2022/11/07 by Francisco Gancedo, Gancedo, Francisco, Rafael Granero-Belinchón +3 · 2 citations
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2211.03437
openalex publication_date 2022/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We study the dynamics of the interface given by two incompressible viscous fluids in the Stokes regime filling a 2D horizontally periodic strip. The fluids are subject to the gravity force and the density difference induces the dynamics of the interface. We derive the contour dynamics formulation for this problem through a x1-periodic version of the Stokeslet. Using this new system, we show local-in-time well-posedness when the initial interface is described by a curve with no self-intersections and C1+γ Hölder regularity, 0<γ<1. This well-posedness result holds regardless of the Rayleigh-Taylor stability of the physical system. In addition, global-in-time existence and decay to the flat stationary state is proved in the Rayleigh-Taylor stable regime for small initial data. Finally, in the Rayleigh-Taylor unstable regime, we construct a wide family of smooth solutions with exponential in time growth for an arbitrary large interval of existence. Remarkably, the initial data leading to this exponential growth possibly lack any symmetry.