NUMERICAL BIFURCATION OF EQUILIBRIA OF NEMATIC CRYSTALS BETWEEN NON-CO-AXIAL CYLINDERS

2001 ◽  
Vol 11 (03) ◽  
pp. 459-473
Author(s):  
FRANÇOIS ALOUGES ◽  
BERNARD D. COLEMAN
Keyword(s):  
Stable Equilibrium ◽  
Circular Cylinders ◽  
Projected Gradient Method ◽  
Director Field ◽  
Finite Element Implementation ◽  
Concentric Circles ◽  
Numerical Bifurcation ◽  
The One ◽  
Bounding Surfaces ◽  
Minimizing Maps

Calculations are presented giving the stable equilibrium states of nematic liquid crystals confined between non-co-axial circular cylinders subject to the condition that the director be normal to the bounding surfaces. The one-constant theory of nematics is employed and emphasis is laid on cases for which the minimizing director field does not lie in the plane perpendicular to the cylinders. Calculations are made of the force that the nematic exerts on the cylinders, tending to make them co-axial. The numerical problem faced is that of finding, for the Dirichlet energy, minimizing maps to S2, from a region in ℝ2 bounded by non-concentric circles. The method employed is a finite-element implementation of a projected gradient method for which the energy decreases at each iteration.

scholarly journals Stable and Metastable Patterns in Chromonic Nematic Liquid Crystal Droplets Forced with Static and Dynamic Magnetic Fields

Crystals ◽  
2020 ◽  
Vol 10 (2) ◽  
pp. 138 ◽  
Author(s):  
Jordi Ignés-Mullol ◽  
Marc Mora ◽  
Berta Martínez-Prat ◽  
Ignasi Vélez-Cerón ◽  
R. Santiago Herrera ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Liquid Crystal ◽  
Liquid Crystals ◽  
Boundary Conditions ◽  
Isotropic Phase ◽  
Director Field ◽  
Line Defects ◽  
The One ◽  
Bulk Structures ◽  
Microscopy Images

Spherical confinement of nematic liquid crystals leads to the formation of equilibrium director field configurations that include point and line defects. Driving these materials with flows or dynamic fields often results in the formation of alternative metastable states. In this article, we study the effect of magnetic field alignment, both under static and dynamic conditions, of nematic gems (nematic droplets in coexistence with the isotropic phase) and emulsified nematic droplets of a lyotropic chromonic liquid crystal. We use a custom polarizing optical microscopy assembly that incorporates a permanent magnet whose strength and orientation can be dynamically changed. By comparing simulated optical patterns with microscopy images, we measure an equilibrium twisted bipolar pattern within nematic gems that is only marginally different from the one reported for emulsified droplets. Both systems evolve to concentric configurations upon application of a static magnetic field, but behave very differently when the field is rotated. While the concentric texture within the emulsified droplets is preserved and only displays asynchronous oscillations for high rotating speeds, the nematic gems transform into a metastable untwisted bipolar configuration that is memorized by the system when the field is removed. Our results demonstrate the importance of boundary conditions in determining the dynamic behavior of confined liquid crystals even for configurations that share similar equilibrium bulk structures.

scholarly journals The rotation of two circular cylinders in a viscous fluid

1922 ◽  
Vol 101 (709) ◽  
pp. 169-174 ◽  
Keyword(s):  
Viscous Fluid ◽  
Boundary Conditions ◽  
Stream Function ◽  
Elastic Stress ◽  
Steady Motion ◽  
Elastic Equilibrium ◽  
Circular Cylinders ◽  
Two Dimensional ◽  
Concentric Circles ◽  
Hydrodynamical Problem

In a previous communication we employed the solution of the equation ∇ 4 ψ = 0 in bipolar co-ordinates defined by α + iβ = log x + i ( y + a )/ x + i ( y - a ) (1) to discuss the problem of the elastic equilibrium of a plate bounded by any two non-concentric circles. There is a well-known analogy between plain elastic stress and two-dimensional steady motion of a viscous fluid, for which the stream-function satisfies ∇ 4 ψ = 0. The boundary conditions are, however, different in the two cases, and the hydrodynamical problem has its own special difficulties.

A systematic theological theory of truth in Kathryn Tanner's Jesus, Humanity and the Trinity: A Brief Systematic Theology

2004 ◽  
Vol 57 (2) ◽  
pp. 203-220 ◽  
Author(s):  
Christine Helmer
Keyword(s):  
World View ◽  
Two Dimensions ◽  
Systematic Theology ◽  
Gift Giving ◽  
Concentric Circles ◽  
Christian Thought ◽  
The One ◽  
Ethical Response ◽  
The Trinity ◽  
Contemporary Context

