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Exact solutions and physical analogies for unidirectional flows

Martin Z. Bazant*

  • Department of Chemical Engineering and Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

  • *Present address: Department of Materials Science and Engineering and SUNCAT Interfacial Science and Catalysis, Stanford University, Stanford, CA 94305, USA.
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Phys. Rev. Fluids 1, 024001 – Published 9 June, 2016

DOI: https://doi.org/10.1103/PhysRevFluids.1.024001

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 17 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 nonconstant 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.

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Phys. Rev. Fluids 1, 024001– Published 9 June, 2016
  • Received 15 January 2016
Article part of CHORUS

©2016 American Physical Society

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