Exact solutions and physical analogies for unidirectional flows

TL;DR

This paper unifies and extends the mathematical understanding of unidirectional flow in fluid mechanics, revealing its physical analogies, methods for exact solutions, and proving key properties like Onsager reciprocity in various geometries.

Contribution

It identifies seventeen physical analogies for unidirectional flow, develops methods for constructing exact solutions, and proves new properties such as Onsager reciprocity for arbitrary geometries.

Findings

Identified 17 physical analogies for unidirectional flow.

Developed methods for constructing exact solutions in complex geometries.

Proved Onsager reciprocity for electrokinetic phenomena in arbitrary-shaped pores.

Abstract

Unidirectional flow is the simplest phenomenon of fluid mechanics. Its mathematical description, the Dirichlet problem for Poisson's equation in two dimensions with constant forcing, arises in many physical contexts, such as the torsion of elastic beams, first solved by de Saint-Venant for complex shapes. Here, the literature is unified and extended by identifying seventeen physical analogies for unidirectional flow and describing their common mathematical structure. Besides classical analogies in fluid and solid mechanics, applications are discussed in stochastic processes (first passage in two dimensions), pattern formation (river growth by erosion), and electrokinetics (ion transport in nanochannels), which also involve Poisson's equation with non-constant forcing. Methods are given to construct approximate geometries that admit exact solutions, by adding harmonic functions to quadratic forms or by truncating eigenfunction expansions. Exact solutions for given geometries are also derived by conformal mapping. We prove that the remarkable geometrical interpretation of Poiseuille flow in an equilateral triangular pipe (the product of the distances from an interior point to the sides) is only shared by parallel plates and unbounded equilateral wedges (with the third side hidden behind the apex). We also prove Onsager reciprocity for linear electrokinetic phenomena in straight pores of arbitrary shape and surface charge, based on the mathematics of unidirectional flow.