  One strategy for arresting propagating detonation waves in pipes is by
imposing a sudden area enlargement, which provides a rapid lateral divergence
of the gases in the reaction zone and attenuates the leading shock. For
sufficiently small tube diameter, the detonation decays to a deflagration and
the shock decays to negligible strengths. This is known as the critical tube
diameter problem. In the present study, we provide a closed form model to
predict the detonation quenching for 2D channels. Whitham's geometric shock
dynamics, coupled with a shock evolution law based on shocks sustained by a
constant source obtained by the shock change equations of Radulescu, is shown
to capture the lateral shock dynamics response to the failure wave originating
at the expansion corner. A criterion for successful detonation transmission to
open space is that the lateral strain rate provided by the failure wave not
exceed the critical strain rate of steady curved detonations. Using the
critical lateral strain rate obtained by He and Clavin, a closed form solution
is obtained for the critical channel opening permitting detonation
transmission. The predicted critical channel width is found in very good
agreement with our recent experiments and simulations of diffracting
H$_2$/O$_2$/Ar detonations.
