Bubble pressure requirements to control the bubbling process in forced co-axial air-water jets

Abstract

We analyse the controlled generation of bubbles of a given size at a determined bubbling rate in a co-flowing water stream forcing the gas flow. The temporal evolution of the bubble size, R(t), the air flow rate, Qa(t), and the pressure evolution inside the bubble, pb(t), during the bubbling process are reported. To that aim, the temporal evolution of the bubble shape and the pressure inside the air feeding chamber, pc(t), where a harmonic perturbation is induced using a loudspeaker, are obtained from high-speed images synchronized with pressure measurements. A model is developed to describe the unsteady motion of the gas stream along the injection needle, coupled with the Rayleigh-Plesset equation for the growing bubble, allowing us to obtain pb(t). Thus, the minimum pressure amplitudes required inside the forming bubble to control their size and bubbling frequency are provided as a function of the gas flow rate, the liquid velocity, uw, and the forcing frequency, ff. Two different behaviors have been observed, depending on the liquid-to-gas velocity ratio, Λ. For small enough values of Λ, the critical pressure amplitude is given by p_s ∼ ρ_a c u_a St_f3, associated to a rapid pressure increase taking place during an interval of time of the order of the acoustic time. However, for larger values of Λ, ps ∼ ρ u_w^2 St_f^3 Λ^{−1/5} We^{−1/4}. Here ρ and ρa are the liquid and gas densities respectively, c the speed of sound in air and St_f = f_f r_o/u_w and We = ρ u_w^2 r_o/σ the Strouhal and Weber numbers, where r_o denotes the outer radius of the injector.