Abstract
This paper concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and the upper fluid is bounded above by a trivial fluid of constant pressure. This is a free boundary problem: the interfaces between the fluids and above the upper fluid are free to move. The fluids are acted on by gravity in the bulk, and at the free interfaces we consider both the case of surface tension and the case of no surface forces. We establish a sharp nonlinear global-in-time stability criterion and give the explicit decay rates to the equilibrium. When the upper fluid is heavier than the lower fluid along the equilibrium interface, we characterize the set of surface tension values in which the equilibrium is nonlinearly stable. Remarkably, this set is non-empty, i.e. sufficiently large surface tension can prevent the onset of the Rayleigh-Taylor instability. When the lower fluid is heavier than the upper fluid, we show that the equilibrium is stable for all non-negative surface tensions and we establish the zero surface tension limit.
Highlights
The surface Γ+(t) = {y3 = + η+(y1, y2, t)} is the moving upper boundary of Ω+(t) where the upper fluid is in contact with the atmosphere, Γ−(t) = {y3 = η−(y1, y2, t)} is the moving internal interface between the two fluids, and Σb = {y3 = −b} is the fixed lower boundary of Ω−(t)
For the sake of clarity we summarize the admissibility conditions that are necessary and sufficient for the existence of an equilibrium: (1) patm ∈ P+(R+), which defines ρ1 := P+−1(patm)
Hataya [11] proved an existence result for a periodic free interface problem with surface tension, perturbed around Couette flow; he showed the local existence of small solution for any physical constants, and the existence of exponentially decaying small solution if the viscosities of the two fluids are sufficiently large and their difference is small
Summary
Hataya [11] proved an existence result for a periodic free interface problem with surface tension, perturbed around Couette flow; he showed the local existence of small solution for any physical constants, and the existence of exponentially decaying small solution if the viscosities of the two fluids are sufficiently large and their difference is small. Kim [33] adapted the two-tier energy methods of [8, 9, 10] to develop the nonlinear RayleighTaylor instability theory for the problem, proving the existence of a sharp stability criterion given in terms of the surface tension coefficient, gravity, periodicity lengths, and ρ. The free boundary problems corresponding to a single horizontally periodic layer of compressible viscous fluid with surface tension have been studied by several authors. In our companion paper [14] we show that the stability criterion is sharp, as in the incompressible case [32, 33], and that the Rayleigh-Taylor instability persists at the nonlinear level (the linear analysis was developed in [7])
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
