Homogeneous shear model

Fig. 9. Illustrating the "homogeneous shear model". In (a) the unit cell built on (a], a2, a3) is sheared and becomes (af, a2, azimuth a. In (b) the corresponding...

Molecular dynamics, in contrast to MC simulations, is a typical model in which hydrodynamic effects are incorporated in the behavior of polymer solutions and may be properly accounted for. In the so-called nonequilibrium molecular dynamics method [54], Newton s equations of a (classical) many-particle problem are iteratively solved whereby quantities of both macroscopic and microscopic interest are expressed in terms of the configurational quantities such as the space coordinates or velocities of all particles. In addition, shear flow may be imposed by the homogeneous shear flow algorithm of Evans [56]. 

The effect of driving shear stresses on the dislocations are studied by superimposing a corresponding homogeneous shear strain on the whole model before relaxation. By repeating these calculations with increasing shear strains, the Peierls barrier is determined from the superimposed strain at which the dislocation starts moving. 

Lumley has solved the equation system for homogeneous shear, and compared the results with homogeneous strain and homogeneous shear experiments. Lumley s model predicts that the time scale T grows without bound, so that homogeneous flows can never attain an equilibrium structure. Champagne et al. (C4) experiments are consistent with Lumley s notion, but Lumley s model does not predict the observed structure very well. Some improvements on Lumley s model based on Eq. (63) are suggested in Section V. 

The mechanics of localized vs. homogeneous shear in metallic glasses was considered first by Spaepen (1977) and later, in more detail, by Argon (1979), who developed a perturbation model of shear localization in the context of flow by repeated STs. A more formal model was presented later by Steif et al. (1982). Similar flow-dilatancy-based models had actually been considered much earlier, independently, for flow of cohensionless media, such as sand and soils (for an overview of these see Anand and Gu (2000)). 

The equation above builds on a shear model that assumes that all material in the distance range O-dgcM oscillates with the crystal and thus contributes fully to the tme sensed mass, whereas the material located further away from the surface does not oscillate with the crystal and does not contribute to the sensed mass. This is clearly a simplification, as are the optical models used for evaluating the ellipsometric thickness. It has been shown that the QCM thickness and the ellipsometric thickness are similar for relatively compact and homogeneous layers [28]. We do not expect this to be the case for more diffuse polymer layers since the ellipsometric thickness is directly influenced by the segment density profile [29], whereas the QCM thickness is influenced by the amount of water that oscillates with the crystal, and tlus quantity is at present an unknown function of the segment density profile. 

Speziale CG, Gatski TB, Fitzmaurice N (1991) An analysis of RNG-based turbulence models for homogeneous shear flow. Phys Fluids A (Fluid Dynamics) 3 (9) 2278-2281... 

The criterion of maintaining equal power per unit volume has been commonly used for dupHcating dispersion qualities on the two scales of mixing. However, this criterion would be conservative if only dispersion homogeneity is desired. The scale-up criterion based on laminar shear mechanism (9) consists of constant > typical for suspension polymerization. The turbulence model gives constant tip speed %ND for scale-up. 

Despite the ability of the GLM to reproduce any realizable Reynolds-stress model, Pope (2002b) has shown that it is not consistent with DNS data for homogeneous turbulent shear flow. In order to overcome this problem, and to incorporate the Reynolds-number effects observed in DNS, a stochastic model for the acceleration can be formulated (Pope 2002a Pope 2003). However, it remains to be seen how well such models will perform for more complex inhomogeneous flows. In particular, further research is needed to determine the functional forms of the coefficient matrices in both homogeneous and inhomogeneous turbulent flows. 

Rogers, M. M., P. Moin, and W. C. Reynolds (1986). The structure and modeling of the hydrodynamic and passive scalar fields in homogeneous turbulent shear flow. Report TF-25, Department of Mechanical Engineering, Stanford University. 

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