Steady advection-diffusion around finite absorbers in two-dimensional potential flows

TL;DR

This paper develops an accurate numerical and analytical framework for solving steady advection-diffusion around finite absorbers in 2D potential flows, providing high-precision solutions and formulas valid across all Péclet numbers.

Contribution

It introduces a novel spectral method combined with asymptotics for high and low Péclet numbers, enabling accurate, uniform solutions for arbitrary shapes in advection-diffusion problems.

Findings

High-accuracy numerical solutions for flux around finite absorbers.

Asymptotic expansions valid from low to high Péclet numbers.

Analytical formulas for Nusselt number with sub-2% error.

Abstract

We perform an exhaustive study of the simplest, nontrivial problem in advection-diffusion -- a finite absorber of arbitrary cross section in a steady two-dimensional potential flow of concentrated fluid. This classical problem has been studied extensively in the theory of solidification from a flowing melt, and it also arises in Advection-Diffusion-Limited Aggregation. In both cases, the fundamental object is the flux to a circular disk, obtained by conformal mapping from more complicated shapes. We construct the first accurate numerical solution using an efficient new method, which involves mapping to the interior of the disk and using a spectral method in polar coordinates. Our method also combines exact asymptotics and an adaptive mesh to handle boundary layers. Starting from a well-known integral equation in streamline coordinates, we also derive new, high-order asymptotic expansions for high and low P\'eclet numbers ($\Pe$). Remarkably, the `high' $\Pe$ expansion remains accurate even for such low $\Pe$ as $10^{-3}$. The two expansions overlap well near $\Pe = 0.1$, allowing the construction of an analytical connection formula that is uniformly accurate for all $\Pe$ and angles on the disk with a maximum relative error of 1.75%. We also obtain an analytical formula for the Nusselt number ($\Nu$) as a function of the P\'eclet number with a maximum relative error of 0.53% for all possible geometries. Because our finite-plate problem can be conformally mapped to other geometries, the general problem of two-dimensional advection-diffusion past an arbitrary finite absorber in a potential flow can be considered effectively solved.