Rectilinear case

The coagulation kernel for the rectilinear case of differential sedimentation is... 

As in tire one-fluid case, the experimental sums are in good agreement with the law of the rectilinear diameter, but the experimental differences fail to give a parabolic shape to tlie coexistence curve. 

The parallelization of crystallites, occurring as a result of fiber drawing, which consists in assuming by crystallite axes-positions more or less mutually parallel, leads to the development of texture within the fiber. In the case of PET fibers, this is a specific texture, different from that of other kinds of chemical fibers. It is called axial-tilted texture. The occurrence of such a texture is proved by the displacement of x-ray reflexes of paratropic lattice planes in relation to the equator of the texture dif-fractogram and by the deviation from the rectilinear arrangement of oblique diffraction planes. With the preservation of the principle of rotational symmetry, the inclination of all the crystallites axes in relation to the fiber axis is a characteristic of such a type of texture. The angle formed by the axes of particular crystallites (the translation direction of space lattice [c]) and the... 

Three common methods of measuring crosslinking (swelling, elastic modulus, and gel point measurements) have recently been critically appraised by Dole (14). A fourth method using a plot of sol + sol against the reciprocal dose has also been used extensively. However, Lyons (23) has pointed out that this relation, even for polyethylenes of closely random distribution, does not have the rectilinear form required by the statistical theory of crosslinking. Flory (19) pointed out many years ago that the extensibility of a crosslinked elastomer should vary as the square root of the distance between crosslinks. More recently Case (4, 5) has calculated that the extensibility of an elastomer is given by ... 

Finally, there are two special cases in which the rectilinear law is observed when the rate-controlling factor is the rate of supply of O2 and when the metal oxide is volatile at the temperature of oxidation. The latter case occurs in the high temperature oxidation of molybdenum, since M0O3 is quite volatile, and in this case dw/dt is negative. 

Unfortunately, this equation underestimates the non-uniformity of the velocity gradient for non-Newtonian liquids, for which the viscosity decreases with increasing velocity gradient. In the previous sections q has been called the shear rate. Velocity gradient and shear rate are strictly the same only in the case of rectilinear flow (d f j). 

Equation (11) is written in the form of Newton s second law and states that the mass times acceleration of a fluid particle is equal to the sum of the forces causing that acceleration. In flow problems that are accelerationless (Dx/Dt = 0) it is sometimes possible to solve Eq. (11) for the stress distribution independently of any knowledge of the velocity field in the system. One special case where this useful feature of these equations occurs is the case of rectilinear pipe flow. In this special case the solution of complex fluid flow problems is greatly simplified because the stress distribution can be discovered before the constitutive relation must be introduced. This means that only a first-order differential equation must be solved rather than a second-order (and often nonlinear) one. The following are the components of Eq. (11) in rectangular Cartesian, cylindrical polar, and spherical polar coordinates ... 

Symmetry, and the Number of Independent Force Constants.—As in harmonic calculations, the rather general discussion of the preceding section can be simplified in particular cases by making use of symmetry, as discussed by Hoy et a/.12 Thus we may choose the curvilinear co-ordinates Jfin linear combinations that span the irreducible representations of the point group we denote such symmetrized curvilinear co-ordinates by the symbol S, and we define them by means of a U matrix exactly analogous to that used for rectilinear coordinates ... 

The energy released after the newly formed AB bond first reaches its equilibrium value is termed repulsive , usually expressed as a percentage of the reaction exoergicity, % R. The remainder (100 — %A—%R) is termed mixed energy release, % M, and corresponds to a non-rectilinear path on the potential surface, i.e. the trajectory cuts the comer. Energy is released between B and C as A approaches B. In the collinear case, this... 

Shear rate Under idealised conditions, the polymer melt subjected to simple shear is contained between two (infinitely extending) parallel walls, one of which is translated parallel to the other at a constant distance. The result of the shear stress (moving wall per unit of surface area) is a velocity gradient in the melt in a direction perpendicular to the wall. If the velocity increases linearly from the one plate to the other one, then the flow is rectilinear. In that case, the velocity gradient is constant (see Fig. 15.1) ... 

The results in Table VII depict how the band structure of carbyne [-CaC] (corresponding to Y - in Table VII) is affected by periodic inclusion of selected atoms or groups which, by introducing a kink into the otherwise rectilinear chain of cart e, would provide greater conformational versatility and therefore possibly improve processability. The chains were considered in their trans planar zig-zag conformation. It is seen that any of the modifica-tions to carb e indicated in Table VII produces an increased E value and a decreased BW value. That electrical conductivity would be adversely affected is reasonable given the disruption of the conjugated system caused 1 such modifications. As yet, no clear relationship is apparent between the specific molecular nature of Y and the effect of its inclusion on E and BW the relatively minor effect for the case Y - CH2 is notewSrthy and under continued study. 

The numerical procedure requires approximating the unknowns in the simple computational domain defined by a stream band B in the mapped domain D. The stream bands are divided into rectangular elements built on two rectilinear streamlines. Meshes are generally refined in the vidnity of contraction sections of the flow domains, as is usually the case. The discretization schemes adopted depend on the problem under consideration. We find it of interest to underline the following points ... 

Pure rotary diffusion of rigid dipoles in two or three dimensions, then, gives exponential decay of polarization with a single relaxation time, provided the sites are uniformly distributed and D is constant. The description of the motion in terms of D alone breaks down, as we shall see, for very short times. A three-dimensional rigid body in any case executes a more complex motion. Even an internally uniform model of rectilinear charge-carrier difiurion automatically shows multiple relaxation. More realistic models must take account of the dynamic s of molecular motion. 

For the particular case of rectilinear motion to which we shall... 

The simulation runs were performed in the grand canonical ensemble, fixing the chemical potential ji, the volume V of the pore and the temperature T. The system typically consisted of 600-700 adsorbed molecules. For the case of attractive pore-wall interaction, the adsorbed molecules formed seven layers parallel to the plane of the pore walls. A rectilinear simulation cell of 10a// by 10a// in the plane parallel to the pore walls was used, consistent with a cutoff of 5a// for the fluid-fluid interaction. The simulation was set up such that insertion, deletion and displacement moves were attempted with equal probability, and the displacement step was adjusted to have a 50% probability of acceptance. Thermodynamic properties were averaged over 100-500 million individual Monte Carlo steps. The length of the simulation was adjusted such that a minimum of fifty times the average number of particles in the system would be inserted and deleted during a single simulation run. 

This relationship was utilized in the interpretation of results by Oldfield and Wright (1963) according to (52) the rectilinear plot of m, — should pass through the origin however, this is usually not the case in practice, since the initial stage is controlled by the surface process. 

---