Other Fluid Models

A polynomial relationship was proposed by Rabinowitsch and Weissenberg it can be written as [20aj  

Another model in which the viscosity is described as a function of shear stress is the Bingham model [20]. This model is used for fluids with a yield stress Xq. Below this yield stress, the viscosity is infinite (no motion) above the yield stress, the viscosity is finite (motion occurs). The Bingham Fluid model is written as  

This model is primarily used for slurries and pastes. The parameters Tq and Po can be reiated empirically to the volume fraction of solids cp, the particle diameter Dp, and the viscosity of the suspending fluid  

Many other fluid models have been proposed. For a more detailed discussion, the reader is referred to the literature [17, 20-22, 54,94]. 

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For other fluid models, the reader is encouraged to refer to Caccamo [99] and references cited therein for detailed discussions. 

Approach used in section 3.2 for power-law and Bingham plastic model fluids can be extended to other fluid models. Even if the relationship between shear stress and shear rate is not known exactly, it is possible to use the following approach to the problem. It depends upon the fact that the shear stress distribution over the pipe cross-section is not a function of the fluid rheology and is given simply by equation (3.2), which can be re-written in terms of the wall shear stress, i.e. 

Fig. 4. Schematic representation of a two-dimensional model to account for the shear modulus of a foam. The foam stmcture is modeled as a coUection of thin films the Plateau borders and any other fluid between the bubbles is ignored. Furthermore, aH the bubbles are taken to be uniform in size and shape.

In the ASTER reactor deposition experiments were performed in order to compare with the 2D model results. Normalized deposition rates are plotted in Figure 22 as a function of radial position for data taken at 25 and 18 Pa. The deposition takes place on a square glass plate. For each pressure two profile measurements were performed, each profile perpendicular to the other (a and b in Fig. 22). A clear discrepancy is present. The use of the simplified deposition model is an explanation for this. Another recent 2D fluid model also shows discrepancies between the measured and calculated deposition rate [257], which are attributed to the relative simplicity of the deposition model. 

For other discussions of two-phase models and numerical solutions, the reader is referred to the following references thermofluid dynamic theory of two-phase flow (Ishii, 1975) formulation of the one-dimensional, six-equation, two-phase flow models (Le Coq et al., 1978) lumped-parameter modeling of one-dimensional, two-phase flow (Wulff, 1978) two-fluid models for two-phase flow and their numerical solutions (Agee et al., 1978) and numerical methods for solving two-phase flow equations (Latrobe, 1978 Agee, 1978 Patanakar, 1980). 

The number of equations to be solved is, among other things, related to the turbulence model chosen (in comparison with the k-e model, the RSM involves five more differential equations). The number of equations further depends on the character of the simulation whether it is 3-D, 21/2-D, or just 2-D (see below, under The domain and the grid ). In the case of two-phase flow simulations, the use of two-fluid models implies doubling the number of NS equations required for single-phase flow. All this may urge the development of more efficient solution algorithms. Recent developments in computer hardware (faster processors, parallel platforms) make this possible indeed. 

One usually distinguishes two types of lattice models. The first type may be called lattice-gas models. In this case, the number of molecules in the system is less than the number of available sites. In other words, there are vacant sites. The second type of lattice models may be called lattice fluids. In this case, all lattice sites are filled exactly by the molecular components in the system the system is considered to be incompressible. It is easily shown that a two-component incompressible lattice fluid model can be mapped on a one-component lattice gas one. In other words, it is possible to interpret vacant sites to be occupied by a ghost ... 

Other fluid dynamic models of slurry flow have also been developed by other workers [57]. Coppeta, Rodgers, Radzak, and coworkers examined slurry flow, both from a simulation point of view, and from an experimental angle [6,10,11]. A special test apparatus is used consisting of flourescent injections of die that is entrained beneath a glass wafer enabling observation of slurry flow patterns and residence time. Such studies are instrumental... 

In Eqs. 12-47/48 the input term is an exterior quantity and not determined by the model itself. If the substance is added to the system through the inflowing water (or other fluid), we can express / by the input concentration I = Q Cm. Inserting this into Eq. 12-49b yields ... 

Equations 1-5 completely define the "hard fluid" model for solvent induced changes in the vibrational frequency of a diatomic (or pseudo-diatomic) solute. The only adjustable parameter in this model is the coefficient Ca appearing in equation 5. The other parameters, such as the diameters of the solute and solvent as well as the solvent density and temperature, are determined using independent measurements and/or parameter correlations (37). The value of Ca can be determined with a minimal amount of experimental data. In particular we use the frequency shift observed in going from the dilute gas to a dense fluid to fix the value of Ca. Having done this, the... 

There are two types of foams closed cell foams and open cell (or reticulated) foams. In open foams, air or other fluids are free to circulate. These are used for filters and as skeletons. They are often made by collapsing the walls of closed cell foams. Closed cell foams are much stiffer and stronger than open cell foams because compression is partially resisted by increased air pressure inside the cells. Figure 19.1 shows that the geometry of open and closed cell foams can modeled by Kelvin tetrakaidecahedra. 

Here we describe the results of a model calculation of the wave-induced pressure forces on a submerged half-cylinder, and compare the results with experimental data. The implications of the comparisons for both the validity of the model and the experimental procedure will be examined. Finally, the application of the model to other fluid flows and to combustion problems will be discussed. 

The present relations differ from the KM approximation since the factor 3 is replaced by the bridge function at zero separation. This feature does not seem to be unreasonable because, from diagrammatic expansions, B (r) = B r)/3 is supposed to be accurate only at very low densities. Eq. (112) presents two advantages at high density i) it provides a closed-form expression for Bother fluids than the HS model and ii) it allows to ensure a consistent calculation of the excess chemical potential by requiring only the use of the pressure consistency condition (the Gibbs-Duhem constraint, no longer required, is nevertheless implicitly satisfied within 1%). 

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