An exact solution to dispersion of a passive scalar by a periodic shear flow

TL;DR

This paper provides an exact analytical solution for shear dispersion of a passive scalar using Mathieu functions, revealing how dispersion transitions from diffusive to propagative behavior depending on flow parameters and eigenvalue coalescence at Exceptional Points.

Contribution

It introduces a novel exact solution involving Mathieu functions for shear dispersion, including analysis of eigenvalue behavior and a continuous closure in wavenumber space.

Findings

Dispersion behaves diffusively for certain parameter ranges.

Eigenvalues coalesce at Exceptional Points, changing dispersion nature.

The derived closure interpolates between diffusion and fractional advection.

Abstract

We present an exact analytical solution to the problem of shear dispersion given a general initial condition. The solution is expressed as an infinite series expansion involving Mathieu functions and their eigenvalues. The eigenvalue system depends on the imaginary parameter $q=2ik$Pe, with $k$ the wavenumber that determines the tracer scale in the initial condition and Pe the P\'{e}clet number. Solutions are valid for all Pe, $t>0$, and $k>0$ except at specific values of $q=q_{\ell}^{EP}$ called Exceptional Points (EPs), the first occurring at $q_{0}^{EP}\approx1.468i$. For values of $q \lessapprox 1.468i$, all the eigenvalues are real, different and eigenfunctions decay with time, thus shear dispersion can be represented as a diffusive process. For values of $q \gtrapprox 1.468i$, pairs of eigenvalues coalesce at EPs becoming complex conjugates, the eigenfunctions propagate and decay with time, and so shear dispersion is no longer a purely diffusive process. The limit $q\rightarrow0$ is approached by the small P\'{e}clet number limit for all finite $k>0$, or equally by the large P\'{e}clet number limit as long as $2k \ll 1/$Pe. The latter implies $k\rightarrow0$, strong separation of scales between the tracer and flow. The limit $q\rightarrow\infty$ results from large P\'{e}clet number for any $k>0$, or from large $k$ and non-vanishing Pe. We derive an exact closure that is continuous in wavenumber space. At small $q$, the closure approaches a diffusion operator with an effective diffusivity proportional to $U_0^2/\kappa$, for flow speed $U_0$ and diffusivity $\kappa$. At large $q$, the closure approaches the sum of an advection operator plus a half-derivative operator (differential operator of fractional order), the latter with coefficient proportional to $\sqrt{\kappa U_0}$.