Abstract
The motion of bubbles and drops through tubes in gravity- and pressure-driven flows is intensively studied numerically and experimentally. The Bretherton asymptotic expressions predict axisymmetric bubbles movement at low velocities. They describe the dependence of capillary (Ca) and Bond (Bo) numbers on the system parameters but are valid only in the ranges 0 < Ca < 0.005 and 0.84 < Bo < 1.04. This paper investigates the gravity-induced motion of bubbles with free or tangentially immobile interfaces in pressure-driven flows. We derive the exact solution of the hydrodynamic problem using the lubrication approximation in the zero- and first-order approximations for pressure and fluid velocity. The respective boundary value problem for the bubble shape is solved numerically to obtain the wetting film thickness, h, between the bubble and the capillary and the dependence of the capillary numbers on the flow parameters and magnitude of gravity. The proposed model expands the applicable solution ranges by 400 and 38 times, respectively (0 < Ca < 2 and 0 < Bo < 7.5), validated with available experimental data. The model's simplicity and transparency open the possibility to generalize this approach including determining new physicochemical properties of liquids and interfaces.
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