56 research outputs found
Local well-posedness and small Deborah limit of a molecule-based -tensor system
In this paper, we consider a hydrodynamic -tensor system for nematic
liquid crystal flow, which is derived from Doi-Onsager molecular theory by the
Bingham closure. We first prove the existence and uniqueness of local strong
solution. Furthermore, by taking Deborah number goes to zero and using the
Hilbert expansion method, we present a rigorous derivation from the
molecule-based -tensor theory to the Ericksen-Leslie theory.Comment: 44 page
Global well-posedness for the dynamical Q-tensor model of liquid crystals
In this paper, we consider a complex fluid modeling nematic liquid crystal
flows, which is described by a system coupling Navier-Stokes equations with a
parabolic Q-tensor system. We first prove the global existence of weak
solutions in dimension three. Furthermore, the global well-posedness of strong
solutions is studied with sufficiently large viscosity of fluid. Finally, we
show a continuous dependence result on the initial data which directly yields
the weak-strong uniqueness of solutions
Conservation-Dissipation Formalism for Soft Matter Physics: II. Application to Non-isothermal Nematic Liquid Crystals
To most existing non-equilibrium theories, the modeling of non-isothermal
processes was a hard task. Intrinsic difficulties involved the non-equilibrium
temperature, the coexistence of conserved energy and dissipative entropy, etc.
In this paper, by taking the non-isothermal flow of nematic liquid crystals as
a typical example, we illustrated that thermodynamically consistent models in
either vectorial or tensorial forms could be constructed within the framework
of Conservation-Dissipation Formalism (CDF). And the classical isothermal
Ericksen-Leslie model and Qian-Sheng model were shown to be special cases of
our new vectorial and tensorial models in the isothermal, incompressible and
stationary limit. Most importantly, from above examples, it was learnt that
mathematical modeling based on CDF could easily solve the issues relating with
non-isothermal situations in a systematic way. The first and second laws of
thermodynamics were satisfied simultaneously. The non-equilibrium temperature
was defined self-consistently through the partial derivative of entropy
function. Relaxation-type constitutive relations were constructed, which gave
rise to the classical linear constitutive relations, like Newton's law and
Fourier's law, in stationary limits. Therefore, CDF was expected to have a
broad scope of applications in soft matter physics, especially under the
complicated situations, such as non-isothermal, compressible and nanoscale
systems.Comment: 29 page
Mathematical problems of nematic liquid crystals: between dynamical and stationary problems
Mathematical studies of nematic liquid crystals address in general two rather different perspectives: That of fluid mechanics and that of calculus of variations. The former focuses on dynamical problems while the latter focuses on stationary ones. The two are usually studied with different mathematical tools and address different questions. The aim of this brief review is to give the practitioners in each area an introduction to some of the results and problems in the other area. Also, aiming to bridge the gap between the two communities, we will present a couple of research topics that generate natural connections between the two areas. This article is part of the theme issue 'Topics in mathematical design of complex materials'
Global strong solutions of the full Navier-Stokes and -tensor system for nematic liquid crystal flows in : existence and long-time behavior
We consider a full Navier-Stokes and -tensor system for incompressible
liquid crystal flows of nematic type. In the two dimensional periodic case, we
prove the existence and uniqueness of global strong solutions that are
uniformly bounded in time. This result is obtained without any smallness
assumption on the physical parameter that measures the ratio between
tumbling and aligning effects of a shear flow exerting over the liquid crystal
directors. Moreover, we show the uniqueness of asymptotic limit for each global
strong solution as time goes to infinity and provide an uniform estimate on the
convergence rate
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