Relaxation of a liquid layer under the action of capillary forces

The theory of creeping motion is used to study the relaxation of an infinite viscous fluid layer (membrane) of nonuniform thickness. The propagation of boundary perturbations in a semi-infinite layer under the action of surface-tension forces is also considered. The layer has at least one common boundary with a gas. It is found that relaxation processes of an infinite layer or the propagation of boundary perturbations inside a bounded layer are non-monotonic, and that wave-like surface perturbations always arise. Relaxation times are determined. Maximum distances are found over which separate regions of the layer can affect each other.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 1, pp. 73–77, January–February, 1970.The author wishes to thank V. G. Levich for discussions.     

Gas Flows with Several Thermal Nonequilibrium Modes

We study gas flows with any finite number of thermal nonequilibrium modes. The equations describing such flows are a hyperbolic system with several relaxation equations. An important feature is entropy increase dictated by physics for any irreversible process. Under physical assumptions we obtain properties of thermodynamic variables relevant to stability. By the energy method we prove global existence and uniqueness for the Cauchy problem when the initial data are small perturbations of constant equilibrium states. We give a precise formulation of the fundamental solution for the linearized system, and use it to obtain large time behavior of solutions to the nonlinear system. In particular, we show that the entropy increases but stays bounded. The resulting asymptotic picture of nonequilibrium flows is in a pointwise sense both in space and in time.     

Nonlinear waves in a viscous compressible fluid with relaxation

The propagation and stability of nonlinear waves in a viscous compressible fluid with relaxation that satisfies a Theological equation of state of Oldroyd type are investigated. An equation that describes the structure of the wave perturbations and its evolution is derived subject to the condition of balance of the nonlinear dissipative and relaxation effects, and its solutions of the solitary wave type are analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 31–35, May–June, 1993.     

Influence of the initial spatially inhomogeneous temperature perturbations on vibrational relaxation dynamics

The vibrational relaxation of a nonequilibrium molecular gas (T_V>T) plays an important role in the physics of gas lasers, laser chemistry [1], and plasma chemistry [2]. This paper is devoted to an analysis of the dynamics of V-T relaxation with spatially inhomogeneous perturbations of the translational temperature taken into account.Translated from Zhurnal Prikladnoi Mekhaniki i Technicheskoi Fiziki, No. 6, pp. 77–80, November–December, 1984.     

Mechanical activation of the transfer process in polymer systems

Mechanical activation (change in reactivity or activation energy) of polymer systems with a locally nonequilibrium (relaxation) response to mechanical perturbations is studied. Mechanical activation in such systems is caused by deviation of the medium microstructure from the state of local thermodynamic equilibrium. A system of equations for shear and uniaxial (elongation) deformation modes is formulated. The influence of local nonequilibrium in the medium microstructure on the mechanism and regular features of the macroscopic transfer process and also the effect of the stressed state of the polymer system and structural, kinetic, and orientation (steric) factors on the reactivity of polymer systems are analyzed. It is demonstrated that the reactivity of systems with a complicated internal structure has an energetic entropic character and depends on the reaction zone size (effect of a “nanoreactor”). The ranges of mechanical activation parameters affecting the reactivity and activation energy of polymer systems are determined for the shear and uniaxial modes of their deformation.     

Dissipativity-based stable controller redesign for nonlinear MIMO switched systems in the presence of perturbations

This paper offers a control redesign method for a class of nonlinear MIMO perturbed dissipative switched systems. The controller for the nominal switched system without perturbation is considered to have been designed based on dissipativity. The nominal model is considered affine with perturbations in both nonlinear terms. The controller is then redesigned such that the dissipativity of the switched system with perturbations is maintained by including a robustifying term in the control law. The design is developed for the two cases of 2-norm and infinity-norm bounded perturbations. The proposed method is applied to a nonlinear MIMO switched system to verify the convergence of the state vector to the origin in the presence of perturbations.     

Transport phenomena in a nonequilibrium dissociating gas

A disturbance of a quasiequilibrium distribution due either to a relaxation process or hydrodynamic perturbations can significantly influence the rate of the relaxation process in a gas. The expression for this rate then contains additional terms proportional to the spatial derivatives of the hydrodynamic variables [1–2]. According to the estimates of [2], the effect is clearly manifested in a nonequilibrium dissociating gas. The present authors have estimated [3] some of the additional terms in the initial stage of dissociation in a pure diatomic gas. In the present paper, expressions are obtained for the coefficients of the additional terms of the hydrodynamic equations in a dissociating diatomic gas for all dissociation laws. Estimates made for the example of oxygen show that the contribution of the additional terms to the hydrodynamic equations is comparable with the contribution of the ordinary Navier-Stokes terms.     

