### Interfacial instability in superposed non-Newtonian fluid layers

Superposed layers of fluid flowing down an inclined plane
are prone to interfacial instability even in the
limit of zero Reynolds number. This situation can be explored by
making use of a lubrication-style approximation of the governing
fluid equations. Two versions of the lubrication theory are presented
for superposed layers of non-Newtonian fluid with power-law rheology.
First, the fluids are assumed to have comparable effective viscosities.
The approximation then furnishes a simplified model
for which the linear stability problem can be solved analytically
and concisely. Weakly nonlinear analysis and numerical computations
indicate that instabilities saturate supercritically beyond onset
and form steady wavetrains. Further from onset, secondary instabilities
arise that destroy trains of widely spaced wave trains. Patterns
of closely spaced waves, on the other hand, coarsen due to wave merger
events. The two mechanisms select steady wavetrains with
wavelengths lying within a prescribed range.
The second lubrication theory assumes that the upper layer is far more viscous
than the lower layer. As a result, the upper fluid flows almost rigidly,
and extensional stresses can become promoted into the leading-order balance
of forces. Interfacial instability still arises in Newtonian fluid
layers, and the nonlinear dynamics is qualitatively unchanged.
Significant complications arise when the upper fluid
is non-Newtonian due to the behaviour of the viscosity at zero
strain rate.

Co-authored with N. J. Balmforth and C. Toniolo

#### Phys. Fluids, 15, 3370--3384, 2003