On a Camassa-Holm type equation describing the dynamics of viscous fluid conduits

2024/12/23 by Rafael Granero-Belinchón, Granero-Belinchón, Rafael
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Fractional Differential Equations Solutions #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2412.17467

openalex publication_date 2024/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note we derive a new nonlocal and nonlinear dispersive equations capturing the main dynamics of a circular interface separating a light, viscous fluid rising buoyantly through a heavy, more viscous, miscible fluid at small Reynolds numbers. This equation that we termed the g-model shares some common structure with the Camassa-Holm equation but has additional nonlocal effects. For this new PDE we study the well-posedness together with the existence of periodic traveling waves. Furthermore, we also show some numerical simulations suggesting the finite time singularity formation.

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