On the Müller paradox for thermal-incompressible media

Gouin, Henri; Muracchini, Augusto; Ruggeri, Tommaso

doi:10.1007/s00161-011-0201-1

Cited by 30 publications

(35 citation statements)

References 15 publications

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“…As done in paper [1] and by taking up the definition of pressure presented by Flory in 1953 for rubber gum [15,16] and extended for hyperelastic media by Gouin and Debiève in 1986 [17] and Rubin in 1988 [18], we can extend the results from fluids to thermo-elastic materials. With this aim we define…”

Section: Generalitiesmentioning

confidence: 81%

A consistent thermodynamical model of incompressible media as limit case of quasi-thermal-incompressible materials

Gouin¹,

Ruggeri²

2012

International Journal of Non-Linear Mechanics

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14 pagesInternational audienceIn this paper we consider the conditions on quasi-thermal-incompressible so that they satisfy all the principles of thermodynamics, including the stability condition associated with the concavity of the chemical potential. We analyze the approximations under which a quasi-thermal-incompressible medium can be considered as incompressible. We find that the pressure cannot exceed a very large critical value and that the compressibility factor must be greater than a lower limit that is very small. The analysis is first done for the case of fluids and then extended to the case of thermoelastic solids

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Section: Generalitiesmentioning

confidence: 81%

A consistent thermodynamical model of incompressible media as limit case of quasi-thermal-incompressible materials

Gouin¹,

Ruggeri²

2012

International Journal of Non-Linear Mechanics

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“…This limiting case leads to two apparent inconsistencies, (cf. [42], [22]). One concerns the inequality (78) 3 , (244) ( ∂ρ ∂T ) 2 < c p ρ 2 T ∂ρ ∂p , and the other is related to the coupled systems of partial differential equations for the variables at hand.…”

Section: Incompressibility In the Context Of Fluid Mixturesmentioning

confidence: 99%

Continuum thermodynamics of chemically reacting fluid mixtures

2014

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We consider viscous, heat conducting mixtures of molecularly miscible chemical species forming a fluid in which the constituents can undergo chemical reactions. Assuming a common temperature for all components, we derive a closed system of partial mass and partial momentum balances plus a mixture balance of internal energy. This is achieved by careful exploitation of the entropy principle and requires appropriate definitions of absolute temperature and chemical potentials, based on an adequate definition of thermal energy excluding diffusive contributions. The resulting interaction forces split into a thermo-mechanical and a chemical part, where the former turns out to be symmetric in case of binary interactions. For chemically reacting systems and as a new result, the chemical interaction force is a contribution being non-symmetric outside of chemical equilibrium. The theory also provides a rigorous derivation of the so-called generalized thermodynamic driving forces, avoiding the use of approximate solutions to the Boltzmann equations. Moreover, using an appropriately extended version of the entropy principle and introducing cross-effects already before closure as entropy invariant couplings between principal dissipative mechanisms, the Onsager symmetry relations become a strict consequence. With a classification of the factors in the binary products of the entropy production according to their parity--instead of the classical partition into so-called fluxes and driving forces--the apparent anti-symmetry of certain couplings is thereby also revealed. If the diffusion velocities are small compared to the speed of sound, the Maxwell-Stefan equations follow in the case without chemistry, thereby neglecting wave phenomena in the diffusive motion. This results in a reduced model with only mass being balanced individually. In the reactive case ..

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“…More elaborate and relatively recent treatments of the Boussinesq approximation are available and some of these are discussed in chapter 15 of the book [414], and in [31,134,159,160,311,359], and in addition [360,361]. Without a Boussinesq approximation to yield a simplified system like (1.42) with a solenoidal velocity field, one is left with treating convection in a compressible fluid, and this is a far more complicated issue, cf.…”

Section: The Balance Of Energy and The Boussinesq Approximationmentioning

confidence: 99%

Convection with Local Thermal Non-Equilibrium and Microfluidic Effects

Straughan

2015

Advances in Mechanics and Mathematics

113

111

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In this chapter we describe work of [366] and of [393] on thermal convection in a Darcy porous material in a vertical layer subject to different temperatures on the vertical walls.For the case of a single temperature the problem where the porous medium occupies a vertical layer subject to differential heating from one side to the other has attracted much attention. This problem has much application to insulation, such as in the area of building design, and is of importance in window double glazing. For the single temperature situation the first proof that a vertical porous slab of Darcy type which is held at fixed but different temperatures on the vertical walls is stable to perturbations from the equilibrium state is due to [156]. His analysis is based on the linear theory and analyses the two-dimensional problem.[409] further analysed this problem but in three-dimensions and he treated the completely nonlinear situation by employing energy-like integral methods. In particular, [409] produced a threshold for the Rayleigh number which guarantees global nonlinear stability, i.e. regardless of how large the initial perturbation may be. He also employed a generalized energy method to demonstrate that the result of [156] is true in the fully three-dimensional case although the initial data must be restricted. Articles dealing with other interesting aspects of convection in a vertical porous slab or pipe involving effects like double diffusion, and LTNE, include the papers by [32, 33, 36, 46, 211, 220, 221, 329, 330, 362] and [460]. The methods of energy stability techniques applied specifically to convection in a vertical porous slab are addressed in [144, 229] and [356], and these are discussed in detail in the book by [414, pp. 126-134].The analogous problem in LTNE to that of [156] was first addressed by [366]. Specifically [366] dealt with the [156] problem of thermal convection in a vertical porous medium, but he employed the theory of local thermal non-equilibrium, allowing for different fluid and solid temperatures. The interesting work of [366] demonstrates that the vertical configuration is always stable according to linear theory, even with the much more complicated LTNE theory. [393] continued the problem of [366] but they analysed the complete nonlinear three-dimensional situation.

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On the Müller paradox for thermal-incompressible media

Cited by 30 publications

References 15 publications

A consistent thermodynamical model of incompressible media as limit case of quasi-thermal-incompressible materials

A consistent thermodynamical model of incompressible media as limit case of quasi-thermal-incompressible materials

Continuum thermodynamics of chemically reacting fluid mixtures

Convection with Local Thermal Non-Equilibrium and Microfluidic Effects

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