# Nonlinearities In The Nematic Stress Tensor

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### EPJE - New Jersey Institute of Technology

that the stress tensor is dependent on the strain rate via a relationship that is generally nonlinear, and that can be, as in the model used in the present work, diﬀerential. Historically, the foundations of the long-wave (or lu-brication) theory were laid in a pioneering work by Reynolds [7], that analyzed the behavior of a viscous liquid

### Hydrodynamic Fluctuations and Instabilities in Ordered

where I is the unit tensor, and fa= and w 0 c 0 are phenomenological constants proportional to the activity of the SPPs. In (5), nematic-elastic torques [14] and viscous forces, both of which are subdominant at small q, have been ignored, as have inertial and other nonlinearities. The pressurelike term despite transverse projection and the

### J. Fluid Mech. (2017), 822, pp. doi:10.1017/jfm.2017.311

by the nematic tensor Q ij D.3q=2/.n in j ij=3/, where q is the magnitude of the nematic order, n i is the director and ij is the Kronecker delta. The conservation equation for the nematic tensor is given by @ tQ ij Cu [email protected] xk Q ij D H ij CR ij; (2.1) where u k are velocity components, t is time, x i are spatial coordinates and is proportional to

### arXiv:cond-mat/0108301v2 [cond-mat.soft] 19 Mar 2002

where I is the unit tensor, and α ∼ fa/ρ and w0 ∼ c0α are phenomenological constants pro-portional to the activity of the SPPs. In (5), nematic elastic torques [13] and viscous forces, both of which are subdominant at small q, have been ignored, as have inertial and other nonlinearities.

### On the uniqueness of heat ow of harmonic maps and

ow of nematic liquid crystals For geometric nonlinear evolution equations or systems with critical nonlinearities, the stress tensor induced by the director

### THE ENIGMATIC ACTIVE NEMATIC

Active stress ∝ Q; current ∝ div Q Simha and SR 2002; Simha, SR, Toner 2003, Hatwalne et al. 2004 Kruse, Juelicher, Joanny, Prost, Voituriez, Sekimoto 2004 Q = orientation tensor (no polarity, only axis) Flow carries, rotates and aligns orientation Orientation distortions generate flow, particle currents

### NUMERICAL APPROXIMATIONS FOR A SMECTIC{A LIQUID CRYSTAL FLOW

Thermotropic LCs can be distinguished into two main di erent phases: Nematic and Smectic. In Nematic phases, the rod-shaped molecules have no positional order, but molecules self-align to have a long-range directional order with their long axes roughly parallel. Thus, the molecules are free to ow and their cen-

### Spontaneous Circulation of Conﬁned Active Suspensions

diffusion tensor, and DðrÞ is a rotational diffusion constant. The ﬂuid has velocity ﬁeld u, rate-of-strain tensor E ðruþruTÞ=2, and vorticity tensor W ð ru ruTÞ=2. The ﬁlament pusher stresslet of strength > 0 generates a stress tensor R p dpðpp I=dÞ that drives ﬂuid ﬂow by the Stokes equation 2r uþr ¼ r with

### Asters,Vortices, and Rotating Spirals in Active Gels of Polar

ics, we include geometric nonlinearities [18]. We thus systematically expand the ﬂuxes dp =dtandu interms of their conjugate forces, h F=p and the stress tensor , respectively, where Fis the polarization free energy. In order to drive the system to nonequilibrium steady states, we introduce a further pair of conjugate

### Electro-mechanical coupling in nematic elastomers: statics

nonlinearities of the stress-strain response. We denote by u(x) = y(x) xthe displacement at a point xof the body , where y is the deformation. We write F = ry for the deformation gradient, and B = FFT for the left Cauchy-Green strain tensor. We denote the nematic director, a unit vector ﬁeld on , by nand set N := n n, where

### Convective Nonlinearities for the Orientational Tensor Order

KEYWORDS: Nematic liquid crystalline polymers, orientational ﬂuctuations, convective nonlinearities, stress tensor 1 Introduction and Results The hydrodynamic description of low-molecular-weight nematic liquid crystals is well es-tablished. It has been derived (Martin et al. 1972) from the fact that the existence of 1

### EcoledesPonts-PekingUniversityjointworkshop

stress tensor as the right-hand side in the momentum equation. feedback and defects in sheared nematic diﬀerence due to orientation nonlinearities in the

### arXiv:0912.2283v1 [cond-mat.soft] 11 Dec 2009

order-parameter tensor has the simple form Q = S 2 cos2θ sin2θ sin2θ −cos2θ , (6) where the scalar order parameter S measures the magnitude of nematic order and θ is the angle from a reference direction. Let us work in the nematic phase, where we can take S = constant and deﬁne θ = 0 along axis of mean macroscopic orientation. Eq. (5) for

