions.

Some subtleties in the weakly nonlinear asymptotics will be pointed out.

The equations generically support instabilities even at zero Reynolds numbers

and these emerge physically from a resonance between the interfaces and manifest

themselves mathematically through hyperbolic to elliptic transitions of the flux part

of the equations. We use the theory of 2x2 systems of conservation laws to derive

a nonlinear stability criterion that can tell us whether a system which is linearly stable,

can (i) become nonlinearly unstable, i.e. a large enough initial condition produces

a large time nonlinear response, or (ii) remains nonlinearly stable, i.e. the solution decays

to zero irrespective of the initial amplitude of the perturbation.

Having described weakly nonlinear solutions we turn to fully nonlinear deformations are also

in the large surface tension limit giving rise to coupled Benney type equations. Their fluxes also

support hyperbolic-elliptic transitions and numerical solutions will be described giving

rise to intricate nonlinear stable traveling waves. Differences between weakly and strongly

nonlinear flows will be described to guide ideas for theory of coupled Benney equations.

The second part is concerned with three-dimensional instabilities of electrified falling

film flows and in particular the multidimensional electrified Kuramoto-Sivashinsky

equation which is the first in a hierarchy of nonlinear models. We will mostly present

computations and end with words of caution in using 2D models instead of 3D ones

in the presence of electric fields.

on the stability of multilayer shear flows of three or more immiscible viscous

fluids caused to flow by gravity and/or a pressure gradient. There are now at

least two free boundaries and asymptotic solutions will be described that

yield a system of coupled partial differential equations for the interfacial positions.

Some subtleties in the weakly nonlinear asymptotics will be pointed out.

The equations generically support instabilities even at zero Reynolds numbers

and these emerge physically from a resonance between the interfaces and manifest

themselves mathematically through hyperbolic to elliptic transitions of the flux part

of the equations. We use the theory of 2x2 systems of conservation laws to derive

a nonlinear stability criterion that can tell us whether a system which is linearly stable,

can (i) become nonlinearly unstable, i.e. a large enough initial condition produces

a large time nonlinear response, or (ii) remains nonlinearly stable, i.e. the solution decays

to zero irrespective of the initial amplitude of the perturbation.

Having described weakly nonlinear solutions we turn to fully nonlinear deformations are also

in the large surface tension limit giving rise to coupled Benney type equations. Their fluxes also

support hyperbolic-elliptic transitions and numerical solutions will be described giving

rise to intricate nonlinear stable traveling waves. Differences between weakly and strongly

nonlinear flows will be described to guide ideas for theory of coupled Benney equations.

The second part is concerned with three-dimensional instabilities of electrified falling

film flows and in particular the multidimensional electrified Kuramoto-Sivashinsky

equation which is the first in a hierarchy of nonlinear models. We will mostly present

computations and end with words of caution in using 2D models instead of 3D ones

in the presence of electric fields.