Inertia correction

A is the friction force coefficient B is the inertia correction coefficient and C is the resistance coefficient due to the Basset force. Equation (6.65) gives... 

Equation (252) represents the generalization of the conventional result based on the Ornstein-Uhlenbeck (inertia-corrected Einstein) theory of the Brownian motion [87] to fractional dynamics. By way of illustration, we show in Fig. 20... 

Eq. (4) reduces to the corresponding normal Fokker-Planck equation for inertia-corrected rotational diffusion considered, for example, in Ref. 91. 

As far as comparison with experimental data is concerned, the fractional Klein-Kramers model under discussion may be suitable for the explanation of dielectric relaxation of dilute solution of polar molecules (such as CHCI3, CH3CI, etc.) in nonpolar glassy solvents (such as decalin at low temperatures see, e.g., Ref. 93). Here, in contrast to the normal diffusion, the model can explain qualitatively the inertia-corrected anomalous (Cole-Cole-like) dielectric relaxation behavior of such solutions at low frequencies. However, one would expect that the model is not applicable at high frequencies (in the far-infrared region), where the librational character of the rotational motion must be taken... 

Lutz also compared his results with those predicted by the fractional Klein-Kramers equation for the probability density function/(x, v, f) in phase space for the inertia-corrected one-dimensional translational Brownian motion in a potential Eof Barkai and Silbey [30], which in the present context is... 

In the introduction we have seen that Barkai and Silbey [13] have proposed recently a generalization of the Klein Kramers equation for the inertia corrected... 

Equation (145) represents the generalization of the a.v.c.f. of the Ornstein Uhlenbeck [21] (inertia-corrected Einstein) theory of the Brownian motion to fractional dynamics. The long-time tail due to the asymptotic (t >> t) t -like dependence [72] of the ((j)(O)(j)(t))o is apparent, as is the stretched exponential behavior at short times (t t). Eor a > 1, ((j)(O)(t)(f))o exhibits oscillations (see Eig. 14) which is consistent with the large excess absorption occurring at high frequencies. 

We form the average of Eqs. (5.20)-(5.22) noting that (L(r)) will vanish throughout because, in the inertia corrected Langevin equation, M is statistically independent of the white noise field h(r). This is not, however, true of the noninertial Langevin equation where the multiplicative noise term L(f) contributes a noise induced drift term to the average (see Section VI). The averages so formed are... 

Low viscosity limited by inertia corrections, secondary flow, and loss of sample at edges... 

Here tbe cross-sectional area of the reservoir is assumed to be larger than the ctq>illary area. Sylvester and Chen (198S) have reviewed the literature and find AT// = 2.0—2.5 and Kc = 3O—SO0 from various experimental studies. For higher Re Reynolds number (eq. 6.2.19), a st fit of their data gave Kc — 133 30. Roger (1987) gives the inertia correction for a power law fluid. 

Lodge (1988,1989) has shown that it is necessary to correct formisalignment of the flush-mounted transducer p and for inertia. Both corrections can be made using measurements firom Newtonian fluids. The misalignment correction probably comes fix>m bending of the flush-mounted transducer diaphragm and must be measured against tu, for each transducer in situ. The inertia correction appears to be linear in stress and Reynolds number. 

With proper inertia correction controlled stress rheometers are very versatile and may replace controlled strain instruments for many applications. The fact that stress and strain are measured on the same shaft in controlled stress instruments allows lower cost and simpler temperature control but also is the source of the inertia limitations. Table 8.2.1 summarizes the advantages and disadvantages of each control mode. 

Section 8.2 described how different rotary rheometers are designed to control and to measure rotation rate, angular position, torque, temperature, and other variables. Equally important is the analysis of these measurements, conversion of the raw millivolts to material functions. Twenty years ago this was all done by hand, but today commercial rheometers spit out materials functions like G and G" in real time. Data analysis software is becoming a more and more important part of rheometer design. We have already seen that the inertia correction algorithms illustrated in Figure 8.2.11 can significantly extend the performance of controlled stress rheometers. 

Fig. 5 a Frequency dependence of Qq for PI-84k, Qq is defined as the plateau value of at small strain amplitudes (Eq. 5 with n = 3). The lines are linear regressions with a fixed slope of 2. The fine for the ARES-G2 black) and the DHR-3 light and dashed) overlap. The DHR-3 data match, if the motor mode is set to stiff, a setting to auto instead caused severe deviations as shown by the open circles, b Frequency dependence of Qq for PIB. The DHR-3 A data was recorded using the correlation acquisition mode, whereas for the DHR-3 B-data, the transient acquisition mode was used. At low angular frequencies Qo is proportional to (Uq. Additionally, the ratio of the sample torque amplitude to the raw torque amphtude Mso/Mro for the DHR-3 A data is shown. When Mso/Mro reaches approx. 0.8, i.e. the inertia contribution makes up 20 % of the total torque, pronounced deviations in Qo are observed when no inertia correction is applied. Data reprinted from [31], copyright 2014, Springer... 

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