This paper raises the truth question to explore the possibility of constructing a system of systematic theology in the contemporary context. By using Schleiermacher's thought as a constructive resource, two dimensions of truth are discussed in dialogue with Kathryn Tanner's book Jesus, Humanity and the Trinity: A Brief Systematic Theology: coherence and correspondence. Addressed in the first section is coherence, along with its auxiliary, comprehensiveness, as a requirement for system. The whole of the Christian world view is grasped by a principle of coherence that reflects both a subjective component of authorial individuality and an objective component of an individual grasp of the ‘spirit of the age’. Furthermore, consensus regarding systematic coherence in Western Christian thought agrees on a metanarrative structuring doctrines from creation to apocalypse. In Tanner's work, the principle of coherence is gift-giving that is traced from an inner-trinitarian narrative to its outer-trinitarian expressions in creation, the incarnation, eschatology, as well as in ethical response. Thematized in the second section is the tricky issue of theological correspondence between claims of redemption on the one hand and the sin, evil, and tragedy of the world on the other. Tanner's book offers one solution to this problem: analogical correspondence that situates doctrinal patterns of gift-giving (such as the incarnation, creation) as concentric circles in varying degrees of correspondence to their inner-trinitarian center. In view of this solution, this paper questions the relation between systematic coherence and the reality of both life's brokenness and the depths of divine self-giving pathos.

Analysis of the Dynamic Behaviour of Flexible Risers Considering Hysteretic Damping

2003 ◽  
Author(s):  
Marcos Q. de Siqueira ◽  
Jose´ Renato M. de Sousa ◽  
Ma´rcio M. Mourelle
Keyword(s):  
Finite Element ◽  
Dynamic Response ◽  
Dynamic Behaviour ◽  
Traditional Approach ◽  
Three Dimensional ◽  
Bending Moment ◽  
Finite Element Implementation ◽  
Hysteretic Damping ◽  
Flexible Risers ◽  
The One

This paper proposes a three-dimensional beam element that can predict the dynamic response of flexible risers considering their bending hysteretic behaviour. The finite element implementation relies on a previously presented co-rotated beam finite element and an analytical formulation for determining the bending moment vs. curvature relationship for flexible risers. A comparison between the dynamic response predicted with the proposed finite element and the one obtained with the traditional approach was performed.

scholarly journals Influence of transverse loading on the stability of a clamped rectangular plate

2018 ◽  
Vol 239 ◽  
pp. 01022
Author(s):  
Mikhail Sukhoterin ◽  
Sergey Baryshnikov ◽  
Tatiana Knysh ◽  
Natalia Pizhurina
Keyword(s):  
Stable Equilibrium ◽  
Compressive Loading ◽  
Transverse Loading ◽  
Before And After ◽  
The Stability ◽  
The One ◽  
The Given ◽  
Small Deformations ◽  
Interior Points ◽  
Clamped Edge

Shapes of a square Kirchhoff plate with a clamped edge are obtained and analyzed, before and after losing stability in the case of a compound bending (uniform transverse loading in combination with edge compressive loading), as well as equilibrium forms and critical loadings only with clamping in the plate’s surface. Hyperbolic trigonometric series are used for solving. It was established that transverse loading causing small deformations does not affect the plate’s stability. The range of the critical state corresponds with an unlimited increase in bends of interior points of a plate. As critical loading, we suggest taking the one at which the bends at the plate’s center tend to infinity the most rapidly. As balanced loading, we suggest taking the one at which the plate acquires a new stable equilibrium form. A range of critical and balanced loadings of a square plate with a clamped edge was presented. The corresponding 3D forms of supercritical equilibrium of the given plate were obtained. A comparison with the results of other authors is given.

Three-Dimensional Flow Pattern Upstream of a Surface-Mounted Rectangular Obstruction

1989 ◽  
Vol 111 (4) ◽  
pp. 449-456 ◽  
Author(s):  
K. Y. M. Lai ◽  
A. H. Makomaski
Keyword(s):  
Saddle Point ◽  
Pressure Coefficient ◽  
Three Dimensional ◽  
Computer Code ◽  
Experimental Results ◽  
Circular Cylinders ◽  
Vortex Model ◽  
Flow Phenomena ◽  
The One ◽  
Corner Vortex

A three-dimensional TEACH-like computer code is developed and employed to study the flow phenomena upstream of a rectangular obstruction placed in a two-dimensional turbulent boundary layer. Satisfactory comparison is obtained with the experimental results of Blair (1984, 1987). The general trends, regarding the dependence of vortex dimensions, wall static pressure distributions and saddle point positions on Re = U∞δ*/ν and on D* = D/δ*, are similar to the experimental results for circular cylinders (Eckerle and Langston, 1986 and Baker, 1980). The position of the saddle point depends on the turbulence intensity in the primary vortex. The pressure coefficient at the foot of the obstruction depends solely on D* if no corner vortex exists. This coefficient is reduced when a corner vortex is present. All the computed flow patterns are similar to the one-vortex model of Eckerle and Langston (1986). The four-vortex model reported by Baker (1980) and Hunt et al. (1978) cannot be found in any of the calculations.

