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, and , where is roller radius), submersion depth (b*), the gap between the rollers (2a*), and strength of air stream flow (measured by Reynolds number, ). It has been observed that the depth of steady entrainment is reduced at compared to 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, ) and gravitational pull (estimated by Archimedes number, ) 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.