The trapping and release of bubbles from a linear pore

We study the effect on bubble motion of the sudden streamwise expansion of a liquid-filled tube of square cross-section, which is formed by a pair of backward-facing and forward-facing steps. When a bubble of finite volume traveling under constant volume flux reaches the backward-facing step that marks the start of the expansion, the extent to which it broadens to partially fill the expansion depends on the balance between viscous and surface tension stresses, measured by the bubble capillary number Ca. The broadening into the expansion of a bubble of initial static length larger than the width of the tube, L > w, is accompanied by its slowing and momentary arrest as the capillary number is reduced towards its critical value for trapping. For Ca < Ca crit the pressure drag forces on the quasi-arrested bubble due to the constant flux flow become insufficient to force the bubble over the forward-facing step that marks the end of the expansion so that the bubble remains trapped. We examine the conditions for trapping by varying bubble volume, flow rate of the carrier fluid and length of expanded region. When this length is equal to the width of the tube, we find that Ca crit depends non-monotonically on the size of the bubble, reaching a maximum value for the smallest bubble volume † that enables the broadened bubble to fill the entire length of the expansion. The trapped bubble configuration depends strongly on the Bond number Bo, the ratio of gravity to surface tension forces, resulting in an order of magnitude increase in the maximum value of Ca crit from micrometric tubes (Bo = 0) to millimetric tubes (Bo = 1), due to reduction in pressure drag because the bubble lifts off the bottom boundary. We verify that a bubble is released if the work of the pressure forces over the length of the expansion exceeds the surface energy required for the trapped bubble to reenter the tube of square cross-section by obtaining good agreement between direct measurements of the pressure drop between tail and tip of the trapped bubble, and the pressure drop estimated from the change in bubble surface energy calculated with a capillary static model. We find that the pressure drop becomes constant for L/w ≥ 2.4, signifying similar broadening into the expansion and Ca crit ∼ (L/w) −1 .