Strain-induced low-frequency relaxation in colloidal DGEBA/SiO2 suspensions

Diglycidyl ether of bisphenol A (DGEBA) is widely exploited as an epoxy resin in adhesives and coatings. In this paper, it is used as an oligomer matrix for silica-filled nanocomposites. Rheological measurements show that the pure matrix obeys power-law relaxation dynamics in the vicinity of the dynamic glass transition of this low-molecular-weight glass former. In the filled systems, a low-frequency relaxation appears additionally to the structural α-process of the matrix. Considering the nanocomposites as Newtonian hard-sphere suspensions at low angular frequencies (or high temperatures), the modified terminal regime behavior of the matrix can be linked to strain-induced perturbations of the isotropic filler distributions. While in the low-frequency regime hydrodynamic stresses relax instantaneously, the Brownian stress relaxation is viscoelastic and can be evidenced by dynamic rheological measurements. At higher angular frequencies, the α-process of the matrix superimposes on the Brownian stress relaxation. In particular, we were able to depict the low-frequency anomaly for concentrated, semi-dilute, and even for dilute suspensions.     

Spatial stability of incompressible attachment-line flow

Linear stability analysis of incompressible attachment-line flow is presented within the spatial framework. The system of perturbation equations is solved using spectral collocation. This system has been solved in the past using the temporal approach and the current results are shown to be in excellent agreement with neutral temporal calculations. Results amenable to direct comparison with experiments are then presented for the case of zero suction.The global solution method utilized for solving the eigenproblem yields, aside from the well-understood primary mode, the full spectrum of least-damped waves. Of those, a new mode, well separated from the continuous spectrum is singled out and discussed. Further, relaxation of the condition of decaying perturbations in the far-field results in the appearance of sinusoidal modes akin to those found in the classical Orr-Sommerfeld problem.Finally, the continuous spectrum is demonstrated to be amenable to asymptotic analysis. Expressions are derived for the location, in parameter space, of the continuous spectrum, as well as for the limiting cases of practical interest. In the large Reynolds number limit the continuous spectrum is demonstrated to be identical to that of the Orr-Sommerfeld equation.     

A note on the stability of Beltrami fields for compressible fluid flows

In this paper we study the stability of the magnetostatic equilibrium through a relaxation of a magnetic field B in perfectly conducting compressible and viscous fluid.We establish stability criterion of a large class of Beltrami flows to any admissible displacement about the equilibrium configuration. We show that the field is stable to any displacement with the same 2π-periodicity as the basic flow, except the case where perturbations with wavelength much greater than the scale of the basic flow are included.     

An integral constitutive law for viscoelastic fluids based on the concept of evolving natural configurations: stability analysis

A general conceptual framework has recently been developed within the context of large deformations for the study of materials (solids and fluids) that either undergo microstructural changes or for which stress states are related to the presence of several relaxation mechanisms. The corner stone of this new approach is that all materials exhibit an infinity of stress-free natural configurations that are evolving in a thermodynamically admissible process. In this paper, that is exploratory in nature, we study the behavior of a non-linear integral viscoelastic constitutive equation (CE) for a fluid assumed to posses an infinity of evolving natural configurations. The CE stability pattern with respect to small perturbations about the rest state is also addressed.     

Über den Relaxationseinfluß auf die Stabilität der ebenen Couette-Strömung

Zusammenfassung Die Stabilität der ebenen Couette-Strömung zweier nicht-newtonscher Fluidmodelle mit Relaxation wird für kleine Störungen in der viskosimetrischen Ebene untersucht. Um die Darstellung der Ergebnisse zu vereinfachen, wird sie auf einFluidmodell mit konstanten Stoffdaten beschränkt.

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The stability of plane Couette flow for two models of non-Newtonian liquid having relaxation properties was investigated for small perturbations in the viscometric plane. To simplify the presentation of the results only models with constant material properties are considered.

     

Finite element simulations of dynamics of multivariant martensitic phase transitions based on Ginzburg–Landau theory

A finite element approach is suggested for the modeling of multivariant stress-induced martensitic phase transitions (PTs) in elastic materials at the nanoscale for the 2-D and 3-D cases, for quasi-static and dynamic formulations. The approach is based on the phase-field theory, which includes the Ginzburg–Landau equations with an advanced thermodynamic potential that captures the main features of macroscopic stress–strain curves. The model consists of a coupled system of the Ginzburg–Landau equations and the static or dynamic elasticity equations, and it describes evolution of distributions of austenite and different martensitic variants in terms of corresponding order parameters. The suggested explicit finite element algorithm allows decoupling of the Ginzburg–Landau and elasticity equations for small time increments. Based on the developed phase-field approach, the simulation of the microstructure evolution for cubic-tetragonal martensitic PT in a NiAl alloy is presented for quasi-statics (i.e., without inertial forces) and dynamic formulations in the 2-D and 3-D cases. The numerical results show the significant influence of inertial effects on microstructure evolution in single- and polycrystalline samples, even for the traditional problem of relaxation of initial perturbations to stationary microstructure.     

