Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

Málek, Josef; Průša, Vít

doi:10.1007/978-3-319-10151-4_1-1

Cited by 21 publications

(30 citation statements)

References 77 publications

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“…where j q denotes the heat flux vector, ρ denotes the density, and U ∶ V = def Tr (UV ⊺ ) denotes the Frobenius dot product on the space of matrices. (For details see Málek and Průša (2017) where we have used the fact that the matrix dot product of a symmetric matrix and a skew-symmetric matrix vanishes. Further, the assumed evolution equation (3.3) for B e implies that…”

Section: Thermodynamicsmentioning

confidence: 99%

A thermodynamic basis for implicit rate-type constitutive relations describing the inelastic response of solids undergoing finite deformation

Cichra,

Průša

2019

Preprint

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Implicit rate-type constitutive relations provide a novel approach to the purely phenomenological description of the inelastic response of solids undergoing finite deformation. However, this type of constitutive relations has been so far considered only in the purely mechanical setting, and the complete thermodynamic basis is largely missing. We address this issue, and we develop a thermodynamic basis for the implicit rate-type constitutive relations. In particular, we focus on the thermodynamic basis for the classical elastic-perfectly plastic response, but the framework is flexible enough to describe another types of inelastic response as well.

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Section: Thermodynamicsmentioning

confidence: 99%

A thermodynamic basis for implicit rate-type constitutive relations describing the inelastic response of solids undergoing finite deformation

Cichra,

Průša

2019

Preprint

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“…Thus, the first essential relation we take into account in this paper connects the definition of the local entropy, we denote by η, and the Helmholtz energy density F. More precisely, the Maxwell's identity (cf. [18], section 2.3) insures η to be defined by means of…”

Section: The Equations Of Motionmentioning

confidence: 99%

“…The above expression of the molecular field h is common in literature, when replacing the free energy density F by the well-known Oseen-Frank energy density (cf. definition (18)). The kinematic transport g is usually formulated as the orthogonal-projection with respect to n of g = γ 1 N + γ 2 Dn, that is…”

Section: The Equations Of Motionmentioning

confidence: 99%

“…It is worth to remark that system (2) coincides to the classical general Ericksen-Leslie system for the evolution of an incompressible nematic, whenever F reduces to the classical Oseen-Frank energy density (as in (18), below), σ L stands for the the classical Leslie tensor (as in (16), below) and moreover (ρ, ϑ , η) are assumed to be constant.…”

Section: The Equations Of Motionmentioning

confidence: 99%

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Non-isothermal General Ericksen–Leslie System: Derivation, Analysis and Thermodynamic Consistency

Anna

Liu

2018

Arch Rational Mech Anal

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article info abstractKeywords: Ericksen-Leslie, nematic liquid crystals, thermodynamics consistency, global wellposedness, Besov regularity.MSC: 35Q30, 35Q35, 35Q79, 76A15.We derive a model describing the evolution of a nematic liquidcrystal material under the action of thermal effects. The first and second laws of thermodynamics lead to an extension of the general Ericksen-Leslie system where the Leslie stress tensor and the Oseen-Frank energy density are considered in their general forms. The work postulate proposed by Ericksen-Leslie is traduced in terms of entropy production. We finally analyze the global-in-time well-posedness of the system for small initial data in the framework of Besov spaces.5.4. The temperature equation 36 5.5. Passage to the limit 42 5.6. Uniqueness 42 6. Appendix 43 References 49 IntroductionThe main aim of this paper is to derive and analyze an evolutionary PDE's-system modeling the dynamics of nematic liquid crystals. The model we are interested in extends the general Ericksen-Leslie theory, allowing a non-constant temperature. We derive such a model in accordance with the main laws of thermodynamics. Nowadays, the engineering and mathematical community is familiar with the concept of nematic liquid crystals. A nematic medium is a compound of fluid molecules, which has a state of matter between an ordinary liquid and a crystal solid. Although the centers of mass can freely translate as in a common fluid, the constitutive molecules present a privileged orientation. This alignment strongly interacts with the underlying flow of the nematic. Reinitzer discovered one of these materials in the 1888 and since then there have been numerous attempts to formulate continuum theories describing the time behavior of the flow. Ericksen and Leslie developed the most widely recognaized model during the 1960's in their pioneeric papers [7,8,22], generalizing the Oseen-Frank theory for the static case [14]. From the first mathematical success in analyzying the model performed by Lin and Liu [23] in the 1991, the dynamics theory of liquid crystal has become the new El Dorado of theoretical studies motivated by real-world applications. The well-posedness analysis in bounded domains [2,20,25] as well as in the whole space [6,15], has especially received high interest in the recent decades, both for what concerns the director theory and the Q-tensor framework. These results are often inspired by the ample literature concerning the Navier-Stokes equations, since any derivation of a consistent model for these anisotropic materials usually starts from the well-known conservation of mass and balances of linear and angular momentum. Despite a wide literature concerning the dynamics of nematic liquid crystals, to the best of our knowledge there are few papers dealing also with thermodynamic effects. Liquid crystals are mostly considered in an isothermal environment, which is sometimes unnatural. Indeed, as described by Stewart in [27], liquid phases are mainly induced by changing the temperature (thermotropic...

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“…To find the constitutive relations specifying the system by the means of the entropy production maximization, we need to determine how the system stores energy and how it produces entropy-to this end, we need to stipulate two scalar functions. The general framework goes as follows, for more details see [19].…”

Section: General Framework Of the Entropy Production Maximization Pro...mentioning

confidence: 99%

Gradient Dynamics and Entropy Production Maximization

Janečka

Pavelka

2018

Journal of Non-Equilibrium Thermodynamics

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Abstract:We compare two methods for modeling dissipative processes, namely gradient dynamics and entropy production maximization. Both methods require similar physical inputs-how energy (or entropy) is stored and how it is dissipated. Gradient dynamics describes irreversible evolution by means of dissipation potential and entropy, it automatically satisfies Onsager reciprocal relations as well as their nonlinear generalization (Maxwell-Onsager relations), and it has statistical interpretation. Entropy production maximization is based on knowledge of free energy (or another thermodynamic potential) and entropy production. It also leads to the linear Onsager reciprocal relations and it has proven successful in thermodynamics of complex materials. Both methods are thermodynamically sound as they ensure approach to equilibrium, and we compare them and discuss their advantages and shortcomings. In particular, conditions under which the two approaches coincide and are capable of providing the same constitutive relations are identified. Besides, a commonly used but not often mentioned step in the entropy production maximization is pinpointed and the condition of incompressibility is incorporated into gradient dynamics.

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Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

Cited by 21 publications

References 77 publications

A thermodynamic basis for implicit rate-type constitutive relations describing the inelastic response of solids undergoing finite deformation

A thermodynamic basis for implicit rate-type constitutive relations describing the inelastic response of solids undergoing finite deformation

Non-isothermal General Ericksen–Leslie System: Derivation, Analysis and Thermodynamic Consistency

Gradient Dynamics and Entropy Production Maximization

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