Longitudinal scaling function

One of the key quantities for a smectic phase is the scaled layer spacing, and this was determined from the periodicity in the longitudinal distribution function gj (r ). This distribution is unattenuated over the distances available from the simulation for the two smectic phases and the periodicity is the same although the width of the distribution is smaller in the smectic B than the... 

The auto-correlation functions can be used to define two characteristic length scales of an isotropic turbulent flow. The longitudinal integral scale is defined by... 

Figure 1 presents measurements of the longitudinal fluctuations in velocity in a uniform, steady air stream as a function of time near the center of the channel, where the turbulence is essentially isotropic. An effort has been made to retain a linear scale of velocity so that the magnitude of the fluctuation may be visualized. Schubauer (S9) made these measurements by means of a hot-wire anemometer (K6) and they indi-... 

Equation (228) is the normalized density correlation function in the Fourier frequency plane and has the same structure as Eq. (210), which is the density correlation function in the Laplace frequency plane. i/ (z) in Eq. (228) is the memory function in the Fourier frequency plane and can be identified as the dynamical longitudinal frequency. Equation (228) provides the expression of the density correlation function in terms of the longitudinal viscosity. On the other hand, t]l itself is dependent on the density correlation function [Eq. (229)]. Thus the density correlation function should be calculated self-consistently. To make the analysis simpler, the frequency and the time are scaled by (cu2)1 2 and (cu2)-1 2, respectively. As the initial guess for rjh the coupling constant X is considered to be weak. Thus rjt in zeroth order is... 

In this chapter, dielectric response of only isotropic medium is considered. However, in a local-order scale, such a medium is actually anisotropic. The anisotropy is characterized by a local axially symmetric potential. Spatial motion of a dipole in such a potential can be represented as a superposition of oscillations (librations) in a symmetry-axis plane and of a dipole s precession about this axis. In our theory this anisotropy is revealed as follows. The spectral function presents a linear combination of the transverse (K ) and the longitudinal (K ) spectral functions, which are found, respectively, for the parallel and the transverse orientations of the potential symmetry axis with... 

If the angle (3 is much less than 1, then, in accord with Figs. 7 and 9, the most part of the rotators move freely under effect of a constant potential U0, since their trajectories do not intersect the conical cavity. A small part of the rotators moves along a trajectory of the type 1 shown in Fig. 10. However, at d > (3—that is, in the most part of such a trajectory—they are affected by the same constant potential U0- Therefore, for this second group of the particles the law of motion is also rather close to the law of free rotation. For the latter the dielectric response is described by Eq. (77). We shall represent this formula as a particular case of the general expression (51), in which the contributions to the spectral function due to longitudinal A) and transverse KL components are determined, respectively, by the first and second terms under summation sign. Free rotators present a medium isotropic in a local-order scale. Therefore, we set = K . Then the second term... 

The cognitive deficits seen in PD patients medicated with mild clinical disability can be less than those seen in PD patients earlier in the course of the disease who are yet to receive medication (Downes et ah, 1989 Owen et ah, 1995). Inferences can also be made on the basis of longitudinal studies. In one large-scale study, Growdon et ah (1998) reported that levodopa improves motor function without impairing cognition in mild, nondemented PD... 

The Miami Prenatal Cocaine Study is a prospective study starting at birth. The effect of prenatal cocaine exposure on language functioning in 476 full-term African-American infants has been evaluated longitudinally at six times, from 4 months to 3 years of age (369). The children were categorized as cocaine-exposed (n = 253) or non-cocaine exposed (n = 223) by maternal self report and bioassays using maternal/infant urine and meconium. The Bay ley Scale of Infant... 

According to von Karmaan and Howarth [179], a consequence of isotropy is that Rij can be expressed in terms of two scalar functions f t,x) and g t, x) identified as the longitudinal and transverse autocorrelation functions, respectively. There are two distinct longitudinal length scales, Af and A/, that can be defined from /, and there are also two corresponding transverse... 

Several length scales of turbulence can also be determined from the measured correlation functions. The Eulerian length scales obtained from f(t, x) and g t,x) are the longitudinal and transverse integral scales (1.312) and (1.313), respectively. These length scales are characteristics of the larger turbulence vortices in the flow. 

Now let us estimate (see Fig 25) the so-called transverse optic-longitudinal optic splitting [8,50] characteristic for ice at v 230 cm-1. Namely, the loss curve e"(v) is shifted on the frequency scale with respect to the energy loss function ... 

Figure 7. Longitudinal dispersion (Dl) divided by the diffusion coefficient (Df) for tracers measured in column experiments as a function of the particle scale Peclet number (Npe). It is defined as the product of the average pore fluid velocity, u, and the grain diameter, d, divided by the free fluid diffusion coefficient, D/. The magnitude of the dispersion is independent of the pore fluid velocity (Vp) for very small Peclet numbers (or fluid velocities). Note that the effective diffusion coefficient in a porous media is smaller than the diffusion coefficient in a free fluid phase due to the tortuosity. The dispersion increases linearly with increasing flow velocity (increasing Peclet number). Modified from Appelo and Postma (1999).

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