Undulations on the surface of elongated bubbles in confined gas-liquid flows

A systematic analysis is presented of the undulations appearing on the surface of long bubbles in confined gas-liquid flows. CFD simulations of the flow are performed with a self-improved version of the open-source solver ESI OpenFOAM (release 2.3.1), for Ca=0.002-0.1 and Re=0.1-1000, where Ca=μU/σ and Re=2ρUR/μ, with μ and ρ being, respectively, the viscosity and density of the liquid, σ the surface tension, U the bubble velocity, and R the tube radius. A model, based on an extension of the classical axisymmetric Bretherton theory, accounting for inertia and for the curvature of the tube's wall, is adopted to better understand the CFD results. The thickness of the liquid film, and the wavelength and decay rate of the undulations extracted from the CFD simulations, agree well with those obtained with the theoretical model. Inertial effects appear when the Weber number of the flow We=CaRe=O(10-1) and are manifest by a larger number of undulation crests that become evident on the surface of the rear meniscus of the bubble. This study demonstrates that the necessary bubble length for a flat liquid film region to exist between the rear and front menisci rapidly increases above 10R when Ca>0.01 and the value of the Reynolds number approaches 1000.