A model of weak viscoelastic nematodynamics

The paper develops a continuum theory of weak viscoelastic nematodynamics of Maxwell type. It can describe the molecular elasticity effects in mono-domain flows of liquid crystalline polymers as well as the viscoelastic effects in suspensions of uniaxially symmetric particles in polymer fluids. Along with viscoelastic and nematic kinematics, the theory employs a general form of weakly elastic thermodynamic potential and the Leslie–Ericksen–Parodi type constitutive equations for viscous nematic liquids, while ignoring inertia effects and the Frank (orientation) elasticity in liquid crystal polymers. In general case, even the simplest Maxwell model has many basic parameters. Nevertheless, recently discovered algebraic properties of nematic operations reveal a general structure of the theory and present it in a simple form. It is shown that the evolution equation for director is also viscoelastic. An example of magnetization exemplifies the action of non-symmetric stresses. When the magnetic field is absent, the theory is reduced to the symmetric, fluid mechanical case with relaxation properties for both the stress and director. Our recent analyses of elastic and viscous soft deformation modes are also extended to the viscoelastic case. The occurrence of possible soft modes minimizes both the free energy and dissipation, and also significantly decreases the number of material parameters. In symmetric linear case, the theory is explicitly presented in terms of anisotropic linear memory functionals. Several analytical results demonstrate a rich behavior predicted by the developed model for steady and unsteady flows in simple shearing and simple elongation.     

简支Mexwell模型粘弹性输流管道的稳定性分析

在弹性输流管道研究的基础上,采用递推格式的有限差分法,对简支Maxwell模型粘弹性输流管道（回转守恒系统）,探讨了其动力特性和稳定性问题,具体分析了材料的松弛时间对无量纲流速与前三阶模态的无量纲频率的实部及虚部之间的变化曲线的影响。发现发散临界流速随松弛时间的减小而降低,随后发生的耦合模态颤振临界流速随松弛时间的减小而增大;甚至在质量比较大时,随着松弛时间的减小,可推迟乃至不发生耦合模态颤振。当无量纲松弛时间达到10^3量级以上时,即可将其按弹性管道处理。甚至在H为10^2量级时,按弹性管道处理也不会带来太大的误差。     

Approximate controllability results for viscoelastic flows with infinitely many relaxation modes

We prove results on approximate controllability for linear viscoelastic flows, with a localized distributed control in the momentum balance equation. The constitutive law is a multimode Maxwell or Jeffreys model with an infinite number of relaxation modes.     

General Solution for the Two-Dimensional System of Static Lame’s Equations with an Asymmetric Elasticity Matrix

We study a two-dimensional system of equations of linear elasticity theory in the case when the symmetric stress and strain tensors are related by an asymmetric matrix of elasticity moduli or elastic compliances. The linear relation between stresses and strains is written in an invariant form which contains three positive eigenmodules in the two-dimensional case. Using a special eigenbasis in the strain space, it is possible to write the constitutive equations with a symmetric matrix, i.e., in the same way as in the case of hyperelasticity. We obtain a representation of the general solution of two-dimensional equations in displacements as a linear combination of the first derivatives of two functions which satisfy two independent harmonic equations. The obtained representation directly implies a generalization of the Kolosov–Muskhelishvili representation of displacements and stresses in terms of two analytic functions of complex variable. We consider all admissible values of elastic parameters, including the case when the system of differential equations may become singular. We provide an example of solving the problem for a plane with a circular hole loaded by constant stresses.     

Transition functions of the Il'yushin approximation method

Transition functions are constructed whose direct substitution for combinations of elastic constants in solutions of problems of elasticity theory permits writing the solution of the corresponding viscoelastic problems. Examples of using the transition functions for obtaining solutions of viscoelastic problems are given: circularly symmetric solid and anular plates, plate with a small circular hole, and viscoelastic half-plane loaded by a concentrated force.M. M. Gubkin Moscow Institute of the Petrochemical and Gas Industry. Moscow Institute of Electronic Engineering. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 405–416, May–June, 1973.     

