Dimensionless relaxation rate

Figure 8 Dimensionless relaxation rate l/pf, as a function of the dimensionless mass action variable X according the kinetic Landau model discussed in the main text, with I, the nucleation rate. Shown is the prediction in the limit where the nucleation reaction is rate limiting. Inset experimental results from measurements on actin (Attri et al., 1991). Notice the zero growth atX = l, the critical polymerization point.)...

The solution of (8 81) for b can be obtained easily. The two terms on the right-hand side of this equation represent the stretching of the drop that is due to the strain rate E, and the restoring force that is due to interfacial tension. The dimensionless relaxation rate, for return to a spherical shape in the absence of a flow, is (5/6)(l/Ca) when the viscosity ratio is small, and (20/19k)(l/Ca) for X 1. Hence, for a relatively inviscid drop, the relaxation rate is determined by the external fluid viscosity, whereas the relaxation rate for a very viscous drop is determined by the internal fluid viscosity. The steady-state solution of (8-81) is... 

It is convenient, however, to use experimentally accessible quantities as variables, i. e. the energy E and a dimensionless relaxation rate R, which can be defined as = rtr = The transformation to the new variables yields for the distribution function [25] ... 

The problem of passing from the dimensionless parameter x = p/lT, to values of 7s in s-1 has been approached in different ways. The simplest idea is to use extrapolation in the x 1(N) dependence to N = 0, leading to 7coi = 0 and 7s = 70 if one assumes that (3.2) holds. This allows us to obtain Tp = 7ox(0) by evaluating the fly-through relaxation rate 70 as a reciprocal transit time of molecules with the most probable velocity through the effective diameter of the laser beam [102]. 

Here, p = AEq o/ eff is the dimensionless energy gap between the upper state and the closest lower-energy state in units of the effective vibrational energy, Veff (cm ). C is the electronic factor, and S is the Huang-Rhys dimensionless excited-state distortion parameter in units of vibrational quanta v ff. As shown in Eq. (2), /c ,p is strongly dependent onp. Additionally, for a given reduced energy gap p, the introduction of even small excited-state distortions, S, can rapidly enhance the radiationless multiphonon relaxation rate such that this dominates the total 0 K relaxation. This model is easily extended to elevated temperatures, where substantial increases in may be observed [7,8]. 

More thorough derivations of formulas equivalent to equation (42) may be found in the literature [42], [43]. The result here shows that the dimensionless decay rate, — a/m, vanishes as cot approaches zero or infinity and attains a maximum value of (ajola Q — l)/4 at cot = I. Thus the dissipation rate is greatest when the reciprocal of the relaxation time equals the frequency of the acoustic oscillations and is negligible for long or short relaxation times (that is, for frozen or equilibrium behavior, respectively compare Section 4.3.4.4). Most vibrational relaxations are too rapid to contribute significantly to a, and most chemical times are either too short or too long values of af /a Q for the few that are not typically produce a —10 co,... 

In the above, X is the chain stretch, which is greater than unity when the flow is fast enough (i.e., y T, > 1) that the retraction process is not complete, and the chain s primitive path therefore becomes stretched. This magnifies the stress, as shown by the multiplier X in the equation for the stress tensor a, Eq. (3-78d). The tensor Q is defined as Q/5, where Q is defined by Eq. (3-70). Convective constraint release is responsible for the last terms in Eqns. (A3-29a) and (A3-29c) these cause the orientation relaxation time r to be shorter than the reptation time Zti and reduce the chain stretch X. Derive the predicted dependence of the dimensionless shear stress On/G and the first normal stress difference M/G on the dimensionless shear rate y for rd/r, = 50 and compare your results with those plotted in Fig. 3-35. 

