Unstable growth of bubbles from a constriction

Abstract

Bubbles and droplets are ubiquitous in many areas of engineering, including microfluidics where they can serve as microreactors for screening of chemical reactions. They are often formed out of a constriction (a microfluidic channel or a cylindrical tube) by blowing a given volume of gas into a liquid phase. It is obviously crucial to be able to control their size, which is not always easy due to the coupling between the volume of the bubble and the gas pressure induced by the Laplace law. In this paper, we examine the size and formation dynamics of soap bubbles blown from a cylindrical tube, which is the paradigm geometry for bubble and droplet formation. To do so, one end of the tube is closed by a soap film, while the other end is connected to a large reservoir of variable volume filled with gas. To inflate the bubble, we reduce the volume of the reservoir, which mimics air inflation through the lung diaphragm or the flow-rate-driven bubble formation in microfluidics geometry such as flow-focusing. As the volume of the reservoir decreases, the soap film curves and takes the form of a spherical cap with a smaller and smaller radius of curvature, which leads to the increase of the gas pressure in the reservoir, according to Laplace's law. This quasistatic process continues until a critical pressure is reached for which the bubble is quasihemispherical. Beyond this pressure, the film undergoes a rapid topological transformation and swells very rapidly (in less than 100 ms) until it reaches its final volume. We describe this instability in particular by showing that this unstable regime appears when a dimensionless number, which depends on the volume of the reservoir, the radius of the tube, surface tension, and external pressure, reaches a critical value. Using a quasistatic model that we solve analytically, we predict the bubble growth dynamics and the amplitude of the unstable height increase for any reservoir volume and constriction size.