Numerical Studies of Viscoelastic Flow Using the Software OpenFOAM

Viscoelastic fluids are a special class of non-Newtonian fluids. There are several types of viscoelastic fluid models, and all of them have a complex rheological response in comparison to Newtonian fluids. This response can be viewed as a combination of viscous and elastic effects and non-linear phenomena. This complex physics makes a numerical simulation a rather challenging task, even in simple test-cases. Studies presented in this paper are numerical studies of the viscoelastic fluid flow in several test cases. These studies have been done in OpenFOAM, an open-source CFD package. Implementation of viscoelastic models and a solver is only available in a community driven version of software (OpenFOAM-ext). One of the goals of research in this paper was to test the solver and models on some simple test cases. We considered start-up and pulsating flows of viscoelastic fluid in a channel and a circular pipe. The important thing is that an analytical solution can be found in these cases, making in possible to test all aspects of numerical simulation in OpenFOAM. Obtained results showed an excellent agreement with the analytical solution for both velocity and stress components. These results encouraged authors' motivation and a choice to use OpenFOAM for simulation of viscoelastic flows. We hope that our research will make a contribution to the OpenFOAM community. Our plan for the further research is a simulation of blood flow in arteries with the viscoelastic solver. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A theory of elasticity with spatial distribution of matter. Three-dimensional complex structure

References [1 and 2] consider a theory of elasticity with spatial distribution of matter for a medium having simple structure and for a one-dimensional medium having complex structure. In the present article the general case of a three-dimensional medium with complex structure is examined. The general scheme of the one-dimensional case [2] is retained; chief attention is directed toward the specific character of the three-dimensional problem. The original micro-model is a complex crystal lattice [3]. In Section 1 this model is generalized to the case of a continuous distribution of matter. The displacements of the mass centers of the unit cells and the micro-strains of the cells are introduced as the kinematic variables. The force variables are the micro-moments. The transition to an exact continuous representation is carried out, and the equations of an elastic medium of complex structure with spatial distribution of matter are derived. The operators corresponding to the continuous theory are expressed in terms of the original microparameters. It is shown that the well known conditions of symmetry of the tensor of elastic constants, which are usually interpreted as the condition of absence of initial stresses [3 and 4], are consequences of the invariance of the elastic energy under translation and rotation. In Section 2 some special models are examined, and the equations of a medium are obtained for the approximation of weak dispersion of matter. These equations contain as a special case the equations of linear nonsymmetric elasticity (couple-stress theory) [5 to 7]. However, in the latter it turns out that the orders of approximation are inconsistent in the various equations from the point of view of the theory of spatial distribution.

In Section 3 the equations of a medium having complex structure are transformed in the acoustic range into equations, one of which contains only a single kinematic variable (the displacement of the mass centers) and the others of which are explicitly solvable for the remaining kinematic variables. The first equation of this set coincides in form with the equation for a medium with simple structure, but differs from it by the presence of a timewise dispersion which is unrelated to energy dissipation. Expressions are written for the energy density, and it is shown that it is possible to introduce a symmetric stress tensor, as in the case of a simple structure.     

On Symmetric Boundary Conditions in the Context of Viscoelastic Fluid Flows

In order to reduce the numerical cost of three dimensional flow problems with geometrical symmetry, the use of symmetric boundary conditions is standard. For Newtonian fluid flow problems this approximation is usually appropriate, particularly when the Reynolds number is small. In the case of viscoelastic fluid flow simulations with stabilization techniques, such as the so-called DEVSS and/or Log-Conformation tensor methods, at high Deborah number flows this implementation is not straightforward, as in the Newtonian case. It is well known that viscoelastic models (e.g. Maxwellian models), show (purely) elastic flow instabilities when the Deborah number is increased above a critical value, even under creeping flow conditions. In this work we present numerical simulations with different stabilization techniques and different differential viscoelastic models at high Deborah number flows. As a test-case, we compare the flow in a full two-dimensional cross-slot geometry to show the asymmetrical behavior of the viscoelastic fluid flow. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Force and energy parameters of the process of viscoelastic polymer flow between calender rolls

Polymer flows between rolls are treated as continua whose rheological behavior is described by a Maxwell model. Analytic expressions in quite simple and convenient form are obtained for the pressure gradient and the contact forces of friction. These are easily converted into relations for the specific pressure, thrust, and effective power for given starting parameters. The solutions obtained correctly reflect a number of qualitative characteristics of the flow of viscoelastic polymers. A comparison with the experimental data shows that the proposed method is suitable for use in engineering practice.Leningrad Lensovet Technological Institute. Translated from Mekhanika Polimerov, No. 4, pp. 734–738, July–August, 1970.     

