Translational velocity

The speed at which a sphere roUs down a cylindrical tube filled with a fluid or down an angled plate covered with a film of the fluid also gives a measure of viscosity. For the cylindrical tube geometry, equation 35, a generalized form of the Stokes equation is used for any given instmment, where p is the translational velocity of the rolling sphere and k is the instmment constant determined by caUbration with standard fluids. 

Flame speed The speed of a flame burning through a flammable mixture of gas and air measured relative to a fixed observer, that is, the sum of the burning and translational velocities of the unbumed gases. 

Chapter 3 is devoted to pressure transformation of the unresolved isotropic Raman scattering spectrum which consists of a single Q-branch much narrower than other branches (shaded in Fig. 0.2(a)). Therefore rotational collapse of the Q-branch is accomplished much earlier than that of the IR spectrum as a whole (e.g. in the gas phase). Attention is concentrated on the isotropic Q-branch of N2, which is significantly narrowed before the broadening produced by weak vibrational dephasing becomes dominant. It is remarkable that isotropic Q-branch collapse is indifferent to orientational relaxation. It is affected solely by rotational energy relaxation. This is an exceptional case of pure frequency modulation similar to the Dicke effect in atomic spectroscopy [13]. The only difference is that the frequency in the Q-branch is quadratic in J whereas in the Doppler contour it is linear in translational velocity v. Consequently the rotational frequency modulation is not Gaussian but is still Markovian and therefore subject to the impact theory. The Keilson-... 

Because non-adiabatic collisions induce transitions between rotational levels, these levels do not participate in the relaxation process independently as in (1.11), but are correlated with each other. The degree of correlation is determined by the kernel of Eq. (1.3). A one-parameter model for such a kernel adopted in Eq. (1.6) meets the requirement formulated in (1.2). Mathematically it is suitable to solve integral equation (1.2) in a general way. The form of the kernel in Eq. (1.6) was first proposed by Keilson and Storer to describe the relaxation of the translational velocity [10]. Later it was employed in a number of other problems [24, 25], including the one under discussion [26, 27]. 

Fig. 1.8. MD calculation of autocorrelation functions of translational velocity (Kv) and the force (KF) acting on a molecule in the liquid [45].

Judging by these results the angular momentum relaxation in a dense medium has the form of damped oscillations of frequency jRo = (Rctc/to)i and decay decrement 1/(2tc). This conclusion is quantitatively verified by computer experiments [45, 54, 55]. Most of them were concerned with calculations of the autocorrelation function of the translational velocity v(t). However the relation between v(t) and the force F t) acting during collisions is the same as that between e> = J/I and M. Therefore, the results are qualitatively similar. In Fig. 1.8 we show the correlation functions of the velocity and force for the liquid state density. Oscillations are clearly seen, which point to a regular character of collisions and non-Markovian nature of velocity changes. 

Fig. 1.14. Comparison of the MD calculations of the correlation functions of the translational velocity and angular momentum in liquid nitrogen [65]. The time is in units of 10-13 s.

If the molecule moves without hindrance in a rigid-walled enclosure (the free enclosure ), as assumed in free volume theories, then rattling back and forth is a free vibration, which could be considered as coherent in such a cell. The transfer time between opposite sides of the cell t0 is roughly the inverse frequency of the vibration. The maximum in the free-path distribution was found theoretically in many cells of different shape [74]. In model distribution (1.121) it appears at a > 2 and shifts to t0 at a - oo (Fig. 1.18). At y — 1 coherent vibration in a cell turns into translational velocity oscillation as well as a molecular libration (Fig. 1.19). 

An effective cross-section translational velocity cross-sections ov and... 

Taylor series 260 torque, correlation functions 28 transfer time, rotational relaxation 51 transitions dipole moment 30 forbidden 30 non-adiabatic 130 translational velocity v 6... 

U is the translational velocity of the vortex ring r is the circulation D is the ring diameter d is the core diameter... 

If the motion of the molecule is one of translation, as it is during sedimentation in a centrifugal field, the velocity of every bead is the same, and in the free-draining case the difference in velocity Aw, for each bead relative to the solvent is the same as the (relative) translational velocity u of the molecule as a whole. Fig. 138 is illustrative of this case. The total force on the molecule is then... 

Translational velocity of a polymer molecule in sedimentation (Chap. XIV). 

Note that the average void fraction of a slug unit depends only on the liquid and gas flow rates, the dispersed velocity ub, the translational velocity w, and the void fraction within the liquid slug, a, and it is independent of the bubble shape or bubble length, the liquid slug length, as well as the film thickness in the film zone (Barnea, 1990). 

Common geometries used to make viscosity measurements over a range of shear rates are Couette, concentric cylinder, or cup and bob systems. The gap between the two cylinders is usually small so that a constant shear rate can be assumed at all points in the gap. When the liquid is in laminar flow, any small element of the liquid moves along lines of constant velocity known as streamlines. The translational velocity of the element is the same as that of the streamline at its centre. There is of course a velocity difference across the element equal to the shear rate and this shearing action means that there is a rotational or vorticity component to the flow field which is numerically equal to the shear rate/2. The geometry is shown in Figure 1.7. 

Although this book is devoted to molecular fluorescence in condensed phases, it is worth mentioning the relevance of fluorescence spectroscopy in supersonic jets (Ito et al., 1988). A gas expanded through an orifice from a high-pressure region into a vacuum is cooled by the well-known Joule-Thomson effect. During expansion, collisions between the gas molecules lead to a dramatic decrease in their translational velocities. Translational temperatures of 1 K or less can be attained in this way. The supersonic jet technique is an alternative low-temperature approach to the solid-phase methods described in Section 3.5.2 all of them have a common aim of improving the spectral resolution. 

Figure 9. Normalized rotational (top) and translational (bottom) components of the solvation velocity TCFs for C153 in acetonitrile (left) and CO2 (right). Also shown are the pure solvent rotational and translational velocity autocorrelations, pygt (i) and (0 ...

The Maroncelh et al. result can easily be obtained from Eqs. (34) and (37). It corresponds to neglecting the center-of-mass translational velocity component of Eq. (37), which is reasonable for low-order multipolar perturbations (see Fig. 4) in the solute charge distribution. 

In the encounter shown in the middle of Fig. 10.3, the relative translational velocity is parallel to that in the top collision but is directed somewhat off-center by an amount b, which is called the impact parameter. (The impact parameter in the top collision is b — 0.) At the point of closest approach, the component of velocity along the line connecting the centers is... 

Seismic velocity techniques for hydrate detection have two components (1) translation of seismic signals to velocity and (2) translation between velocity and detection of hydrates. The first component is beyond the scope of this monograph. However, a brief consideration will be given to advances in translating velocity to the detection of hydrates. 

For these reactions the 0 K activation enthalpy and the room temperature activation enthalpies and free energies are almost the same, and so are the 0 K reaction enthalpy and the room temperature reaction enthalpies and free energies. This is presumably so because these are unimolecular reactions, in which the relative translational velocities of reacting molecules are not a factor. 

Figure 7.1 Most modern NMR techniques are based on the fact, that the phase (p of the precessing transverse magnetisation M t) kann be measured. By use of the Fourier transformation the phase provides access to NMR spectra, images, and parameters of translational motion like velocity v and acceleration a. Spectroscopic parameters as well as components of translational velocity and acceleration can be used for generating contrast in NMR imaging. In the drawing the magnetisation M(t) has been generated from Mz by use of a 90° pulse of the B1 radio-frequency (rf) field in y direction...

---