Compressibility-induced destabilisation of falling liquid films: an integral approach

We revisit the classical 2D problem of a gravity-driven liquid layer down an inclined plate (Kapitza, 1948), relaxing the usual assumption of homogeneous fluid. We set out to answer three major issues. When the fluid density is allowed to vary, (i) how does this feature structurally affect the formulation of a low dimensional depth-averaged model? (ii) To what extent and (iii) by virtue of which physical mechanism does compressibility participate in the long-wave interfacial instability? To provide the relevant answers, (i) we first make use of a second-order asymptotic expansion in the shallowness parameter to develop a weakly compressible boundary-layer system: starting from a two-equation momentum-integrated model, an additional barotropic equation of state is required for closure purposes. In this respect, (ii) a temporal linear stability analysis is performed: it is revealed that compressibility plays a destabilising role whose magnitude is enhanced at intermediately tilted configurations, and the more the Reynolds number approaches the critical threshold in the incompressible limit. (iii) We finally interpret the ensuing dispersion relation under the convenient framework of two-wave hierarchy, initiated by Whitham (1974): the primary instability gets promoted by the flow compressibility as it contributes to deceleration of dynamic waves most significantly in the low inertia regime. Indeed, compressibility locally acts as a further boost to the inertia-based mechanism of Kapitza instability by amplifying flow-rate variations within the liquid film.