Viscosity coefficients computer simulation

Note that the Eqs. (1), (2), and (8) are really and essentially different due to the absence or presence of different turbulent transport terms. Only by incorporating dedicated formulations for the SGS eddy viscosity can one attain that LES yield the same flow field as DNS. RANS-based simulations with their turbulent viscosity coefficient, however, essentially deliver steady flow fields and as such are never capable of delivering the same velocity fields as the inherently transient LES or DNS, irrespectively of the refinement of the computational grid ... 

The Mazur function can be used to calculate R ) and (R ) (n = 2,3,...), but its usefulness extends no more than that. Calculation of such important global polymer properties as mean-square radius of gyration, particle scattering function, intrinsic viscosity, and diffusion coefficient needs an analytical expression for l+,j(R), the distribution function for the distance R between specified beads i and j. We find no a priori reason for this function to be approximated by the Mazur function. Knowledge about Wij has increased in recent years mainly on the basis of computer simulation experiments. Some of the typical results from such work are described below. 

Whether, however, one can conclude that computer simulation is a practical tool for calculating transport coefficients in general is open to question. It is more difficult to obtain the viscosity and thermal conductivity to within a reasonable error than it is to deter-min and self-diffusion, and the latter is not a trivial calculation. ... 

However, a different approach has been introduced recently by Hoover ( 7), and by other authors, which could eventually make computer simulation a viable tool. Rather than evaluate the correlation function, it is suggested that a laboratory experiment, which would lead to the transport coefficient, be simulated. For example, a flow system can be set up on the computer to represent a shear flux produced by a velocity gradient. Numerical evaluation of the flux for a given gradient gives the viscosity coefficient. 

Monte Carlo simulations require less computer time to execute each iteration than a molecular dynamics simulation on the same system. However, Monte Carlo simulations are more limited in that they cannot yield time-dependent information, such as diffusion coefficients or viscosity. As with molecular dynamics, constant NVT simulations are most common, but constant NPT simulations are possible using a coordinate scaling step. Calculations that are not constant N can be constructed by including probabilities for particle creation and annihilation. These calculations present technical difficulties due to having very low probabilities for creation and annihilation, thus requiring very large collections of molecules and long simulation times. 

Molecular dynamics calculations are more time-consuming than Monte Carlo calculations. This is because energy derivatives must be computed and used to solve the equations of motion. Molecular dynamics simulations are capable of yielding all the same properties as are obtained from Monte Carlo calculations. The advantage of molecular dynamics is that it is capable of modeling time-dependent properties, which can not be computed with Monte Carlo simulations. This is how diffusion coefficients must be computed. It is also possible to use shearing boundaries in order to obtain a viscosity. Molec-... 

As computer power continues to increase over the next few years, there can be real hope that atomistic simulations will have major uses in the prediction of phases, phase transition temperatures, and key material properties such as diffusion coefficients, elastic constants, viscosities and the details of surface adsorption. 

We have presented EMD and NEMD simulation algorithms for the study of transport properties of liquid crystals. Their transport properties are richer than those of isotropic fluids. For example, in a uniaxially symmetric nematic liquid crystal the thermal conductivity has two independent components and the viscosity has seven. So far the different algorithms have been applied to various variants of the Gay-Beme fluid. This is a very simple model but the qualitative features resembles those of real liquid crystals and it is useful for the development of molecular dynamics algorithms for transport coefficients. These algorithms are completely general and can be applied to more realistic model systems. If the speed of electronic computers continues to increase at the present rate it will become possible to study such systems and to obtain agreement with experimental measurements in the near future. 

The kinetic theory model was extended to include the effect of the mass transfer coefficient between the liquid and the gas and the water gas shift reaction in the slurry bubble column reactor. The computed granular temperature was around 30 cm /sec and the computed catalyst viscosity was closed to 1.0 cp. The volumetric mass transfer coefficient estimated by the simulation has a good agreement with experimental values shown in the literature. The optimum particle size was determined for maximum methanol production in a SBCR. The size was about 60 - 70 microns, found for maximum granular temperature. This particle size is similar to FCC particle used in petroleum refining. 

Thus, the task of calculating the diffusion coefficient of a particle is reduced to the task of computing its drag coefficient y. For comphcated particle shapes, one can simulate Stokes flow around the particle, using a finite element simulation, to calculate y. The simplest diffusion translational and rotational diffusion coefficients are those of a spherical particle its one-dimensional translational diffusion coeffi-cient is and its one-dimensional rotational diffusion coefficient is, where n is the dynamical viscosity of the fluid and r is the radius of the sphere. 

In our simulations published earlier [20,71,84], we map the DPD interactions onto the macroscopic parameters of fluid, for example, viscosity, compressibility and diffusion coefficient, by using continuum limit equations obtained from kinetic theory. For example, for a given density Pk and sound speed cp- in fcth DPD fluid, we computed the scaling factor 11 from the following continuum equations [81]... 

The model species, total mass, momentum, and energy continuity equations are similar to those presented in Section 13.7 on fluidized bed reactors. Constant values of the gas and liquid phase densities, viscosities, and diffusivities were assumed, as well as constant values of the interphase mass transfer coefficient and the reaction rate coefficient. The interphase momentum transfer was modelled in terms of the Eotvos number as in Clift et al. [1978]. The Reynolds-Averaged Navier-Stokes approach was taken and a standard Computational Fluid Dynamics solver was used. In the continuous liquid phase, turbulence, that is, fluctuations in the flow field at the micro-scale, was accounted for using a standard single phase k-e model (see Chapter 12). Its applicability has been considered in detail by Sokolichin and Eigenberger [1999]. No turbulence model was used for the dispersed gas phase. Meso-scale fluctuations around the statistically stationary state occur and were explicitly calculated. This requires a transient simulation and sufficiently fine spatial and temporal grids. 

NatureWorks s injection moldable grade of Ingeo in the capillary rheometer test has shown a good fit into the Cross-WLF viscosity model (see Table 6.2). There are seven coefficients in the model and it is readily embedded into Moldflow software for injection molding simulation (Moldflow Plastic Labs 2007). Moldflow is computer software that is widely used across the plastic injection molding industry to predict and optimize the... 

Since momenta are available in MD configurations, dynamical observables (e.g., diffusion coefficient, viscosity) can be also computed, and in this respect MD can be considered more versatile than the MC method. The basic microcanonical MD technique has been readily extended to perform simulations at constant temperature and pressure, and a considerable number of thermostats [110] and barostats [96] are currently available. 

Basic requirements on feasible systems and approaches for computational modeling of fuel cell materials are (i) the computational approach must be consistent with fundamental physical principles, that is, it must obey the laws of thermodynamics, statistical mechanics, electrodynamics, classical mechanics, and quantum mechanics (ii) the structural model must provide a sufficiently detailed representation of the real system it must include the appropriate set of species and represent the composition of interest, specified in terms of mass or volume fractions of components (iii) asymptotic limits, corresponding to uniform and pure phases of system components, as well as basic thermodynamic and kinetic properties must be reproduced, for example, density, viscosity, dielectric properties, self-diffusion coefficients, and correlation functions (iv) the simulation must be able to treat systems of sufficient size and simulation time in order to provide meaningful results for properties of interest and (v) the main results of a simulation must be consistent with experimental findings on structure and transport properties. 

---