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The reciprocal theorem in fluid dynamics and transport phenomena

Published online by Cambridge University Press:  30 September 2019

Hassan Masoud Open the ORCID record for Hassan Masoud [Opens in a new window]  and
Howard A. Stone Open the ORCID record for Howard A. Stone [Opens in a new window]
Hassan Masoud*
Affiliation:
Department of Mechanical Engineering–Engineering Mechanics, Michigan Technological University, Houghton, Michigan 49931, USA
Howard A. Stone*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA
*
†Email addresses for correspondence: hmasoud@mtu.edu, hastone@princeton.edu
†Email addresses for correspondence: hmasoud@mtu.edu, hastone@princeton.edu
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Abstract

In the study of fluid dynamics and transport phenomena, key quantities of interest are often the force and torque on objects and total rate of heat/mass transfer from them. Conventionally, these integrated quantities are determined by first solving the governing equations for the detailed distribution of the field variables (i.e. velocity, pressure, temperature, concentration, etc.) and then integrating the variables or their derivatives on the surface of the objects. On the other hand, the divergence form of the conservation equations opens the door for establishing integral identities that can be used for directly calculating the integrated quantities without requiring the detailed knowledge of the distribution of the primary variables. This shortcut approach constitutes the idea of the reciprocal theorem, whose closest relative is Green’s second identity, which readers may recall from studies of partial differential equations. Despite its importance and practicality, the theorem may not be so familiar to many in the research community. Ironically, some believe that the extreme simplicity and generality of the theorem are responsible for suppressing its application! In this Perspectives piece, we provide a pedagogical introduction to the concept and application of the reciprocal theorem, with the hope of facilitating its use. Specifically, a brief history on the development of the theorem is given as a background, followed by the discussion of the main ideas in the context of elementary boundary-value problems. After that, we demonstrate how the reciprocal theorem can be utilized to solve fundamental problems in low-Reynolds-number hydrodynamics, aerodynamics, acoustics and heat/mass transfer, including convection. Throughout the article, we strive to make the materials accessible to early career researchers while keeping it interesting for more experienced scientists and engineers.

JFM classification

Mathematical Foundations: Mathematical Foundations Low-Reynolds-number flows: Low-Reynolds-number flows Multiphase and Particle-laden Flows: Particle/fluid flow
Type
JFM Perspectives
Information
Journal of Fluid Mechanics , Volume 879 , 25 November 2019 , P1
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© 2019 Cambridge University Press 

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