Explicit Velocity Fields in Bubbly Taylor–Couette Flow with Buoyancy on Gas Bubbles

Explicit expressions for bubbly Taylor–Couette flow fields are rarely available in the literature. The present work aims to derive explicit expressions for bubble velocity fields in laminar gas–liquid Taylor–Couette flow between two rotating coaxial cylinders with the buoyancy effect on gas bubbles. It is assumed that the angular velocity of the rotating cylinder(s) is moderately low and the bubble radius is relatively small so that the Stokes number of bubbles is small enough and, consequently, the radial bubble migration is ignorable and the bubble volume fraction can be treated as being constant in a limited period of time. Explicit leading-order solutions are derived for the spiral rising bubble velocity field in the dilute limit. Unlike the heavy particles dominated by the Stokes drag, the added mass and lift forces are shown to be relevant for the bubbly flows. The radial bubble velocity field is discussed in detail for several cases of major interest under the condition that the added mass coefficient is equal to the lift force coefficient, as assumed by some authors in the literature. Our results show that the radial-to-azimuthal velocity ratio of bubbles is linearly proportional to the Stokes number of bubbles and can be controlled by the angular velocity of the rotating cylinder(s) and the bubble radius so that the assumption of ignorable radial bubble migration can be reasonably justified within a limited period of time (for example, in the first few tens of revolutions of the rotating cylinder(s)).