The problem is reduced to finding the phase trajectories of the equation system (104) at the (g, 0)-plane at different y values (dimensionless reaction rate) and values of p (relationship of the rates of relaxation g and heat removal at T = Tq). Dependence of the solution on x and in the physically justified ranges of their variation (tj > I at q qi ij< 1) turns out to be relatively weak. The authors of ref 234 applied the well-known method of analysis of specific trajectories changing at the bifurcational values of parameters [237], In the general case, the system of equations (104) has four singular points. The inflammation condition has the form... 

Alternatively, we may introduce a model of intermittent continuum, in which is a smooth function on average but experiences rapid variation on a smaller scale of tot. To this end, we may adopt the following simple model characterized by the two dimensionless parameters rj, 772 and the mean relaxation rate Tq [Pg.553]

Figure 17. Time-resolved fluorescence spectra of a solute with one vibrational mode in ethanol at 247 K.68 The various frames show the fluorescence spectrum measured at successively later times after the application of a 1 ps excitation pulse. Each spectrum is labeled with the observation time. The steady-state fluorescence spectrum is given by the dashed curve in the bottom frame. In the electronic ground state, the solute vibrational frequency is400cm 1, and in the excited state, the frequency is 380 cm 1. The dimensionless displacement is 1.4. The permanent dipole moment changes by 10 Debye upon electronic excitation. The Onsager radius is 3A. The longitudinal dielectric relaxation time, xL, is 150 ps.

Cl and 37ci nmr signals in a two-site system where the chlorine nuclei are undergoing exchange between two sites, A and B, one of which B is much less populated than the other. The exchange rate is characterized by the dimensionless parcimeter [cf. Eq, (5.29)] and the difference in chemical shift between the two sites is characterized by the dimensionless parameter n [cf. Eq, (5.30)]. (C indicates that the rate parameter is referred to the relaxation rate, 1/T, of Cl)... 

FIGU RE 7.24 Trouton ratio (t e/t o) versus dimensionless time (f/Xj,) for three different polyisobutylene solutions in low-molecular-weight polybutene at different dimensionless extension rates (s Lp). Both the time and the extension rate are made dimensionless using the longest relaxation times for the solutions provided in the inset. (Data from Tirtaatmadja, V., and T. Sridhar, J. RheoL, 37, 1081, 1993.)... 

This characterizes the time taken for the restoration of the equilibrium microstructure after a disturbance caused, for example, by convective motion, i.e., this is the relaxation time of the microstructure. The time scale of shear flow is given by the reciprocal of the shear rate, 7. The dimensionless group formed by the ratio (tD ff/tShear) is the Peclet number... 

The realm in which relaxation oscillations arise for equations such as the present scheme is that in which the different participants vary on quite different timescales. If we take the rate equations for the concentration of A and the temperature rise in their dimensionless form we have... 

With the average elongational strain rate of the flow field between the eddies and the relaxation time of the polymer molecules, one can define a dimensionless characteristic number, the Deborah number, which represents the ratio of a characteristic time of flow and a characteristic time of the polymer molecule, and thus one can transfer considerations in porous media flow to the turbulent flow region. 

Kinetic studies of ECE processes (sometimes called a DISP mechanism when the second electron transfer occurs in bulk solution) [3] are often best performed using a constant-potential technique such as chronoamperometry. The advantages of this method include (1) relative freedom from double-layer and uncompensated iR effects, and (2) a new value of the rate constant each time the current is sampled. However, unlike certain large-amplitude relaxation techniques, an accurately known, diffusion-controlled value of it1/2/CA is required of each solution before a determination of the rate constant can be made. In the present case, diffusion-controlled values of it1/2/CA corresponding to n = 2 and n = 4 are obtained in strongly acidic media (i.e., when kt can be made small) and in solutions of intermediate pH (i.e., when kt can be made large), respectively. The experimental rate constant is then determined from a dimensionless working curve for the proposed reaction scheme in which the apparent value of n (napp) is plotted as a function of kt. 

