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Unformatted text preview: Chapter 5 The Entropy Inequality and Constitutive Equations In the 1950’s and 1960’s, thermodynamics was combined with classical continuum me chanics to form a rational method for developing constitutive equations. These methods were pioneered by A. C. Eringen [3] and C. Truesdell [11] and their students. Today there exist many extensions to the theory presented, with names such as ”Irreversible Thermodynamics” and ”Extended Thermodynamics”. (NEED MORE DETAILS) 5.1 Entropy Inequality We begin with the field equation for entropy: hatwide Λ = ρ Dη Dt − ∇ · φ − ρb ≥ . (5.1) We need to relate the terms in this equation with terms appearing in the conservation of energy. To do this, we make the assumptions: φ = − q T b = h T , (5.2) i.e., that the entropy flux is due only to heat flux, and entropy generation is due only to heat generation (units check). When this holds we call this a simple material [3], and it appears to hold for most processes which are not too far from equilibrium. In extended thermodynamics, it is assumed that φ is a function of other variables as well, and the results are more applicable to rapidly evolving processes (processes far from equilibrium) [5]. Using assumptions (5.2) in (5.1) we get hatwide Λ = ρ Dη Dt + ∇ · parenleftbigg q T parenrightbigg − ρ h T = ρ Dη Dt + 1 T ∇ · q − 1 T 2 q · ∇ T − ρ h T ≥ . (5.3) 91 92 CHAPTER 5. THE ENTROPY INEQUALITY AND CONSTITUTIVE EQUATIONS Now rewriting the energy equation as 1 T ( − ∇ · q + ρh ) = 1 T ρ De Dt − 1 T t : d , (5.4) we can use the above expression in (5.3) to get a new form of the entropy inequality: T hatwide Λ = Tρ Dη Dt − ρ De Dt + t : d − 1 T q · ∇ T ≥ . (5.5) Note that we’ve eliminated the external supplies, b and h , which is good, since a consti tutive equation should not depend on an externally supplied source. Because temperature is more easily measured than entropy, we use a Legendre trans formation to replace energy, e , which is a function of entropy, by the Helmholtz potential, A , which is a function of temperature. Using e = A + Tη (where A now has units of energy per unit mass) we have by the chain rule De Dt = DA Dt + DT Dt η + T Dη Dt (5.6) which allows us to rewrite (5.5) as T hatwide Λ = − ρη DT Dt − ρ DA Dt + t : d − 1 T q · ∇ T ≥ . (5.7) Now note that we have made no assumptions on the specific material yet. Because we need the same number of equations as unknowns (otherwise we could not guarantee a unique solution), let’s make a table listing the unknowns and corresponding equations: Table 5.1: Unknown Variables and Corresponding Equations Unknown Equation ρ conservation of mass v conservation of momentum T conservation of energy t , A , q constitutive equations Specific entropy, η , can be determined by knowing A ( T ). Thus we see that we need constitutive equations for the variables t , A , and q , in order to close the system (i.e., have the same number of equations as unknowns). The conservation of angular momentum,the same number of equations as unknowns)....
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This note was uploaded on 11/11/2009 for the course MATH 6735 taught by Professor Bennethum during the Fall '06 term at University of Colombo.
 Fall '06
 Bennethum
 Equations, The Land

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