Pipe Flow of Suspensions Containing Bubbles

The steady laminar pipe flow of a suspension with a gas volume fraction ∅ ≤ 0.5 and small or intermediate bubble deformations in long, and straight sections of a circular pipe is calculated. The calculations are based on the constitutive equation that was originally derived for dilute emulsions and further developed for concentrated suspensions containing bubbles. In contrast to the literature, an analytical procedure is used to determine the solution of a pipe flow more accurately. The results are presented and discussed with respect to the Reynolds number Re and capillary number Ca. If Ca < 1 or Ca > 1, a bubble suspension has a parabolic velocity profile indicating a Newtonian rheology. If Ca ≈ 1, two regimes of flow are observed in agreement with the literature; that is, an inner plug flow where deformation rates are low and an outer flow where deformation rates are high. These results imply that, if Ca < 1, the Reynolds number Re decreases with increasing gas volume fraction ∅ and that, if Ca ≥ 1, the opposite effect occurs; that is, the Reynolds number Re increases with increasing gas volume fraction.