Gaseous Entrainment Dynamics in a Viscous Pool Due to Combined Influence of Asymmetric Rotational Field and Crossflow of Air

Efforts are made to perform simulations to describe the gaseous entrainment dynamics in a viscous liquid pool due to the combined influence of asymmetric converging rotational field and continuous freestream flow of air. A pair of counter-rotating and equal sized rollers is placed inside the pool along a horizontal line. Gerris is an open-source solver, which is employed to carry out the present computational study. Complex interfacial configurations are illustrated with the influence of relevant input parameters, such as rotation of rollers 1 and 2 (measured by Capillary number, $Ca1=Rω1μl/σ$ and $Ca2=Rω2μl/σ$⁠, where $R=D/2$ is roller radius), submersion depth (b*), the gap between the rollers (2a*), and strength of air stream flow (measured by Reynolds number, $Reflow=ρgUD/μg$⁠). It has been observed that the depth of steady entrainment is reduced at $Reflow≠0$ compared to $Reflow=0$ because the hydrodynamic force acts as an opposing force to viscous pumping and rotating inertia. A complete understanding of disintegration of and subsequent accumulation of gaseous bubbles from the cusp tip is characterized in detail. In addition, the influence of viscous drag (specified by Morton number, $Mo=gμl4(ρl−ρg)/(ρl2σ3)$⁠) and gravitational pull (estimated by Archimedes number, $Ar=gD3ρl2/μ2$⁠) on the phase contours are also reported. Finally, an analytical formulation is proposed to analyze the structure of entrainment, and this model reports an excellent match with the numerical findings.