Normalized distance

Extensional flows occur when fluid deformation is the result of a stretching motion. Extensional viscosity is related to the stress required for the stretching. This stress is necessary to increase the normalized distance between two material entities in the same plane when the separation is s and the relative velocity is ds/dt. The deformation rate is the extensional strain rate, which is given by equation 13 (108) ... 

Mark, H., "Normalized distances for Qualitative Near-Infrared Reflectance Analysis", Anal. Chem. 1986 (58) 379-384. 

It might be expected that the strain in cyclopropane, in which the C-C-C bond angles are distorted from 109°28 to 60°, would weaken the bonds and thus lead to an increase in the bond distance. This effect is not observed, however, the carbon-carbon distance in cyclopropane being the same as in the other hydrocarbons to within the accuracy of the investigation. There is even some small indication that the C-C distance in cyclic aliphatic hydrocarbons is slightly smaller (by about 0.01 A.) than the normal distance, the three values reported being 1.53, 1.52, and 1.53 A. 

The surface force apparatus (SFA) is a device that detects the variations of normal and tangential forces resulting from the molecule interactions, as a function of normal distance between two curved surfaces in relative motion. SFA has been successfully used over the past years for investigating various surface phenomena, such as adhesion, rheology of confined liquid and polymers, colloid stability, and boundary friction. The first SFA was invented in 1969 by Tabor and Winterton [23] and was further developed in 1972 by Israela-chivili and Tabor [24]. The device was employed for direct measurement of the van der Waals forces in the air or vacuum between molecularly smooth mica surfaces in the distance range of 1.5-130 nm. The results confirmed the prediction of the Lifshitz theory on van der Waals interactions down to the separations as small as 1.5 nm. 

The Eulerian approach requires a measurement of the temperature or the progress variable at many sample points at a given normal distance from the ignition plane, at a given time elapsed since ignition. The progress variable introduced here can be for instance a normalized temperature or concentration that varies from... 

In the absence of chemical reactions in solution, depends only on the normalized distance between the centers of the disks d/r, where d is the center-to-center separation). The theory developed for two identical coplanar disks gives the following approximation (valid at d/r 2) [23] ... 

Cheremisinoff and Davis (1979) relaxed these two assumptions by using a correlation developed by Cohen and Hanratty (1968) for the interfacial shear stress, using von Karman s and Deissler s eddy viscosity expressions for solving the liquid-phase momentum equations while still using the hydraulic diameter concept for the gas phase. They assumed, however, that the velocity profile is a function only of the radius, r, or the normal distance from the wall, y, and that the shear stress is constant, t = tw. ... 

A related geometry results from the use217 of 2,6-pyridine dicarboxylate in Ph2SnC7H3N04-H20. The two phenyl groups are at normal distances and trans (CSnC = 172°). The tridentate ligand occupies three sites in the equatorial plane and the coordinated... 

Figure 3. Evolution of normalized NILS hydrogen concentration C, IC vs. normalized distance R lb ahead of the crack tip for crack size r/ = 1.9 mm (a) near crack tip solution, (b) solution over the entire uncracked ligament. The parameter b denotes the crack tip opening displacement which varies with time as the hydrogen pressure increases toward its final value of 15 MPa over 1 sec. The parameter C =2.659x10 H atoms/m ( = 3.142x10 H atoms per solvent atoms) denotes the hydrogen concentration on the inner wall-surface and crack faces in equilibrium with the hydrogen gas.

Figure 4. Plot of the full-field solution for the normalized hydrostatic stress

Figure 6. Normalized NILS hydrogen concentration CL / C at steady state vs. normalized distance R/b from the crack tip for the full-field (crack depth wh=0.2) and MBL (domain size L=h-a) solutions under zero hydrogen flux conditions on the OD surface and remote boundary, respectively. The parameter b denotes the crack tip opening displacement for each case. The inset shows the concentrations near the crack tip.

Even a molecularly smooth single-crystal face represents a potential energy surface that depends on the lateral position x, y) of the water molecule in addition to the dependence on the normal distance z. One simple way to introduce this surface corrugation is by adding the lattice periodicity. An example of this approach is given by Berkowitz and co-workers for the interaction between water and the 100 and 111 faces of the Pt crystal. In this case, the full (x, y, z) dependent potential was determined by a fit to the full atomistic model of Heinzinger and co-workers (see later discussion). 