Influence of Vibrational Relaxation on the Pulsation Activity in Flows of an Excited Diatomic Gas

The influence of vibrational relaxation on the nonlinear evolution of a large vortex structure in a shear flow of a highly nonequilibrium diatomic gas is studied. Calculations are performed using the equations of twotemperature gas dynamics for a viscous heatconducting gas. Relaxation of the temperature of vibrational levels of gas molecules to equilibrium is described by the Landau–Teller equation. The contribution of the relaxation of rotational levels is taken into account by the bulk viscosity in the stress tensor. It is shown that in the presence of only the relaxation process with no viscous dissipation, the damping of the kinetic energy of perturbations and Reynolds stresses increases by up to 10 % compared to the case of thermal equilibrium. For high (actually attainable) degrees of excitation of the vibrational mode, moderate dynamic and bulk viscosities, and a typical relaxation time comparable to flow time, the relative effect of perturbation damping reaches 15%.     

Instability of Streaming Viscoelastic Fluids in the Presence of ��Effective Interfacial Tension�� Through Porous Medium

The ??effective interfacial tension?? effect on the instability of the plane interface between two uniform, superposed, and streaming Rivlin?CEricksen viscoelastic fluids through a porous medium is considered. The case of two uniform streaming Rivlin?CEricksen viscoelastic fluids separated by a horizontal boundary is studied. In the absence of ??effective interfacial tension??, stability/instability of the system as well as perturbations transverse to the direction of streaming are found to be unaffected by the presence of streaming if the perturbations in the direction of streaming are ignored, whereas for perturbations in all other directions, there exists instability for a certain wave number range. ??Effective interfacial tension?? is able to suppress this Kelvin?CHelmholtz instability for small wavelength perturbations, the medium porosity reduces the stability range given in terms of a difference in streaming velocities.     

A new constitutive equation that models extensional flow strain hardening based on evolving natural configurations: stability analysis

Polymer melt viscoelastic fluids often exhibit in elongational flows a significant increase in the elongational viscosity known as strain hardening. This phenomenon could be related to polydispersity, e.g. the presence of a small fraction of very high molecular weight chains whose time frame relaxation spectrum is different from the small chains one. In the present work, we present a fully objective constitutive equation (CE) to primarily model extensional strain hardening based on the new concept of multiple configuration materials. Next, we analyze the CE stability properties with respect to small perturbations about the rest state.     

Gas Dynamics in Thermal Nonequilibrium¶and General Hyperbolic Systems¶with Relaxation

We study gas flow in vibrational nonequilibrium. The model is a 4Ǹ nonlinear hyperbolic system with relaxation. Under physical assumptions, properties of thermodynamic variables relevant to stability are obtained, global existence for Cauchy problems with smooth and small data is established, and large time behavior is studied in the pointwise sense. We formulate the fundamental solution in a systematic way for a general linear system with relaxation. The fundamental solution provides insights to the behavior of the nonlinear system, and is crucial to obtain our pointwise asymptotic picture for the nonequilibrium flow. We also clarify in a general setting the relation between subcharacteristic conditions and a dissipative criterion that was originally proposed for hyperbolic-parabolic systems and has now proved to be important also for hyperbolic systems with relaxation.     

Stability of elastic and viscoelastic systems under a random perturbations of their parameters

Summary The present paper is concerned with the investigation of the almost sure stability of elastic and viscoelastic systems, when their parameters assume a random wide-band stationary process. The parameters are parametric loads, characteristics of external damping and material viscosity. With the help of Liapunov's direct method, the sufficient condition of the almost sure asymptotic stability for distributed parameter systems with respect to perturbations of initial conditions of an arbitrary form is obtained. It is shown that, in some cases, this condition coincides with a similar condition derived from the assumption that the form of sure and required perturbations coincides with the first eigenfunction of system oscillations. However, an example is given for the stability of a viscoelastic rod, when the perturbations of initial conditions are more dangerous, if their form differs from the first eigenfunction.This research was sponsored by the Russian Foundation of Fundamental Research of the Russian Academy of Sciences under Grant 94-01-01522.     

Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations

In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous dynamical systems. This is accomplished through a detailed analysis of the structure of the invariant sets and its behavior under perturbation. We prove that a bounded hyperbolic global solutions persists under singular perturbations and that their nonlinear unstable manifold behave continuously. To accomplish this, we need to establish results on roughness of exponential dichotomies under these singular perturbations. Our results imply that, if the limiting pullback attractor of a nonautonomous dynamical system is the closure of a countable union of unstable manifolds of global bounded hyperbolic solutions, then it behaves continuously (upper and lower) under singular perturbations.