  We construct a kinetic model for matter-radiation interactions whose
hydrodynamic gradient expansion can be computed analytically up to infinite
order in derivatives, in the fully nonlinear regime, and for arbitrary flows.
The frequency dependence of the opacity of matter is chosen to mimic the
relaxation time of a self-interacting scalar field. In this way, the transient
sector simulates that of a realistic quantum field theory. The gradient series
is found to diverge for most flows, in agreement with previous findings. We
identify, for the model at hand, the mechanism at the origin of the divergence,
and we provide a successful regularization scheme. Additionally, we propose a
universal qualitative framework for predicting the breakdown of the gradient
expansion of an arbitrary microscopic system undergoing a given flow. This
framework correctly recovers all previously known instances of gradient
expansion divergence. As a new prediction, we show that the gradient expansion
diverges when the energy-dependent mean free path is unbounded above.
