A shear flow with a fixed profile mixes a passive scalar through the interplay of advection and diffusion. The mixing rate depends on the flow's critical points — locations where the shear velocity vanishes and the stretching mechanism stalls. Enhanced dissipation quantifies how advection accelerates the decay beyond what diffusion alone would achieve, but critical points limit the enhancement.
The paper considers the simplest non-autonomous generalization: the shear profile translates rigidly at speed c. The flow field is the same at every instant, just shifted. For a stationary observer, the critical points move, spending finite time at each spatial location before sliding away.
The mixing rate interpolates between two known regimes. When c is small compared to diffusivity, the critical points dominate as in the stationary case. When c is large, the flow averages out and the solution approaches pure heat equation decay — advection becomes irrelevant. The interesting regime is intermediate: c = c₀ν^ℓ for ℓ between 1/3 and 3/4, where the decay rate is ν^((1+2ℓ)/5). This rate is faster than the stationary flow and slower than the heat equation, with the exponent continuously controlled by the translation speed.
Translation weakens critical points by limiting their dwell time. A critical point that passes through a region quickly cannot stall the mixing as effectively as one that sits there permanently. The mixing rate becomes a continuous function of how long the obstruction persists — and the obstruction is the same in every frame, just moving.