Uniqueness of director configuration states for liquid crystals in the case of weak anchoring

2019 ◽  
Vol 84 (4) ◽  
pp. 748-762
Author(s):  
Carlo Greco
Keyword(s):  
Magnetic Field ◽  
Liquid Crystal ◽  
External Field ◽  
Thin Layer ◽  
Elastic Constants ◽  
Cholesteric Liquid Crystal ◽  
Stable Configuration ◽  
Director Field ◽  
Critical Thresholds ◽  
Bounding Surfaces

Abstract In a thin layer of liquid crystal the configuration of the director field depends on the interaction between the elastic properties of the material, the thickness $d$ of the layer, the boundary conditions and the external fields that may have been applied. Suitable combinations of these factors can give rise to distorted configurations (Freedericksz transitions). In this paper we assume the Oseen-Frank model for the energy and that the director field depends only on the direction orthogonal to the layer; we assume also weak anchoring conditions at the two bounding surfaces, and we mainly study the problem of uniqueness of such distorted configurations. More precisely, we first consider the nematic case in the presence of a magnetic field $\mathbf H$, and we prove the uniqueness of the stable configuration provided the magnitude of $\mathbf H$ is between two critical thresholds, simplifying some results already known in the literature, and calculating explicitly the critical thresholds. Then we study the case of a cholesteric liquid crystal without external field. In this case the director field tends to form a right-angle helicoid around a twist axis orthogonal to the layer, and we have distorted configurations (namely oblique helicoid) for suitable value of $d$. Also in this case, with suitable restrictions on the elastic constants in the Oseen-Frank energy, we find two critical thresholds for $d$, and we prove the existence of only one stable director configuration if $d$ is between them.

On static equilibrium of a hemispheroid

2014 ◽  
Vol 98 (541) ◽  
pp. 73-78
Author(s):  
Subhranil De
Keyword(s):  
Stable Equilibrium ◽  
Flat Surface ◽  
Static Equilibrium ◽  
Equatorial Radius ◽  
Solid Objects ◽  
Coffee Table ◽  
The One

In the course of a coffee-table conversation with my friends regarding the nature of static equilibrium of different solid objects the situation involving a uniform hemisphere came up. Intuition (and perhaps experience) tells that a uniform hemisphere as shown in Figure 1 resting on a flat surface will be at stable equilibrium, and so will an oblate hemispheroid as shown in Figure 2. Things get complicated when we move to a prolate hemispheroid like the one shown in Figure 3, for the nature of its equilibrium is less obvious. The intuition does come to mind though that if the prolate hemispheroid is made indefinitely taller, keeping its equatorial radius fixed, then the equilibrium should eventually become unstable. Intrigued, we decided to probe into the matter quantitatively.

Orientational director effects in nematic liquid crystals with small Ericksen number

2006 ◽  
Vol 462 (2073) ◽  
pp. 2737-2758 ◽  
Author(s):  
C Atkinson ◽  
P.J.S Pereira
Keyword(s):  
Liquid Crystals ◽  
Differential Equations ◽  
Elastic Constant ◽  
Nematic Liquid Crystals ◽  
Nonlinear Partial Differential Equations ◽  
Infinite Plate ◽  
Director Field ◽  
The Many ◽  
The One ◽  
Strong Anchoring

Orientational director effects in nematic liquid crystals with small Ericksen number are investigated. The director field is disturbed by a semi-infinite plate on y =0 and and moving with a constant velocity U . Strong anchoring conditions at the plate are assumed. The resulting equations are a system of nonlinear partial differential equations for a nematic in the one elastic constant approximation. These equations are reduced to a coupled set of nonlinear ordinary differential equations by a suitable transformation. No such transformation seems possible for the many elastic constant case. The resulting equations are solved using analytical methods and strict bounding solutions obtained. These strict analytical solutions are compared with Picard iterated solutions.

scholarly journals The plastico-viscous deformation of right circular cylinders of soft metal under variable load axially directed

1934 ◽  
Vol 233 (721-730) ◽  
pp. 217-245
Keyword(s):  
The Other ◽  
Circular Cylinders ◽  
Variable Load ◽  
Similar Nature ◽  
Viscous Deformation ◽  
Standard Of Comparison ◽  
Soft Metal ◽  
The Subject ◽  
The Many ◽  
The One

1. Soft metals are of service to man in many ways, extended use following each advance in knowledge. The present investigation had its origin in an attempt to account analytically for the many apparently irreconcilable properties exhibited by right circular cylinders of soft copper when subjected to appreciable strain under heavy crushing loads. During the course of the work it became clear that if advance at all were to be made general methods of analysis would require to be adopted ; and that results forthcoming would be applicable to materials, of a similar nature, other than the one directly used as a standard of comparison. The presentation of the subject matter has accordingly been arranged to give prominence to theoretical and rational aspects, references to experiment being rather for comparison than for support to the argument. Two publications will be found of service in the present connection, the one illustrative of a cogent point in the theory, the other descriptive of practice. They will be referred to as I, and II, respectively.

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