### HAL archive ouverte

HAL Id: jpa-00208832 https://hal.archives-ouvertes.fr/jpa-00208832 Submitted on 1 Jan 1978 HAL is a multi-disciplinary open access archive for the deposit and

### Regular and chaotic states in a local map description of

nematic, in a uniform shear ﬂow 11 13 Nonlinear relaxation equations for the symmetric, trace-less second rank tensor Q characterizing local order in a sheared nematic have been derived 11 18 Assuming spa-tial uniformity, a system of ﬁve coupled ordinary differential equations ODEs for the ﬁve independent components of Q

### Please note - -ORCA

(2.3), a>0is the temperature-dependent stretch parameter, ⊗ denotes the usual tensor product of two vectors, and I=diag(1,1,1)is the identity tensor. Here, it is assumed that a is spatially-independent (i.e., no differential swelling). For an ideal nematic solid, the anisotropy ratio r =a1/3/a−1/6=a1/2 is the same in all directions

### Published online 24 August 2011 Emergence of coherent

tive particle stress tensor modelling the effects of the force dipoles exerted by the particles on the surround-ing ﬂuid. Such a model was proposed by Simha & Ramaswamy [30], who extended phenomenological equations for liquid crystals to account for active stres-ses, and used them to investigate the stability of aligned suspensions.

### www.math.purdue.edu

REGULARITY AND EXISTENCE OF GLOBAL SOLUTIONS TO THE ERICKSEN-LESLIE SYSTEM IN R2 JINRUI HUANG, FANGHUA LIN, AND CHANGYOU WANG Abstract. In this paper, we rst establish the regular

### Exact results for sheared polar active suspensions with

The stress tensor is given by ˙ ij = 2 E ij + ˙r + ˙a; (7) where the rst term is the dissipative contribution ( is the uid viscosity), ˙r ij is reversible contribution (as in passive LCs), and ˙a ij is the active contribution. The reversible stress is given by ˙r ij= + 2 (p ih j+ ph) + 1 2 (ph ph); (8) where is the pressure. The active

### J. Fluid Mech. (2018), 841, pp. https://doi.org/10.1017/jfm

where p is the pressure, and the higher-order stress tensor ˙ D. 0 2r2 C 4r4/Trv C.rv/>U; (1.2) with r2n r2/n, n >2, accounts effectively for both passive contributions from the intrinsic solvent ﬂuid viscosity and active contributions representing the stresses exerted by the microswimmers on the ﬂuid. By construction, equations (1.1)

### N.#Admal

N.#Admal: A Decomposition of the Atomistic Stress Tensor into an Elastic and a Plastic Component In this talk, we propose an additive decomposition of the

### Spatiotemporal complexity of electroconvection patterns in

nematic liquid crystals is a prime paradigm for pattern nonlinearities and with the time dependence of the applied stress tensor depends in a complicated

### The Ratcheting behaviour of stainless steel pressurized

The nonlinearities are given as a recall term in the Prager rule [41]: 2 ddd 3 p XC X= −ε γε p (1) where X is the back stress tensor, dε p is the equivalent plastic strain rate, C and γ are two material de-pendent coefficients Frederick kinematic hardening modelin the Armstrong, and γ=0 stands for the linear

### Universities of Leeds, Sheffield and York http://eprints

stress ﬁbers [27]. However, there is another form of mechanical anisotropy that arises, not from the geometric microstructure of the network, but rather from the nonlinear response of individual ﬁbers. Consider applying an anisotropic prestress to an isotropic ﬁber network. This prestress can emerge sponta-

### Nonlinear Stress - Strain Behavior of Nematic Elastomers

Fig.3: The same stress-strain data points of Urayama et al. and the theoretical line obtained by the present model (with the nonlinear elastic experimental contributions added) now in the representation of the nominal stress as a function of the true strain.

### A Calculation of Orientational Relaxation in Nematic Liquid

density, the vector F any external body force acting, and t the stress tensor which here is asymmetric. The second is equivalent to conservation of angular momentum, and C- is a constant inertial coefficient, G a generalized body force arising from any body couple present due to magnetic or electric fields, g a generalized intrinsic body

### On well-posedness of Ericksen-Leslie's parabolic-hyperbolic

Inertial model in Q-tensor framework It is the system coupled a forced incompressible Navier-Stokes equations, modeling the ﬂow, with a hyperbolic convection-diffusion system for matrix-valued functions that model the evolution of the orientations of the nematic molecules. The inertial term is responsible for the hyperbolic character. Qian

### Complex dynamics of a sheared nematic fluid

Jul 18, 2020 by a nematic alignment tensor field reproduces several statistical features found in rheochaos and elastic turbulence. We numerically analyse the full non-linear hydrodynamic equations of a sheared nematic fluid under shear stress and strain rate controlled situations, incorporating spatial heterogeneity only in the gradient direction.

