Spin-velocity density function

The spin-displacement density function, c (z, Z), and the normalized displacement distribution function, P(z, Z), can be converted readily into the joint spin-velocity density function, q(z, vn), and the normalized velocity distribution function, P(z, vn), respectively, with the net velocity vn defined as vn = Z/A. Once the velocity density function is determined for each of the volume elements, the superficial average velocity, v, is calculated by [23] ... 

An example of the spin-velocity density function is demonstrated in Figure 4.1.6. A velocity imaging experiment was performed on water flowing through a 6-mm diameter tube. The velocity density function was spatially resolved along the axial direction of the tube, denoted by z in the figure. It is observed that the velocity density function has a steep peak at zero velocity when the fluid is not flowing, but is shifted to a positive velocity when the flow rate was increased to 2.5 mL min-1. 

Fig. 12. The joint spin-velocity density function, p(z)P (v , -), as a function of position z for water flow in the rectangular Bentheimer sandstone sample (voxel size is 0 94 mm)...

The porosity cj)(z) for the voxel at z, is determined as described in Section 4.1.4.1. Note that the observed spin-displacement density function, q(z, vn), is not actually associated with the intrinsic value of spin density, q(z), due to the transverse and longitudinal relaxation. However, this does not affect the calculated average velocity given in Eq. (4.1.23) because the spin density terms in the denominator and nominator cancel each other. 

In order to obtain spatial resolution of the molecular translations within each voxel, it is necessary to combine the velocity-encoding gradient sequence shown in Section 3.1.1 with a standard imaging sequence such as frequency or phase encoding. The displacement distribution function, P A (R), must now be generalized to a spatial-displacement joint density function Aso that it describes the total number of spins located at r with displacement R during time A 28... 

The electric monopole interaction is a function of the s electron densities at the nucleus. This results in a displacement of the spectrum and is expressed as the velocity of the source (mm s ) necessary to counteract the displacement. This isomer (or chemical) shift, 6, provides information about the coordination number, the valency and spin state of the iron in the compound. 

The similarity in form between the two real equations implied by the single-body spin-0 Schrddinger equation in the position representation (wave mechanics) and the equations of fluid mechanics with potential flow in its Eulerian formulation was first pointed out by Madelung in 1926 [1]. In this analogy, the probability density is proportional to the fluid density, and the phase of the wave function is a velocity potential. A novel feature of the quantum fluid is the appearance of quantum stresses, which are usually represented through the quantum potential. To achieve mathematical equivalence of the models, the hydrodynamic variables have to satisfy... 

Debye-Bueche function, 178 Debye-Waller factor, 20, 24 densification aerogels, 216 process, 70 silica xerogels sound velocity, 104 thermal treatment effect, 101 density measurement, 191 deposition methods dip-coating, 66 spin, 66 detectors photodiodes, 23 scintillation, 23... 

The description of the behavior of particulate materials relies on constitutive equations, functions of stress, strain, and other physical quantities describing the system. It is rather difficult to extract macroscopic observables like the stress from experiments, e.g. in a two-dimensional (2D) geometry with photo-elastic material, where stress is visualized via crossed polarizers [6, 7]. The alternative is, to perform discrete element simulations [2, 4] and to average over the microscopic quantities in the simulation, in order to obtain some averaged macroscopic quantity. The averages over scalar quantities like density, velocity and particle-spin are strai tforward, but for the stress and the deformation gradient, one finds slightly different definitions in the literature [3, 8-11]. 

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