Modeling approaches coordinate system

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

Although the foregoing example in Sec. 4.2.1 is based on a linear coordinate system, the methods apply equally to other systems, represented by cylindrical and spherical coordinates. An example of diffusion in a spherical coordinate system is provided by simulation example BEAD. Here the only additional complication in the basic modelling approach is the need to describe the geometry of the system, in terms of the changing area for diffusional flow through the bead. 

Once the initial state of the system has been set up, the equations of motion (e.g., in the form of the Newton s equations) are solved numerically by means of the finite difference approach. Known atomic positions, velocities, and forces at time t are used to obtain the coordinates and velocities at time t + At, after which the procedure is repeated. The value of the time step that can be used depends on the specific model of the system and the accuracy of the integration algorithm. In general, the time step should be smaller than t/lO, where t is the minimum characteristic time in the system, e.g., the period of the highest frequency vibration. 

Proton transfer at Rg = 2.52 A can be treated using the approach developed in the previous section, see Eqs. (9.13) and (9.14). According to Refs. [35, 56] for a model H-bonded system with Rg 2.52 A no vibrational assistance of the proton tunneling occurs. This effect can be explained using the <0m R 0m> matrix element for different m, see Eq. (9.15). The excitation of the low-frequency vibrations leads to an increase in the effective 0- -0 distance (Table 9.1), i.e. to an increase in the effective barrier along the proton coordinate. 

---