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

Abstract

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...

Non-isothermal electrokinetics: energetic variational approach

“…To emphasize, our approach guarantees the resulting equations satisfying fundamental laws of thermodynamics and it can be generalized to a variaty of complex fluid systems such as liquid crystal [6].…”

Thermostatics vs. Electrodynamics

“…The thermal dynamics being developed by Chun Liu's group promises advances in this endeavor once it marries mechanical models, chemical reactions and electrodynamics [158][159][160][161][162][163].…”

“…Chun Liu and his collaborators have started to develop a general approach to what they sometimes call non-isothermal systems. They are in fact creating a thermal dynamics [158][159][160][161][162][163] in which everything can interact with everything else, and flows always exist. Chemical equilibrium, and thermodynamic limits are conspicuous by their absence, making comparison with existing literature tricky.…”

Concentration-cancellation in the Ericksen–Leslie model

“…The energetic variational approach is executed in e.g. [7,23]. From the mathematical point of view, (1.1) preserves the major mathematical challenges of the full dynamic Ericksen-Leslie model.…”