Modeling of the bubbling process in a planar co-flow configuration

•Simple theoretical model that incorporates the main features of the process.•The model does not require any adjustable parameter.•The initial velocity of the neck is determined from a previous scaling law.•The temporal evolution of bubble thicknesses and pressures is obtained.•Good agreement with previous experimental and numerical results.

This work presents an analytical model developed to describe the bubbling regime resulting from the injection of an air sheet of thickness 2Ho with a mean velocity ua   between two water streams of thickness Hw−Ho,Hw−Ho, moving at a uniform velocity uw. Based on previous experimental and numerical characterizations of this flow, the gas stream is modeled as a two-dimensional sheet divided into three different parts in the streamwise direction: a neck that moves downstream at the water velocity, a gas ligament attached to the injector upstream of the neck, and a forming bubble downstream of the neck, whose uniform dimensionless half-thicknesses are ηn(τ), ηl(τ), ηb(τ) respectively, and the corresponding pressures are given by Πn(τ), Πl(τ  ), and Πb(τ)=Πn(τ)Πb(τ)=Πn(τ). Lengths are made dimensionless with Ho  , and pressures with ρaua2, where ρa is the air density. In a reference frame moving with the water velocity, and imposing a negative pressure caused by the sudden expansion of the air stream at the outlet of the injector, a set of algebraic-differential equations are deduced, that can be numerically integrated to obtain the temporal evolution of the interface positions and gas pressures, as well as of the gas flow rate through the neck. The model shows a good agreement with previous experimental and numerical results for a given value of the initial velocity of the collapsing neck, determined by an iterative method that matches the bubbling time with that given by Gutiérrez-Montes et al. (2013), τbc=9.1Λ(ρw/ρa)(h−1)/[We(β−β2)]. Here Λ=uw/uaΛ=uw/ua is the water-to-air velocity ratio, We=ρwuw2Ho/σ the Weber number, h=Hw/Hoh=Hw/Ho the water-to-air thickness ratio and (1−β)=(Ho−Hi)/Ho(1−β)=(Ho−Hi)/Ho the dimensionless wall thickness of the air injector.