Fig. 3.12 (a) A pom-pom with three arms at each branch point (q = 3). At short times the polymer chains are confined to the Doi-Edwards tuhe. Sc is the dimensionless length of branch point retraction into the tube X is the stretch ratio where L is the curvilinear length of the crossbar and Lq is the curvilinear equilibrium length, (b) Relaxation process of a long-chain-branched molecule such as LDPE. At a given flow rate e the molecule contains an unrelaxed core of relaxation times t > g 1 connected to an outer fuzz of relaxed material of relaxation t < g 1, behaving as solvent. [Reprinted by permission from N. J. Inkson et al., J. Rheol., 43(4), 873 (1999).]... 

In order to elucidate the correlation method it may be recalled that the viscosity 77 approaches asymptotically to the constant value r c with decreasing shear rate q. Similarly, the characteristic time t approaches a constant value xQ and the shear modulus G has a limiting value G0 at low shear rates. Bueche already proposed that the relationship between 77 and q be expressed in a dimensionless form by plotting 77/r]0 as a function of qx. According to Vinogradov, also the ratio t/tq is a function of qxQ. If the zero shear rate viscosity and first normal stress are determined, then a time constant x0 may be calculated with the aid of Eqs. (15.60). This time constant is sometimes used as relaxation time, in order to be able to produce general correlations between viscosity, shear modulus and recoverable shear strain as functions of shear rate. 

The data suggest that both the initial deposition rate and the asymptotic deposit mass are both dependent upon the bulk velocity u raised to the power 0.6 - 0.7. The results were also compared with the mass transfer rates of Cleaver and Yates [1975] and Metzner and Friend [1958]. Although the dimensionless particle relaxation times (see Section 7.3) were below 0.1, the inertial deposition rates calculated from the theory of Cleaver and Yates were of an order of magnitude higher than the difiusional rates calculated and indeed measured. The measured power on velocity of 0.7 compared to a theoretical value of 0.875 for difrusion and 2 for inertial particle transfer, suggest a diffiision controlled mechanism. 

Shear stress or particle relaxation time Dimensionless particle relaxation time Rate of deposition Particle flux Particle volume... 

The four parameters of this model, namely, rjo, 7oo, 2, and n, are the apparent viscosity at zero shear rate, apparent viscosity at infinite shear rate, time shear relaxation constant, and the exponential index, respectively. The parameter 1 has a unit of time and can assume any value in the range (0, oo). The index n is dimensionless, with 0 < n < 1. Equation 8 is graphically represented in Fig. 1 for illustration. Various parameters assumed for obtaining the plots appearing in Fig. 1 are as follows rjo = 900, rj o = 0.1, and the nondimensional time shear relaxation... 

Clearly, the variable suffices to characterize the equilibrium behavior determined by = 0. It should be mentioned that can be also be interpreted as a density or concentration variable according to = (1 — c/c )/(l - ck/c ) where c stands for the concentration in lyotropic liquid crystals. For the full nonequilibrium system, times and shear rates are made dimensionless with a convenient reference time. The relaxation time of the alignment in the isotropic phase is Tai4o (1 - T /T) showing a pre-transitional increase. This relaxation time, at coexistence temperature Tk, is used as a reference time... 

There is a single dimensionless group, XVjL, which is known as the Weissenberg number, denoted by various authors as We or Wi. (We is more common, but it can lead to confusion with the Weber number, so Wi will be used here.) The shear rate in any viscometric flow is equal to a constant multiplied by V/L, so it readily follows that the ratio of the first normal stress difference to the shear stress is equal to twice that constant multiphed by Wi. Hence, Wi can be interpreted as the relative magnitude of elastic (normal) stresses to shear stresses in a viscometric flow. The ratio of the shear stress to the shear modulus, G, is sometimes known as the recoverable shear and is denoted Sr. Sr differs from Wi for a Maxwell fluid only by the constant that multiplies F jL to form the shear rate for a given flow. In fact, many authors define Wi as the product of the relaxation time and the shear rate, in which case Wi = Sr. It is important to keep the various definitions of Wi in mind when comparing results from different authors. 

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