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Relaxation equations examples

Finally, note that the method used by Kadanoff and Swift is a very general scheme. For example, the expression of ris[1H is similar to the expression of viscosity derived later by Geszti [39]. In addition, the projection operator technique used in their study is the same used to derive the relaxation equation [20], and the expression of Ly and Uy are equivalent to the elements of the frequency and memory kernel matrices, respectively. [Pg.89]

We should pay special attention to the last relation in (8.20), which is a relaxation equation for the variable One can find examples of relaxation equations in Section 2.7 for dilute solutions of polymers and in Chapter 7 for concentrated solutions and melts of polymers. The presence of internal variables and equations for their change are specific features of the liquids we consider in this monograph. [Pg.162]

The set of internal variables is usually determined when considering a particular system in more detail. For concentrated solutions and melts of polymers, for example, a set of relaxation equation for internal variables were determined in the previous chapter. One can see that all the internal variables for the entangled systems are tensors of the second rank, while, to describe viscoelasticity of weakly entangled systems, one needs in a set of conformational variables xfk which characterise the deviations of the form and size of macromolecular coils from the equilibrium values. To describe behaviour of strongly entangled systems, one needs both in the set of conformational variables and in the other set of orientational variables w fc which are connected with the mean orientation of the segments of the macromolecules. [Pg.165]

As an example, we shall consider simple shear when z/12 0, and find components of the tensor of the recoverable displacement gradients A12, An, A22, A33 the components of the tensor are calculated from the relaxation equations (9.49) or (9.58). In this case the matrix of the deformation tensor is determined as follows... [Pg.197]

In general, it is more convenient to determine the moments from equations which can be derived directly from the diffusion equation (F.18). For example, on multiplying equation (F.18) by pipk and integrating with respect to all the variables, we find the relaxation equation... [Pg.234]

We use the expansion of the distribution function (F.21) and the relaxation equations (F.25) and (F.28) to calculate the second-order moments of coordinates in steady-state and non-steady-state cases in the form of a series expansion for low values of the velocity gradients. Calculations are simple but tedious. As a first step of calculations, we demonstrate the mean values of the products of different variables or moment of equilibrium distribution functions. They are defined, for example, as... [Pg.235]

Our findings lead us in a number of useful directions. One of these directions is a generalization of our basic instantaneous approach to dynamics. Our original linear INM formalism assumed the potential energy was instantaneously harmonic, but that the coupling was instantaneously linear [Equation (15)]. We still need to retain the harmonic character of the potential to justify the existence of independent normal modes (at least inside the band), but we are free to represent the coupling by any instantaneously nonlinear function we wish. A rather accurate choice for vibrational relaxation, for example, is the instantaneous exponential form ... [Pg.195]

The relaxation of certain properties of the system can often be described by simple phenomenological equations called relaxation equations. In chemical kinetics, for example, the constrained state may be a mixture of gases in metastable equilibrium—for example, hydrogen and oxygen. A spark is then introduced and the gas mixture reacts. The concentration of the reactants and products change with time until a new equilibrium state is achieved. The relaxation equations are the familiar phenomenological equations of chemical kinetics and the relaxation times are related to the chemical rate constants. [Pg.227]

The conservation equations do not by themselves constitute a closed set of relaxation equations. In order to close Eq. (10.3.8) we must specify a constitutive relation relating the flux Jfi(r, t) to the density A(r, t). As an example we consider the simple example of diffusion in a binary mixture, in which there are no chemical reactions. Then Eq. (10.3.8) applies with no source term. According to Fick s second law the local average current of the solute is... [Pg.232]

In this eventuality the set A is called an orthogonal set of properties. It is sometimes useful to use orthogonal sets. For example, using orthogonal sets, it is readily proved that the relaxation equations are stable, and the resulting time-correlation matrix decays to zero... [Pg.291]

