Extra motions

Because of the extra motion into the fifth dimension, an extra mass term appears in 4D descriptions. Considering compactihed radius Rc 10-12—10-13 cm this extra mass is m 100 MeV similar values appear in Ref. [5], A quite recent approach of hadron spectra by Arkhipov [11] is also worthwhile to consult with. 

All the above theories attribute the enhanced friction in entangled systems to extra motions. In a crude way they speak to the main issue, which is the loss of relaxation pathways due to connectedness in the chain environment. However, all are rather arbitrary they select certain types of motion for examination and exclude others, while in fact the motions must be cooperative and interdependent. In addition, the theories have features which appear to be incorrect, or at least inconsistent, even within the limited realm of motions examined. 

This type of motion is shown as an extra motion because the motion proceeds in different coordinates in comparison to the first shown deformational motion. However, the vibration frequency and the altering of the dipole moment by these last two vibrations are the same. Such type of motions, where the vibration frequency of two or more motion types is the same, is called downgraded motion. A special value, called degradation number, is given for characterization of such motions. This value shows the number of molecules that have the same vibration frequencies. For example, the degradation number for the linear molecule shown above is two. 

For example, when creaming occurs, this extra motion (in addition to Brownian motion) may increase the rate at which droplets encounter one another. This is... 

The principal dilTerence from liquid-state NMR is that the interactions which are averaged by molecular motion on the NMR timescale in liquids lead, because of their anisotropic nature, to much wider lines in solids. Extra infonnation is, in principle, available but is often masked by the lower resolution. Thus, many of the teclmiques developed for liquid-state NMR are not currently feasible in the solid state. Furthemiore, the increased linewidth and the methods used to achieve high resolution put more demands on the spectrometer. Nevertheless, the field of solid-state NMR is advancing rapidly, with a steady stream of new experiments forthcoming. 

The equations of motion are integrated using a modified velocity Verlet algorithm. The modification is required because the force depends upon the velocity the extra step involves... 

All of the described differential viscoelastic constitutive equations are implicit relations between the extra stress and the rate of deformation tensors. Therefore, unlike the generalized Newtonian flows, these equations cannot be used to eliminate the extra stress in the equation of motion and should be solved simultaneously with the governing flow equations. 

Application of the weighted residual method to the solution of incompressible non-Newtonian equations of continuity and motion can be based on a variety of different schemes. Tn what follows general outlines and the formulation of the working equations of these schemes are explained. In these formulations Cauchy s equation of motion, which includes the extra stress derivatives (Equation (1.4)), is used to preseiwe the generality of the derivations. However, velocity and pressure are the only field unknowns which are obtainable from the solution of the equations of continuity and motion. The extra stress in Cauchy s equation of motion is either substituted in terms of velocity gradients or calculated via a viscoelastic constitutive equation in a separate step. 

Normally, the extra stress in the equation of motion is substituted in terms of velocity gradients and hence this equation includes second order derivatives of... 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

After the substitution of Cauchy stress via Equation (3.20) and the viscous part of the extra stress in terms of rate of defonnation, the equation of motion is written as... 

The (CEF) model (see Chapter 1) provides a simple means for obtaining useful results for steady-state viscometric flow of polymeric fluids (Tanner, 1985). In this approach the extra stress in the equation of motion is replaced by explicit relationships in terms of rate of strain components. For example, assuming a zero second normal stress difference for veiy slow flow regimes such relationships arc written as (Mitsoulis et at., 1985)... 

A similar approximation should be applied to the components of the equation of motion and the significant terms (with respect to ) consistent with the expanded constitutive equation identified. This analy.sis shows that only FI and A appear in the zero-order terms and hence should be evaluated up to the second order. Furthermore, all of the remaining terms in Equation (5.29), except for S, appear only in second-order terms of the approximate equations of motion and only their leading zero-order terms need to be evaluated to preserve the consistency of the governing equations. The term E, which only appears in the higlier-order terms of the expanded equations of motion, can be evaluated approximately using only the viscous terms. Therefore the final set of the extra stress components used in conjunction with the components of the equation of motion are... 

Bom and Oppenheimer showed that, to a very good approximation, these extra terms were of the order of m /M and so the motions of the electron and the nuclei could indeed he considered separately for many purposes. 

The first term on the right is the potential energy of the solute molecule at the center of the cavity and the second term is the average of the extra potential energy of the molecule due to its motion in the field w(r). By substituting the values of w(o) and d In g/dT following from the L-J-D theory, one obtains... 

Due to the gradual decay of the hydrodynamic interaction (Eq. 27), the extra velocity component at bead i results from the motion of all the remaining beads and it is presumed that vj depends linearly on the hydrodynamic forces acting on these beads ... 

In the slow exchange limit, where 22 > A2, two Lorentzians, centered at cb + A and bb — A, respectively, are observed with width 2/Tf + 20. The exchange imposed by the molecular motion thus causes an extra broadening of the lines observed in absence of motion. 

In the ultraslow exchange limit, O -4 1/T, the line shape becomes insensitive to the motion because the extra broadening is much smaller than the natural line width. 

There exist a series of beautiful spectroscopy experiments that have been carried out over a number of years in the Lineberger (1), Brauman (2), and Beauchamp (3) laboratories in which electronically stable negative molecular ions prepared in excited vibrational-rotational states are observed to eject their extra electron. For the anions considered in those experiments, it is unlikely that the anion and neutral-molecule potential energy surfaces undergo crossings at geometries accessed by their vibrational motions in these experiments, so it is believed that the mechanism of electron ejection must involve vibration-rotation... 

Enolate anions (4e) that have been heated by infiared multiple photon absorption for which torsional motion about the H2C-C bond, which destabilizes the 7t orbital containing the extra electron, is the mode contributing most to vibration-to-electronic energy transfer and thus to ejection. 

The flow is induced in the following way. External forces are applied on the particles of each reservoir In order to ke p the average y velocities of the reservoirs constant. The Imposed motion of the reservoirs shears the liquid slab. The work supplied In order to keep the reservoirs moving eventually Is dissipated and heats up the liquid. In order to remove this extra heat from the system the velocities of the reservoir molecules are scaled at each time step so as to keep the average reservoir temperatures constant. The Imposed shear rate Is obviously... 

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