Effects of inertia and stratification in incompressible ideal fluids: pressure imbalances by rigid confinement

Consequences of density stratification are studied for an ideal (Euler) incompressible fluid, confined to move under gravity between rigid lids but otherwise free to move along horizontal directions. Initial conditions that generate horizontal pressure imbalances in a laterally unbounded domain are examined. The aim is to show analytically the existence of classes of initial data for which total horizontal momentum evolves in time, even though only vertical forces act on the fluid in this set-up. A simple class of such initial conditions, leading to momentum evolution, is identified by systematic asymptotic expansions of the governing inhomogeneous Euler equations in the small-density-variation limit. These results for Euler equations are compared and confirmed with long-wave asymptotic models, which can handle arbitrary density variations and provide closed-form mathematical expressions for limiting cases. In particular, the role of wave dispersion arising from the fluid inertia is captured by the long-wave models, even for short-time dynamics emanating from initial conditions outside the models’ asymptotic range of validity. These results are compared with direct numerical simulations for variable-density Euler fluids, which further validate the numerical algorithms and the analysis.