The nonlinear stability of viscous, immiscible multilayer flows in plane channels
driven both by a pressure gradient and gravity is studied. Three fluid phases are
present with two interfaces. Weakly nonlinear models of coupled evolution equations
for the interfacial positions are derived and studied for inertialess, stably stratified
flows in channels at small inclination angles. Interfacial tension is demoted and
high-wavenumber stabilisation enters due to density stratification through second-order
dissipation terms rather than the fourth-order ones found for strong interfacial
tension. An asymptotic analysis is carried out to demonstrate how these models arise.
The governing equations are 2 × 2 systems of second-order semi-linear parabolic
partial differential equations (PDEs) that can exhibit inertialess instabilities due to
interaction between the interfaces. Mathematically this takes place due to a transition
of the nonlinear flux function from hyperbolic to elliptic behaviour. The concept
of hyperbolic invariant regions, found in nonlinear parabolic systems, is used to
analyse this inertialess mechanism and to derive a transition criterion to predict the
large-time nonlinear state of the system. The criterion is shown to predict nonlinear
stability or instability of flows that are stable initially, i.e. the initial nonlinear fluxes
are hyperbolic. Stability requires the hyperbolicity to persist at large times, whereas
instability sets in when ellipticity is encountered as the system evolves. In the former
case the solution decays asymptotically to its uniform base state, while in the latter
case nonlinear travelling waves can emerge that could not be predicted by a linear
stability analysis. The nonlinear analysis predicts threshold initial disturbances above
which instability emerges.