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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.
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Flux-based modeling of heat and mass transfer in multicomponent systems
In the present work, the macroscopic governing equations governing the heat and mass transfer for a general multicomponent system are derived via a systematic nonequilibrium thermodynamics framework. In contrast to previous approaches, the relative (with respect to the mass average velocity) component mass fluxes (relative species momenta) and the heat flux are treated explicitly, in complete analogy with the momentum flux. The framework followed here, in addition to allowing for the description of relaxation phenomena in heat and mass transfer, establishes to the fullest the analogy between all transport processes, momentum, heat, and mass transfer, toward which R. B. Bird contributed so much with his work. The inclusion of heat flux-based momentum as an additional variable allows for the description of relaxation phenomena in heat transfer as well as of mixed (Soret and Dufour) effects, coupling heat and mass transfer. The resulting models are Galilean invariant, thereby resolving a conundrum in the field, and always respect the second law of thermodynamics, for appropriate selection of transport parameters. The general flux-based dynamic equations reduce to the traditional transport equations in the limit when mass species and heat relaxation effects are negligible and are fully consistent with the equations established from the application of kinetic theory in the limit of dilute gases. As an added benefit, for the particular example case of hyperbolic diffusion we illustrate the application of the proposed models as a method to allow the use of powerful numerical solvers normally not available for solving mass transfer models more generally.
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- Award ID(s):
- 1804911
- PAR ID:
- 10364314
- Publisher / Repository:
- American Institute of Physics
- Date Published:
- Journal Name:
- Physics of Fluids
- Volume:
- 34
- Issue:
- 3
- ISSN:
- 1070-6631
- Page Range / eLocation ID:
- Article No. 033113
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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