  We develop a simple dynamical model for the evolution of gas in the centres
of barred spiral galaxies, using the Milky Way's Central Molecular Zone (CMZ,
i.e., the central few hundred pc) as a case study. We show that, in the
presence of a galactic bar, gas in a disc in the central regions of a galaxy
will be driven inwards by angular momentum transport induced by acoustic
instabilities within the bar's inner Lindblad resonance. This transport process
drives turbulence within the gas that temporarily keeps it strongly
gravitationally stable and prevents the onset of rapid star formation. However,
at some point the rotation curve must transition from approximately flat to
approximately solid body, and the resulting reduction in shear reduces the
transport rates and causes gas to build up, eventually producing a
gravitationally-unstable region that is subject to rapid and violent star
formation. For the observed rotation curve of the Milky Way, the accumulation
happens $\sim 100$ pc from the centre of the Galaxy, in good agreement with the
observed location of gas clouds and young star clusters in the CMZ. The
characteristic timescale for gas accumulation and star formation is of order
$10-20$ Myr. We argue that similar phenomena should be ubiquitous in other
barred spiral galaxies.
