Bubble motion in a potential flow within a Venturi

The motion of a bubble within a liquid-filled Venturi is computed using a simple force balance which considers pressure forces, added mass, and steady drag acting on the bubble. The bubble is small compared to the pipe radius, and interactions between the bubble and the pipe wall are neglected. Interfacial tension is assumed to be sufficiently strong that the bubble remains spherical. The liquid velocity is assumed either to be a simple one-dimensional flow with velocity inversely proportional to the pipe cross-sectional area, or to be an axisymmetric potential flow. A bubble on the axis of the Venturi remains on the axis. In the absence of drag, the bubble moves through the Venturi more rapidly than the liquid. If drag is small, the model predicts that the bubble becomes trapped within the Venturi. If drag is large, relative motion between the liquid and the bubble is suppressed, and the bubble flows through the Venturi without oscillation. Off the axis, a bubble in the converging section of the Venturi accelerates towards the centreline more rapidly than the liquid. In the absence of drag, if bubbles are distributed uniformly across the cross-section of the pipe upstream of the Venturi, they will be concentrated close to the axis in the throat. The bubbles eventually cross the axis and hit the far wall of the Venturi, at which point trajectory computations were stopped. If drag acts on the bubbles, the resulting combination of oscillatory axial motion and radial motion causes the bubbles to move towards the walls of the Venturi, where the potential flow is fast and pressures are small. Order of magnitude estimates suggest that such oscillations would not be observed for air bubbles in water, since the bubbles would be deformed and drag would become large.