A high-Reynolds-number asymptotic theory is developed for
linear instability waves
in a two-dimensional incompressible boundary layer on a flat surface coated
with
a thin film of a different fluid. The focus in this study is on the influence
of the
film flow on the lower-branch Tollmien–Schlichting waves,
and also on the effect of
boundary-layer/potential flow interaction on interfacial
instabilities. Accordingly, the
film thickness is assumed to be comparable to the thickness of a viscous
sublayer in a
three-tier asymptotic structure of lower-branch Tollmien–Schlichting
disturbances. A
fully nonlinear viscous/inviscid interaction formulation is derived,
and computational and analytical solutions for small disturbances are obtained
for both Tollmien–Schlichting and interfacial instabilities for a
range of
density and viscosity ratios of
the fluids, and for various values of the surface tension coefficient and
the Froude
number. It is shown that the interfacial instability contains
the fastest growing modes
and an upper-branch neutral point within the chosen flow regime if the
film viscosity
is greater than the viscosity of the ambient fluid. For a less viscous
film the theory
predicts a lower neutral branch of shorter-scale interfacial waves. The
film flow is
found to have a strong effect on the Tollmien–Schlichting
instability, the most dramatic
outcome being a powerful destabilization of the flow due to a
linear resonance between
growing Tollmien–Schlichting and decaying capillary modes. Increased
film viscosity also destabilizes Tollmien–Schlichting disturbances,
with the
maximum growth rate
shifted towards shorter waves. Qualitative and quantitative comparisons
are made
with experimental observations by Ludwieg & Hornung (1989).
Instabilities in a high-Reynolds-number
boundary layer on a film-coated surface
