2017/03/15 by Mathias Wilke, Wilke, Mathias · 4 citations
Engineering · Mathematics · #35B35 #35R35 Primary #76D03 #76D05 #76D45 #76E17 #Analysis of PDEs (math.AP) #Bifurcation #Boundary (topology) #Boundary value problem #Capillary surface #Classical mechanics #Compressibility #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Instability #Mathematical analysis #Mathematics #Mechanics #Navier-Stokes equation solutions #Nonlinear system #Optics #Physics #Rayleigh scattering #Rayleigh–Taylor instability #Stability and Controllability of Differential Equations #Surface tension #Thermodynamics #math.AP #msc:35B35 #msc:35R35 #msc:76D03 #msc:76D05 #msc:76D45 #msc:76E17
paper · pdf · doi:10.48550/arxiv.1703.05214
published in arXiv (Cornell University) (Cornell University) · 132 pages
arxiv created 2017/03/15 · openalex publication_date 2017/03/15 · arxiv updated 2017/03/16 · openalex created_date 2022/10/01 · openalex updated_date 2026/08/04
This article is concerned with the dynamic behaviour of two immiscible and\nincompressible fluids in a cylindrical domain, which are separated by a sharp\ninterface. In case that the heavy fluid is situated on top of the light fluid,\none expects that the fluid on top sags down into the lower phase. This effect\nis known as the Rayleigh-Taylor-Instability. We present a rather complete\nanalysis of the corresponding free boundary problem which involves a contact\nangle. Our main result implies the existence of a critical surface tension with\nthe following property. In case that the surface tension of the interface\nseparating the two fluids is smaller than the critical surface tension, one has\nRayleigh-Taylor-Instability. On the contrary, if the interface has a greater\nsurface tension than the critical value, the instability effect does not occur\nand one has exponential stability of a flat interface. The last part of this\narticle is concerned with the bifurcation of nontrivial equilibria in multiple\neigenvalues. The invariance of the corresponding bifurcation equation with\nrespect to rotations and reflections yields the existence of bifurcating\nsubcritical equilibria. Finally it is proven that the bifurcating equilibria\nare unstable.\n