Multi-scale analysis of compressible viscous and rotating fluids

TL;DR

This paper investigates the asymptotic behavior of compressible viscous rotating fluids under various scaling limits, revealing simplified 2-D models and the influence of centrifugal forces in different regimes.

Contribution

It provides a rigorous analysis of the singular limits of the compressible Navier-Stokes equations with rotation, deriving new reduced models depending on the relative scaling of Mach and Rossby numbers.

Findings

Limit with Rossby number dominating yields 2-D incompressible Navier-Stokes equations.

Equal Mach and Rossby numbers lead to a linear, radially symmetric 2-D equation.

Centrifugal force effects are incorporated into the analysis.

Abstract

We study a singular limit for the compressible Navier-Stokes system when the Mach and Rossby numbers are proportional to certain powers of a small parameter $\ep$. If the Rossby number dominates the Mach number, the limit problem is represented by the 2-D incompressible Navier-Stokes system describing the horizontal motion of vertical averages of the velocity field. If they are of the same order then the limit problem turns out to be a linear, 2-D equation with a unique radially symmetric solution. The effect of the centrifugal force is taken into account.