Rigorous justification of Taylor dispersion via center manifolds and hypocoercivity

TL;DR

This paper rigorously justifies Taylor dispersion by decomposing the PDE solution into an approximate part governed by a finite-dimensional center manifold, and a rapidly decaying remainder, using spectral and hypocoercivity techniques.

Contribution

It introduces a rigorous PDE analysis of Taylor dispersion, employing center manifold theory and hypocoercivity to justify the phenomenon's diffusion rate.

Findings

The solution splits into an approximate part and a rapidly decaying remainder.

The approximate solution's dynamics are governed by a finite-dimensional system exhibiting diffusion.

The remainder term decays faster than the main diffusion behavior.

Abstract

Taylor diffusion (or dispersion) refers to a phenomenon discovered experimentally by Taylor in the 1950s where a solute dropped into a pipe with a background shear flow experiences diffusion at a rate proportional to $1/\nu$, which is much faster than what would be produced by the static fluid if its viscosity is $0 < \nu \ll 1$. This phenomenon is analyzed rigorously using the linear PDE governing the evolution of the solute. It is shown that the solution can be split into two pieces, an approximate solution and a remainder term. The approximate solution is governed by an infinite-dimensional system of ODEs that possesses a finite-dimensional center manifold, on which the dynamics correspond to diffusion at a rate proportional to $1/\nu$. The remainder term is shown to decay at a rate that is much faster than the leading order behavior of the approximate solution. This is proven using a spectral decomposition in Fourier space and a hypocoercive estimate to control the intermediate Fourier modes.