Highly rotating viscous compressible fluids in presence of capillarity effects

TL;DR

This paper investigates the simultaneous incompressible and high rotation limits of a Navier-Stokes-Korteweg system with Coriolis force, considering both constant and vanishing capillarity effects, and identifies the resulting limit equations.

Contribution

It provides a rigorous analysis of the singular limits of viscous compressible fluids with capillarity under rotation, including anisotropic scaling and different capillarity regimes.

Findings

Limit equations are 2-D incompressible Navier-Stokes in the orthogonal variables.

Different capillarity regimes lead to slightly different limit equations.

The analysis employs the RAGE theorem for the proofs.

Abstract

In this paper we study a singular limit problem for a Navier-Stokes-Korteweg system with Coriolis force, in the domain $\R^2\times\,]0,1[\,$ and for general ill-prepared initial data. Taking the Mach and the Rossby numbers to be proportional to a small parameter $\veps$ going to $0$, we perform the incompressible and high rotation limits simultaneously. Moreover, we consider both the constant capillarity and vanishing capillarity regimes. In this last case, the limit problem is identified as a $2$-D incompressible Navier-Stokes equation in the variables orthogonal to the rotation axis. If the capillarity is constant, instead, the limit equation slightly changes, keeping however a similar structure. Various rates at which the capillarity coefficient can vanish are also considered: in most cases this will produce an anisotropic scaling in the system, for which a different analysis is needed. The proof of the results is based on suitable applications of the RAGE theorem.