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On the thermodynamics of some generalized second-grade fluids

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Abstract

The generalized second-grade fluids, which have been used for modeling the creep of ice and the flow of coal-water and coal-oil slurries, are among the simplest non-Newtonian fluid models that can describe shear-thinning/thickening and exhibit normal stress effects. In this article, we conduct thermodynamic analysis on a class of generalized second-grade fluids, one distinguishing feature of which is the existence of a constitutive function Φ that describes frictional heating. We work within the framework of Serrin’s original formulation of neoclassical thermodynamics, where internal energy and entropy functions, if they exist for a continuous body at all, are to be derived from the classical First Law and (quantitatively reformulated) Second Law of thermodynamics for cycles. For the class of generalized second-grade fluids in question, we show from the First Law that an internal energy density u exists, and we derive the equation of energy balance; from the Second Law, we demonstrate the existence of an entropy density s and derive the Clausius–Duhem inequality that it satisfies. We obtain explicit expressions for u, s and the frictional heating Φ, and derive thermodynamic restrictions on the material functions of temperature μ, α 1, and α 2 that appear in the constitutive relation for the Cauchy stress. For the special case of second-grade fluids, our expressions for u and s agree with those which Dunn and Fosdick [6] derived under the theoretical framework of the rational thermodynamics of Coleman and Noll.

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Authors and Affiliations

  1. Department of Mathematics, University of Kentucky, Lexington, KY, 40506-0027, USA

    Chi-Sing Man

  2. US Department of Energy, National Energy Technology Laboratory, Box 10940, Pittsburgh, PA, 15236, USA

    Mehrdad Massoudi

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  1. Chi-Sing Man
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  2. Mehrdad Massoudi
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Correspondence to Chi-Sing Man.

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Communicated by R. Abeyaratne.

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Man, CS., Massoudi, M. On the thermodynamics of some generalized second-grade fluids. Continuum Mech. Thermodyn. 22, 27–46 (2010). https://doi.org/10.1007/s00161-009-0123-3

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