  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 gas in 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 gas
pressure increases, the soap film curves and takes the form of a spherical cap
with an increasingly smaller radius of curvature. This quasi-static process
continues until a critical pressure is reached for which the bubble is
quasi-hemispherical. Beyond this pressure, the film undergoes a rapid
topological transformation and swells very rapidly (in less than a hundred ms)
until it reaches its final volume. We describe this instability in particular
by showing that this unstable regime appears when a dimensionless number -
whose expression we specify - reaches a critical value. Using a quasi-static
model that we solve analytically, we predict the bubble growth dynamics and the
final height of the bubble produced for any reservoir volume and constriction
size.