The relaxation equation (equation 13) and thus the four relaxation strategies can be easily extended to higher dimensions. Take the 2-D example. In this case, the analog of equation (13) becomes ... [Pg.2090]

For example, if the molecular structure of one or both members of the RP is unknown, the hyperfine coupling constants and -factors can be measured from the spectrum and used to characterize them, in a fashion similar to steady-state EPR. Sometimes there is a marked difference in spin relaxation times between two radicals, and this can be measured by collecting the time dependence of the CIDEP signal and fitting it to a kinetic model using modified Bloch equations [64]. [Pg.1616]

Depending on the method of pumping, the population of may be achieved by — Sq or S2 — Sq absorption processes, labelled 1 and 2 in Figure 9.18, or both. Following either process collisional relaxation to the lower vibrational levels of is rapid by process 3 or 4 for example the vibrational-rotational relaxation of process 3 takes of the order of 10 ps. Following relaxation the distribution among the levels of is that corresponding to thermal equilibrium, that is, there is a Boltzmann population (Equation 2.11). [Pg.360]

Equation (2-3.7) suggests that at very low values of conductivity (/c 0.01 pS/m), charge will relax extremely slowly from a liquid. Eilters for example would have to be an hour or more upstream of tanks before the charge would dissipate to 5% of its initial value. [Pg.211]

Example 2.18 A particular grade of polypropylene can have its relaxation modulus described by the equation... [Pg.118]

This theory is adequate to explain practically all oscillatory phenomena in relaxation-oscillation schemes (e.g., multivibrators, etc.) and, very often, to predict the cases in which the initial analytical oscillation becomes of a piece-wise analytic type if a certain parameter is changed. In fact, after the differential equations are formed, the critical lines T(xc,ye) = 0 are determined as well as the direction of Mandelstam s jumps. Thus the whole picture of the trajectories becomes manifest and one can form a general view of the whole situation. The reader can find numerous examples of these diagrams in Andronov and Chaikin s book4 as well as in Reference 6 (pp. 618-647). [Pg.387]

The second use of Equations (2.36) is to eliminate some of the composition variables from rate expressions. For example, 0i-A(a,b) can be converted to i A a) if Equation (2.36) can be applied to each and every point in the reactor. Reactors for which this is possible are said to preserve local stoichiometry. This does not apply to real reactors if there are internal mixing or separation processes, such as molecular diffusion, that distinguish between types of molecules. Neither does it apply to multiple reactions, although this restriction can be relaxed through use of the reaction coordinate method described in the next section. [Pg.67]

An accurate calculation of the heat conductivity requires solving a kinetic equation for the phonons coupled with the multilevel systems, which would account for thermal saturation effects and so on. We encountered one example of such saturation in the expression (21) for the scattering strength by a two-level system, where the factor of tanh((3co/2) reflected the difference between thermal populations of the two states. Neglecting these effects should lead to an error on the order of unity for the thermal frequencies. Within this single relaxation time approximation for each phonon frequency, the Fermi golden rule yields, for the scattering rate of a phonon with Ha kgT,... [Pg.157]

Rose et al. (1958) and Hanson and Sommerville (1963) have applied relaxation methods to the solution of the unsteady-state equations to obtain the steady-state values. The application of this method to the design of multistage columns is described by Hanson and Sommerville (1963). They give a program listing and worked examples for a distillation column with side-streams, and for a reboiled absorber. [Pg.545]

Distributions of relaxation or retardation times are useful and important both theoretically and practicably, because // can be calculated from /.. (and vice versa) and because from such distributions other types of viscoelastic properties can be calculated. For example, dynamic modulus data can be calculated from experimentally measured stress relaxation data via the resulting // spectrum, or H can be inverted to L, from which creep can be calculated. Alternatively, rather than going from one measured property function to the spectrum to a desired property function [e.g., Eft) — // In Schwarzl has presented a series of easy-to-use approximate equations, including estimated error limits, for converting from one property function to another (11). [Pg.72]


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Relaxation equation

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