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Steady-state heat conduction in quiescent fluids: Incompleteness of the Navier–Stokes–Fourier equations

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  • Brenner, Howard

Abstract

Linear irreversible thermodynamic principles are used to demonstrate, by counterexample, the existence of a fundamental incompleteness in the basic pre-constitutive mass, momentum, and energy equations governing fluid mechanics and transport phenomena in continua. The demonstration is effected by addressing the elementary case of steady-state heat conduction (and transport processes in general) occurring in quiescent fluids. The counterexample questions the universal assumption of equality of the four physically different velocities entering into the basic pre-constitutive mass, momentum, and energy conservation equations. Explicitly, it is argued that such equality is an implicit constitutive assumption rather than an established empirical fact of unquestioned authority. Such equality, if indeed true, would require formal proof of its validity, currently absent from the literature. In fact, our counterexample shows the assumption of equality to be false. As the current set of pre-constitutive conservation equations appearing in textbooks are regarded as applicable both to continua and noncontinua (e.g., rarefied gases), our elementary counterexample negating belief in the equality of all four velocities impacts on all aspects of fluid mechanics and transport processes, continua and noncontinua alike.

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  • Brenner, Howard, 2011. "Steady-state heat conduction in quiescent fluids: Incompleteness of the Navier–Stokes–Fourier equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(20), pages 3216-3244.
  • Handle: RePEc:eee:phsmap:v:390:y:2011:i:20:p:3216-3244
    DOI: 10.1016/j.physa.2011.04.023
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    References listed on IDEAS

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    1. Bardow, André & Christian Öttinger, Hans, 2007. "Consequences of the Brenner modification to the Navier–Stokes equations for dynamic light scattering," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 373(C), pages 88-96.
    2. Brenner, Howard, 2005. "Kinematics of volume transport," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 349(1), pages 11-59.
    3. Brenner, Howard, 2005. "Navier–Stokes revisited," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 349(1), pages 60-132.
    4. Bedeaux, Dick & Kjelstrup, Signe & Christian Öttinger, Hans, 2006. "On a possible difference between the barycentric velocity and the velocity that gives translational momentum in fluids," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 371(2), pages 177-187.
    5. Brenner, Howard, 2006. "Fluid mechanics revisited," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 370(2), pages 190-224.
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    Cited by:

    1. Brenner, Howard, 2013. "Bivelocity hydrodynamics. Diffuse mass flux vs. diffuse volume flux," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(4), pages 558-566.
    2. Brenner, Howard, 2011. "Derivation of constitutive data for flowing fluids from comparable data for quiescent fluids," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(21), pages 3645-3661.

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