Remarks on flows of elastic fluids along cylindrical boundaries

A study is undertaken to ascertain non-Newtonian effects in steady flows of elastic fluids along circular cylinders when there is extraction across the boundary surfaces. Exact representations for the velocity and stress fields are derived when the fluid prototypes are generalized Maxwell prototypes (exhibiting stress relaxation). Whilst the scheme of study has physical importance, it is the nature of the solution which is of foremost interest here, it being shown that the presence of elasticity (of the type considered) has a marked effect on the transport velocity.     

Symmetries in equations of incompressible viscoelastic Maxwell medium<Superscript>*</Superscript>

We consider unsteady flows of incompressible viscoelastic Maxwell medium with upper, low, and Jaumann convective derivatives in the rheological constitutive law. We give characteristics of a system of equations that describe a three-dimensional motion of such a medium for all three types of convective derivative. In the general case, due to the incompressibility condition, the equations of motion have both real and complex characteristics. We study group properties of this system in the two-dimensional case. On this basis, we choose submodels of the Maxwell model that can be reduced to hyperbolic ones. The properties of the hyperbolic submodels obtained depend on the choice of the invariant derivative in the rheological relation. We also present concrete examples of invariant solutions.     

Stability analysis of cantilever carbon nanotubes subjected to partially distributed tangential force and viscoelastic foundation

Dynamic instability of cantilever carbon nanotubes conveying fluid embedded in viscoelastic foundation under a partially distributed tangential force is investigated based on nonlocal elasticity theory and Euler–Bernouli beam theory. The present study has incorporated the effects of nonlocal parameter, Knudsen number, surface effects and magnetic field. And two main parameters have also considered, namely partially distributed tangential force and foundation. It is assumed that viscoelastic foundation has modeled as Kelvin–Voigt, Maxwell and Standard linear solid types. The size-dependent governing equation of transverse vibration is derived using Hamilton’s variational principle and discretized by the Galerkin truncation method. A detailed parameter study is carried out, indicating the stability behavior of the nanotubes. In the light of numerical results, it is shown that variables considered in nondimensional equations have significant effects on natural frequencies and flutter velocities, especially for the foundation distribution length and model as well as the partially distributed tangential force.     

On the direct connection of rheological elements in nonlinear continuum mechanics

The structure of complicated phenomenological material models at finite strains is often exemplified with the help of rheological elements. Thereby, simple material behaviour, i.e. elasticity or viscous and plastic flow, are composes by components. In our approach, we directly apply this concept to obtain material models at finite strains. Towards this end, the thermodynamically consistent material behaviour of single elements is defined first. Subsequently, the elements are connected by evaluation of stress equilibria equations formulated on interconnecting configurations. The basic equations of this concept are presented using the example of nonlinear viscoelasticity of Maxwell type. The model results from a series connection of an elastic and a viscous element, whereas both are formulated in a thermodynamically consistent way within the framework on nonlinear continuum mechanics. Furthermore, an approach of numerical implementation using the stress equilibria is suggested. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Acoustic Torque Acting upon Nematic Liquid Crystals

There is plenty of experimental evidence that the propagation of an ultrasonic wave in a nematic liquid crystal affects the director n, which represents the average molecular orientation, thus producing detectable optical effects. There have been several attempts to explain these observations on the basis of a coherent variational theory. We consider here a general theory for nematoacoustics that incorporates flow effects and that has been recently proposed in E.G.?Virga (Phys. Rev. E 80:031705, 2009). An explicit application of the proposed theory to a simple computable case was given in G.?De?Matteis and E.G.?Virga (Phys. Rev. E 83:011703, 2011) by linearizing the corresponding balance equations derived from the basic theory. This study was done in order to estimate phenomenological parameters involved in the theory and by using available experimental data. After reviewing the results previously obtained, as a further application of the governing equations, we consider here an approximate equation for the flowless nematodynamics by introducing a second-order acoustic torque acting upon the nematic and we solve the obtained equation in a standard geometry of confined liquid crystal.     

A New Class of Nonlinear Waves in Parallel Flows

Waves in parallel shear flows are found to have different characteristics depending on whether nonlinear or viscous effects dominate near the critical layer. In this paper a nonlinear theory is developed which gives rise to a class of disturbances not found in the classical viscous theory. It is suggested that the modes found from such an analysis may be of importance in the breakdown of laminar flow due to free stream disturbances.     