### arXiv:1212.0043v2 [math.AP] 26 Mar 2013

with Ginzburg-Landau type approximation modeling nematic liquid crystal ﬂows. First, by overcoming the diﬃculties from lack of maximum principle for the director equation and high order nonlinearities for the stress tensor, we prove existence of global-in-time weak solutions

### 41; ' # '9& *#4 &

ik is the viscous stress tensor, i are the viscosity Leslie coefficients, ik ¼ 1,i¼ k; ik ¼ 0,i6¼ k, and F is the free energy density of SALC. Typically, SALC is sup-

### $FWLYHQHPDWLFVRQDVXEVWUDWH *LDQWQXPEHU IOXFWXDWLRQVDQGORQJ

described by the traceless symmetric tensor ﬁeld Q with components Qij(r,t) at point r and time t. For uniaxial nematics, to which we restrict our attention here, we can write Q =[nn − (1/d)I]S, where the unit vector n is the director ﬁeld, I is the unit tensor, and the conventional scalar order parameter S [18] measures the degree of

### Phase Transition in Nematic Elastomers

Nematic elastomers are essentially nematic liquid crystals embedded in rubbery polymer melt. Examples of nematic elastomers include artificial muscles and contact lenses. Fig. 1: Nematic, Smectic-C and Tanaka gels with hard-rod dispersion We shall be investigating the kind of phase transition encountered in nematic elastomers, but it is

### EnhancedMixingatLowReynoldsNumbersThroughElasticTurbulence

The symmetric traceless stress tensor s is given by the sum of two terms, one involving the sec-ond Newtonian viscosity h¥, the other one the ex-tra stress p, viscoelastic effects can be included

### Stokes' second problem and reduction of inertia in active fluids

The stress tensor σ(t,x) comprises sufﬁciently low so that energy transfer due to nonlinearities arising from polar or nematic ordering potentials [54] can be

### Dynamics of the Orientational Tensor Order Parameter in

the nematic and the isotropic phase. Furthermore we give the form of the appropri-ate orientational-elastic stresses in the stress tensor. They are completely ﬁxed by the orientation dynamics and no choices are left. Thus, there are again diﬀerent expressions for the isotropic and the nematic phase. In particular, a simple stress-optical

### Revisit the Stress-Optical Rule for Entangled Flexible Chains

For flexible polymer chains, the deviatoric parts of the stress tensor and optical anisotropy tensor are proportional to each other in both linear and nonlinear regime, unless the chain is highly stretched in the latter regime and the finite extensible nonlinear elasticity (FENE) effect becomes important. 1,2)

### A pseudo-anelastic model for stress softening in liquid rspa

liquid crystals, nematic solids, large deformation, stress softening, residual strain, cyclic loads Author for correspondence: L. Angela Mihai e-mail: [email protected] A pseudo-anelastic model for stress softening in liquid crystal elastomers L. Angela Mihai1 and Alain Goriely2 1School of Mathematics, Cardiff University,

### 033. - shelf2.library.cmu.edu

order-parameter tensor field for nematic liquid crystals confined to finite cylinders by minimizing the Landau-deGennes free-energy functional subject to combinations of strong and weak surface anchoring and to external electric and magnetic fields. The problem is discretized by a finite-element method using piecewise-linear

### Preface p. xiii Introduction to Liquid Crystals p. 1

General Stress Tensor for Nematic Liquid Crystals p. 55 Flows with Fixed Director Axis Orientation p. 55 Flows with Director Axis Reorientation p. 57 Field-Induced Director Axis Reorientation Effects p. 58 Field-Induced Reorientation without Flow Coupling: Freedericksz Transition p. 58 Reorientation with Flow Coupling p. 61 References p. 62

### Contents

5 The twisted nematic display 304 6 Fluctuations and light scattering 306 6.3 Smectic liquid crystals 308 1 The elastic free energy 309 2 Fluctuations 312 3 Nonlinearities 314 4 The nematic-to-smectic-A transition 315 6.4 Elasticity of solids: strain and elastic energy 316 1 The strain tensor 316 2 The elastic free energy 318 3 Isotropic and