Taylor dispersion and large-deviation structure in stagnation-point flows without homogenisation

Abstract

We study the long-time dispersion of a passive scalar in linearly accelerating flows whose strain rate $v(\eta )$ 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 $U={\Lambda }^{\prime }(0)$ and log-space diffusivity $D=\frac{1}{2}{\Lambda }^{\prime \prime }(0)$: Taylor dispersion emerges without homogenisation, and only because the strain varies transversely. A Mellin-transform analysis, based on the principal eigenvalue $\Lambda (q)$ of the transverse advection-diffusion operator augmented by $qv$, gives the large-time structure in terms of the log-speed $x=\ln |\mathbf{\xi }|/t$, where $\mathbf{\xi }\in {\mathbb{R}}^d$ 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 $\Lambda (q)$) Behind the rear tail, below $x_{\ast }={\Lambda }^{\prime }(-d)<U$, a plateau independent of $\mathbf{\xi }$ decays at the rate $-\Lambda (-d)$. It carries the highest concentrations but only a fraction $e^{-I(x_{\ast })t}$ 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.