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Register a new userPartial regularity of a nematic liquid crystal model with kinematic transport effects
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In this paper, we will establish the global existence of a suitable weak solution to the Erickson--Leslie system modeling hydrodynamics of nematic liquid crystal flows with kinematic transports for molecules of various shapes in ${mathbb{R}^3}$, which is smooth away from a closed set of (parabolic) Hausdorff dimension at most $frac{15}{7}$.
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Liquid crystal droplets are of great interest from physics and applications. Rigorous mathematical analysis is challenging as the problem involves harmonic maps (and in general the Oseen-Frank model), free interfaces and topological defects which could be either inside the droplet or on its surface along with some intriguing boundary anchoring conditions for the orientation configurations. In this paper, through a study of the phase transition between the isotropic and nematic states of liquid crystal based on the Ericksen model, we can show, when the size of droplet is much larger in comparison with the ratio of the Frank constants to the surface tension, a $Gamma$-convergence theorem for minimizers. This $Gamma$-limit is in fact the sharp interface limit for the phase transition between the isotropic and nematic regions when the small parameter $varepsilon$, corresponding to the transition layer width, goes to zero. This limiting process not only provides a geometric description of the shape of the droplet as one would expect, and surprisingly it also gives the anchoring conditions for the orientations of liquid crystals on the surface of the droplet depending on material constants. In particular, homeotropic, tangential, and even free boundary conditions as assumed in earlier phenomenological modelings arise naturally provided that the surface tension, Frank and Ericksen constants are in suitable ranges.
In this paper, we study the active hydrodynamics, described in the Q-tensor liquid crystal framework. We prove the existence of global weak solutions in dimension two and three, with suitable initial datas. By using Littlewood-Paley decomposition, we also get the higher regularity of the weak solutions and the uniqueness of weak-strong solutions in dimension two.
For any smooth domain $Omegasubset mathbb{R}^3$, we establish the existence of a global weak solution $(mathbf{u},mathbf{d}, theta)$ to the simplified, non-isothermal Ericksen-Leslie system modeling the hydrodynamic motion of nematic liquid crystals with variable temperature for any initial and boundary data $(mathbf{u}_0, mathbf{d}_0, theta_0)inmathbf{H}times H^1(Omega, mathbb{S}^2)times L^1(Omega)$, with $ mathbf{d}_0(Omega)subsetmathbb{S}_+^2$ (the upper half sphere) and $displaystyleinf_Omega theta_0>0$.
We create controllable active particles in the form of metal-dielectric Janus colloids which acquire motility through a nematic liquid crystal film by transducing the energy of an imposed perpendicular AC electric field. We achieve complete command over trajectories by varying field amplitude and frequency, piloting the colloids at will in the plane spanned by the axes of the particle and the nematic. The underlying mechanism exploits the sensitivity of electro-osmotic flow to the asymmetries of the particle surface and the liquid-crystal defect structure. We present a calculation of the dipolar force density produced by the interplay of the electric field with director anchoring and the contrasting electrostatic boundary conditions on the two hemispheres, that accounts for the dielectric-forward (metal-forward) motion of the colloids due to induced puller (pusher) force dipoles. These findings open unexplored directions for the use of colloids and liquid crystals in controlled transport, assembly and collective dynamics.
We consider a mathematical model that describes the flow of a Nematic Liquid Crystal (NLC) film placed on a flat substrate, across which a spatially-varying electric potential is applied. Due to their polar nature, NLC molecules interact with the (nonuniform) electric field generated, leading to instability of a flat film. Implementation of the long wave scaling leads to a partial differential equation that predicts the subsequent time evolution of the thin film. This equation is coupled to a boundary value problem that describes the interaction between the local molecular orientation of the NLC (the director field) and the electric potential. We investigate numerically the behavior of an initially flat film for a range of film heights and surface anchoring conditions.
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