The Small Vorticity Nonlinear Critical Layer for Kelvin Modes on a Vortex

We consider in this paper the propagation of neutral modes along a vortex with velocity profile being the radial coordinate. In the linear stability theory governing such flows, the boundary in parameter space separating stable and unstable regions is usually comprised of modes that are singular at some value of r denoted r_c , the critical point. The singularity can be dealt with by adding viscous and/or nonlinear effects within a thin critical layer centered on the critical point. At high Reynolds numbers, the case of most interest in applications, nonlinearity is essential, but it develops that viscosity, treated here as a small perturbation, still plays a subtle role. After first presenting the scaling for the general case, we formulate a nonlinear critical layer theory valid when the critical point occurs far enough from the center of the vortex so that the vorticity there is small. Solutions are found having no phase change across the critical layer thus permitting the existence of modes not possible in a linear theory. It is found that both the axial and azimuthal mean vorticity are different on either side of the critical layer as a result of the wave–mean flow interaction. A long wave analysis with O (1) vorticity leads to similar conclusions.     

On time-dependent behavior of cross-ply laminated strips with viscoelastic interfaces

The two-dimensional problem of a simply supported laminated orthotropic strip with viscoelastic interfaces under static loading is studied. State-space formulations are developed based on the exact elasticity equations governing orthotropic media and the Kelvin–Voigt constitutive relation of interfaces. Since the response of the strip is time-dependent, the power series expansion technique is adopted to model the variations of elastic fields with time. Results show that the response of the laminated strip with viscoelastic interfaces changes remarkably with time, which is also significantly different from that of a plate with perfect interfaces or with viscous interfaces. Note that from the present analysis, the response for a laminated plate with spring-like interfaces or with viscous interfaces can be easily obtained because they are just two particular cases of the present Kelvin–Voigt model.     

Maxwell equations in complex form of Majorana–Oppenheimer,solutions with cylindric symmetry in Riemann <Emphasis Type="Italic">S</Emphasis><Subscript>3</Subscript> and Lobachevsky <Emphasis Type="Italic">H</Emphasis><Subscript>3</Subscript> spaces

Complex formalism of Riemann–Silberstein–Majorana–Oppenheimer in Maxwell electrodynamics is extended to the case of arbitrary pseudo-Riemannian space-time in accordance with the tetrad recipe of Tetrode–Weyl–Fock–Ivanenko. In this approach, the Maxwell equations are solved exactly on the background of static cosmological Einstein model, parameterized by special cylindrical coordinates and realized as a Riemann space of constant positive curvature. A discrete frequency spectrum for electromagnetic modes depending on the curvature radius of space and three parameters is found, and corresponding basis electromagnetic solutions have been constructed explicitly. In the case of elliptical model a part of the constructed solutions should be rejected by continuity considerations. Similar treatment is given for the Maxwell equations in hyperbolic Lobachevsky model, the complete basis of electromagnetic solutions in corresponding cylindrical coordinates has been constructed as well, no quantization of frequencies of electromagnetic modes arises.     

Fundamentals of the theory of non-fabric glass-reinforced plastics

Starting from the known mechanical properties of the components (glass reinforcement and resin) the stress-strain relations for an anisotropic viscoelastic material (glass-reinforced plastic) are determined. Using Volterra's principle [2] of replacing the elastic constants in the solution of the problem of the theory of elasticity [5] by integral operators, relations for the given material are obtained and found to be in fairly good agreement with the experimental data. In a separate study the high-temperature behavior of GRP is investigated by calculating the integral operators, which are functions of the viscous properties of the resin and the temperature.Mekhanika polimerov, Vol. 1, No. 1, pp. 151–158, 1965     

Dispersion analysis of generalized thermo elastic plate of polygonal cross-sections

In this paper, wave propagation in generalized thermo elastic plate of polygonal cross-sectional plate is studied using the Fourier expansion collocation method. The equations of motion based on two-dimensional theory of elasticity is applied under the plane strain assumption of generalized thermo elastic plate of polygonal cross-sections composed of homogeneous isotropic material. The frequency equations are obtained by satisfying the boundary conditions along the surface of the polygonal plate using Fourier expansion collocation method. The numerical calculations are carried out for triangular, square, pentagonal and hexagonal cross sectional plates. Dispersion curves are drawn using the computed non-dimensional frequencies for longitudinal and flexural (symmetric and antisymmetric) modes of vibration.