Body force pseudo

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. 

KLEIN - We do indeed use a semi-empirical model for the various interaction potentials. First, we model the ammonia inter molecular potential with an effective pair potential which ignores many body polarization. Models of this type are remarkably successful in explaining the physical properties of polar fluids. Of course, we really should include many body forces, but at this stage we ignore them. The ammonia potential is fitted to the heat of evaporation and the zero-pressure density. The electron-alkali metal (Lithium) potential is represented by the Shaw pseudo potential fitted to the ionization energy. This is the simplest and crudest model possible. We have explored the effect of using (a) Heine-Abarenkov, (b) Ashcroft, and (c) Phillips-Kleinman forms. Our results are not very sensitive to the choice of pseudo potential. (In the case of Cs metal, which I did not discuss, the sensitivity to the potential is crucial). 

Fig. 4.1.7 Two-body classical scattering in a spherically symmetric potential. The relative motion of two atoms may be described as the motion of one pseudo-atom , with the reduced mass p = m-Ams/(rnA+rn-s), relative to a fixed center of force. Two trajectories are shown for the first trajectory, the final and initial relative velocity vector and the associated deflection angle x are shown. This trajectory corresponds to the impact point (b, = 0), whereas the second trajectory corresponds to the impact point (b, 4> = ir).

The components of the tensor A and the pseudo-tensor C have a simple interpretation. For example, the i j component of A is just the i component of the force on the body for translation with unit velocity in the j direction. To see this, we may express (7 20a) in component form ... 

The time evolution in the N body reacting fluid is, in general, given by the Liouville operator introduced earlier. If, however, we make the additional assumption that the strongly repulsive solute-solvent and solvent-solvent forces can be approximated by effective hard-sphere interactions, the theory can be formulated in a way that greatly simplifies the calculation. This can be accomplished by the use of the pseudo-Liouville representation for the dynamics in a hard-sphere system. In a hard-sphere system, the time evolution of a dynamic variable is given by the pseudo-Liouville equation... 

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