This relation is the ground relation of diagnosis and the easiest to use. It consists to check that two sensors providing measurements of the same variable give the same values. Given two sensors and S2, providing values Vi and V2 of the variable, the residual r is defined by the normalized distance d between Vi and V2 ... 

A value of 7 lower than 0 or higher than 1 shows an error from one of the inputs of the relation. The residual built from this relation is then a normalized distance between the computed value of g and the nearest authorized bound i.e., 0 or 1). 

Thus the screw has a narrower normal distance between flights at the screw root because the helix angle is larger and because the lead remains the same. 

A dimensionless lag parameter, j, describes the normalized distance between any two increments ... 

Ax normal distance between the helix axes of A and B (no limit)... 

Stmcture Refinement in Previously Characterized Systems. Even when a protein crystal stmcture is known, EXAFS measurements can provide useful information. An example is the Fe-S protein mbredoxin, for which initid crystallographic results suggested that the Fe was coordinated by three sulfurs at normal distances (2.24-2.34 A) and one sulfur at a very short distance (2.05 A) (18). Detailed EXAFS characterization of this Fe site was possible because of the simplicity of the site (only... 

Reinders and his coworkers assume that the silver atoms sublimated onto the glass plate adhere to it at the point of contact. They then calculate on a probability basis the number of atom pairs, triplets, etc. expected per unit area from the total number of atoms impinging upon that area. They assume that a pair is formed only if the second atom strikes the first or if it strikes the glass at a distance from the first not greater than the normal distance between adjacent atoms in the silver crystal. Experimentally, they find that the number of developed nuclei corresponds closely with the calculated number of initial centers containing four or more atoms. A recalculation of their data based on a more accurate probability treatment (Berg, 43) does not materially change the result. 

The Coulombic electrostatic force is a very long-range force. When the distance is doubled, the force decreases to 50% when a molecule is removed to 10 times the normal distance, the attractive force declines to 10%. 

First-principles calculations of an STM, including a real tip and a real sample, clearly show that within the normal tip-sample distances (3.-6 A from nucleus to nucleus), in the gap region, the local electronic density resembles neither that of the tip nor that of the sample. Substantial local modifications are induced by the strong interaction. An example is the system of an A1 sample with an A1 tip, calculated by Ciraci, Baratoff, and Batra (1990a), as shown in Fig. 8.1. As the tip-sample distance is reduced to 8 bohr, the electron density begins to show a substantial concentration in the middle of the gap. This phenomenon becomes much more pronounced when the tip-sample distance is reduced to about 7 bohr. These distances are exactly the normal distances where atom-resolved images are obtained. 

Figure 10.6a represents a molecule at O located a normal distance z from the surface of a bulk sample of the same material. The bulk portion is assumed to have a planar face but otherwise is of infinite extension. The molecule is located a distance x from all the molecules in the ring-shaped volume element shown in the figure. The volume of this ring is given by dV = 2-ny dy d. Therefore the increment of interaction between the molecule and the block due to the molecules a distance x from the point O is given by... 

Antimony(III) halides are Lewis adds and will form adducts with Lewis bases. An example is provided by Lipka and Mootz,149 who prepared an adduct of diphenyl with SbCl3, 2SbCl3-PhPh (51). A molecule of SbCl3 is bonded to each of the biphenyl rings, but there are two differently bonded antimony atoms. One which forms the chain is six-coordinate there are three chlorine atoms at a normal distance, 2.36 A (av), and two at the longer distances of 3.443... 

Equation (12.36) has a particularly simple geometrical interpretation, as shown in Fig. 12.1. If we drop perpendiculars from each vector X), Y) to the vector Z) to obtain the normal distances of each vector X), Y) from the common Z direction [denoted Xz and Yz (dashed lines) in Fig. 12.1], then (dX/dY)z is seen to be given by... 

Figure 12.1 Geometrical evaluation of the thermodynamic derivative V = (dX/dY)z according to (12.36), showing the normal distances (Xz, Yz, dashed lines) of dropped perpendiculars from the vectors X, Y to the Z-axis, whose ratio gives the numerical value (V = Xz/Yz) of the derivative.

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