Global Solution to the Three-Dimensional Incompressible Flow of Liquid Crystals

The equations for the three-dimensional incompressible flow of liquid crystals are considered in a smooth bounded domain. The existence and uniqueness of the global strong solution with small initial data are established. It is also proved that when the strong solution exists, all the global weak solutions constructed in [16] must be equal to the unique strong solution

Landau-De Gennes theory of nematic liquid\ud crystals: the Oseen-Frank limit and beyond

We study global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in W 1,2 , to a global minimizer predicted by the Oseen-Frank theory for uniaxial nematic liquid crystals with constant order parameter. Moreover, the convergence is uniform in the interior of the domain, away from the singularities of the limiting Oseen-Frank global minimizer. We obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the Landau-De Gennes global minimizer.\ud \ud \ud We also study the interplay between biaxiality and uniaxiality in Landau-De Gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions

Blow up criterion for compressible nematic liquid crystal flows in dimension three

In this paper, we consider the short time strong solution to a simplified hydrodynamic flow modeling the compressible, nematic liquid crystal materials in dimension three. We establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of both the maximum norm of the deformation tensor of velocity gradient and the square of maximum norm of gradient of liquid crystal director field.Comment: 22 page

Hydrodynamics of domain growth in nematic liquid crystals

We study the growth of aligned domains in nematic liquid crystals. Results are obtained solving the Beris-Edwards equations of motion using the lattice Boltzmann approach. Spatial anisotropy in the domain growth is shown to be a consequence of the flow induced by the changing order parameter field (backflow). The generalization of the results to the growth of a cylindrical domain, which involves the dynamics of a defect ring, is discussed.Comment: 12 revtex-style pages, including 12 figures; small changes before publicatio

Well-Posedness of Nematic Liquid Crystal Flow in Luloc3(R3)L^3_{\hbox{uloc}}(\R^3)

In this paper, we establish the local well-posedness for the Cauchy problem of the simplified version of hydrodynamic flow of nematic liquid crystals (\ref{LLF}) in $\mathbb R^3$ for any initial data $(u_0,d_0)$ having small $L^3_{\hbox{uloc}}$-norm of $(u_0,\nabla d_0)$. Here $L^3_{\hbox{uloc}}(\mathbb R^3)$ is the space of uniformly locally $L^3$-integrable functions. For any initial data $(u_0, d_0)$ with small $\displaystyle |(u_0,\nabla d_0)|_{L^3(\mathbb R^3)}$, we show that there exists a unique, global solution to (\ref{LLF}) which is smooth for $t>0$ and has monotone deceasing $L^3$-energy for $t\ge 0$.Comment: 29 page

Weak-strong uniqueness property for the full Navier-Stokes-Fourier system

The Navier-Stokes-Fourier system describing the motion of a compressible, viscous, and heat conducting fluid is known to possess global-in-time weak solutions for any initial data of finite energy. We show that a weak solution coincides with the strong solution, emanating from the same initial data, as long as the latter exists. In particular, strong solutions are unique within the class of weak solutions

A comparison of Finite Elements for Nonlinear Beams: The absolute nodal coordinate and geometrically exact formulations

Two of the most popular finite element formulations for solving nonlinear beams are the absolute nodal coordinate and the geometrically exact approaches. Both can be applied to problems with very large deformations and strains, but they differ substantially at the continuous and the discrete levels. In addition, implementation and run-time computational costs also vary significantly. In the current work, we summarize the main features of the two formulations, highlighting their differences and similarities, and perform numerical benchmarks to assess their accuracy and robustness. The article concludes with recommendations for the choice of one formulation over the other

Liquid crystals and their defects

These lecture notes discuss classical models of liquid crystals, and the different ways in which defects are described according to the different models.Comment: CIME lecture course, Cetraro, 201

A Review of Fiber-Reinforced Injection Molding: Flow Kinematics and Particle Orientation

The existing flow and particle orientation models applicable to fiber- reinforced injection molding are reviewed. After a brief description of injection molding, previous studies on the flow kinematics and fiber reinforcement are presented. Basics of Hele-Shaw flows are described Including the commonly used viscosity models and foun tain flow effects. Some of the existing models for particle orientation are analyzed with particular emphasis on the amsotropic description of the material system. Concentration regions for short fiber suspensions are defined and relevant constitutive equations are dis cussed. A few example solutions are also given which describe the three-dimensional ori entation field for the filling of a sudden expansion cavity, depicting skin-core orientation structure.Yeshttps://us.sagepub.com/en-us/nam/manuscript-submission-guideline