Abstract:

Simple shear flows such as plane Couette
flows are known to be linearly stable for all Reynolds numbers.
If more than one immiscible fluids are present there can an
interfacial instability that produces traveling waves or more
complex nonlinear dynamics such as spatiotemporal chaos. The
instabilities require non-zero Reynolds numbers and have been
reported in experiments.

We have managed to describe these using
multiscale asymptotic analysis and agreement with both direct
numerical simulations and experiments is very good. When
multiple layers are present (applications include coating flows)
there are now at least two free interfaces. Asymptotic solutions
will be presented that yield a system of coupled partial
differential equations for the interfacial positions.

The equations are parabolic with fourth
order diffusion when surface tension is presentor second order
diffusion when surface tension is absent and the fluids are
stably stratified.

The equations generically support
instabilities even at zero Reynolds numbers. These emerge
physically from an interaction between the interfaces and
manifest themselves mathematically through hyperbolic to
elliptic transitions of the fluxes 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 (i.e. the initial conditions are in the
hyperbolic region of the flux function) 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 will consider fully nonlinear deformations in the
large surface tension limit to derive coupled Benney type
equations. Their fluxes also

support hyperbolic-elliptic transitions
and numerical solutions will be described giving rise to
intricate nonlinear traveling waves. Transitional flows are
harder to find than in

the weakly nonlinear models, but examples
will be given where linearly stable initial conditions
transition into elliptic regions to sustain energy growth and
saturation to nonlinear states.