Taylor dispersion and large-deviation structure in stagnation-point flows without homogenisation
We study the long-time dispersion of a passive scalar in linearly accelerating flows whose strain rate $v(η)$ varies transversely: the axially strained flow in a porous tube with wall injection, and radially strained stagnation-point flows over a wall (Homann and von Kármán flows). Exponential stretching outpaces cross-sectional diffusion, so the scalar never homogenises across the section and classical cross-sectional averaging fails. Instead, in the logarithm of the stretched coordinate the problem becomes a Taylor dispersion problem, with mean log-stretching rate $U=Λ'(0)$ and log-space diffusivity $D=\tfrac{1}{2}Λ''(0)$: Taylor dispersion emerges without homogenisation, and only because the strain varies transversely. A Mellin-transform analysis, based on the principal eigenvalue $Λ(q)$ of the transverse advection-diffusion operator augmented by $qv$, gives the large-time structure in terms of the log-speed $x=\ln|\boldsymbolξ|/t$, where $\boldsymbolξ\in \mathbb R^d$ denotes the stretching directions ($d=1$ axial and $d=2$ radial). Essentially all the scalar mass lies in a log-normal bulk at $x=U$, flanked by forward and rear large-deviation tails with mass density $e^{-I(x)t}$, where $I(x)$ is the rate function (i.e., the Legendre transform of $Λ(q)$) Behind the rear tail, below $x_*=Λ'(-d)<U$, a plateau independent of $\boldsymbolξ$ decays at the rate $-Λ(-d)$. It carries the highest concentrations but only a fraction $e^{-I(x_*)t}$ of the mass. A thin transition zone joins the plateau to the rear tail. For uniformly strained flows, as in the Burgers vortex, all regions collapse onto a single Gaussian.