Solute dispersion in axially strained tube flows: Large-time asymptotics and Ornstein-Uhlenbeck Gaussian profiles

The dispersion of a passive scalar in an axially strained flow in a slender tube is studied, with particular focus on large-time asymptotics following the approach of~\cite{rajamanickam2020dispersion}. For times exceeding the cross-sectional diffusion timescale, the scalar field forms an axial (Ornstein--Uhlenbeck) Gaussian profile whose variance increases exponentially with the local axial strain, which itself varies with radial location, while radial diffusion only slowly modulates the overall amplitude. In other words, the scalar is dominated by strong axial stretching, completely overwhelming radial diffusion. In striking contrast to classical Taylor dispersion, where radial diffusion rapidly homogenizes the profile and axial convection appears only as a small correction, here axial stretching governs the dominant dynamics, producing a fundamentally different transport mechanism.