Exact solutions and physical analogies for unidirectional flows

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.

now 17 analogies, 100 refs

References in corpus (4)

• Reexamination of Hagen-Poiseuille flow: shape-dependence of the hydraulic resistance in microchannels
• Analysis of electrolyte transport through charged nanopores
• Dynamics of Conformal Maps for a Class of Non-Laplacian Growth Phenomena
• Fingered growth in channel geometry: A Loewner equation approach

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• Diffusion towards a nanoforest of absorbing pillars
• Diffusion in a fluid flow generated by a source at the apex of a wedge
• Electrolyte flows under magnetic fields: Manning-like counterion condensation in one dimension