Inviscid models

The kinematic constraint imposed by a rigid body ensures that the flow goes around each body and this flow (on the upstream portion of the body) is strongly determined by the body shape. This blocking effect is not a feature of distributed drag models - its importance is illustrated here using inviscid models (see Eames, Hunt Belcher [163] for a more comprehensive description). We show in Section 7.4 how inviscid blocking may be included into future computational models. 

The inviscid model (7.35) suggests that the interstitial Eulerian velocity is related to the bubble rise velocity (v) and bulk fluid velocity U through,... 

Prediction of liquid recirculation based on the early inviscid models of Freedman and Davidson (1969) and Whalley and Davidson (1974) is only applicable to shallow tanks stirred with gas plumes. Extending the so-called principle of minimization of maximum vorticity to tall columns, lead Joshi and Sharma (1979) to conclude that multiple circulation cells of height equal to column diameter are present (Figure 3a). 

Brayshaw, M.D., 1990. Numerical model for the inviscid flow of a fluid in a hydrocyclone to demonstrate the effects of changes in the vorticity function of the flow field on particle classification. International Journal of Mineral Processing, 29, 51. 

Fig. 11. Comparison of Kumar and Kuloor s model (K18) with the data and model of Davidson and Schuler (D9) for bubble formation in inviscid liquids.

This model makes use of the same concepts as for inviscid liquids except that the viscous resistance is now taken into consideration. Further, the... 

The model for inviscid liquids is equally well applicable to viscous liquids also, provided that the resistance due to viscous drag is included in the analysis. As a first approximation, the viscous drag may be evaluated by a Stokes resistance term, since the bubble is not followed by a wake. Thus we proceed as before, first evaluating the force-balance bubble volume Kr and then the total bubble volume by reference to the detachment stage. 

Fig. 15. Comparison of the model (S3) with the data collected for bubble formation in inviscid liquids under constant pressure conditions.

This model is a modification of the model developed by Kumar and Kuloor (K18) for bubble formation in inviscid fluids in the absence of surface-tension effects. The need for modification arises because the bubble forming nozzles actually used to collect data on bubble formation in fluidized beds differ from the orifice plates in that they do not have a flat base. Under such conditions the bubble must be assumed to be moving in an infinite medium and the value of 1/2 is more justified than the value 11/16. 

The model proposed for constant flow conditions is again used here to explain the data. The modified equations for inviscid fluids are ... 

A SECOND GRADIENT MODEL FOR DEFORMABLE POROUS MATRICES FILLED WITH AN INVISCID FLUID... 

Favelukis et al. (37,38) dealt with the problem of droplet deformation in exten-sional flow with both Newtonian and non-Newtonian Power Law model fluids, as wellas bubble breakup. For the Newtonian case, they find that as an inviscid droplet (or bubble) deforms, the dimensionless surface area is proportional to the capillary number... 

It would seem natural to ask how both drag forces and inviscid blocking may be included in current incompressible computational models of multibody flows. To demonstrate this for the case of an isolated body, we first note that that the inertial terms in the momentum equation may be expressed as... 

Figure 7.7 Normalised interstitial velocity versus voidage. The closed symbols are experimental measurements. The inviscid prediction (7.48), is plotted as a full line, while the wake corrected model (7.49) is plotted as open symbols.

A number of authors, however, have represented forest or urban canopy layers by porous regions of distributed force (or drag) [206, 213, 217, 318, 320, 576, 662], The advantage of such an approach is that it permits inclusion of a canopy sublayer without the use of excessive and costly grid resolution. Yamada [662] and Shaw and Schumann [576] introduced the approach in order to add vegetation to meso-scale models of complex terrain. Jeram et al. [320] used the concept in 2d calculations for inviscid flow and constant eddy diffusivity flow estimates of the up and downwind penetration of flow within simple urban areas. 

Auton [7], Thomas et al [152] and Auton et al [8] determined a lift force due to inviscid flow around a sphere. In an Eulerian model formulation this lift force parameterization is usually approximated for dilute suspensions, giving ... 

Flow around single cylinders is the elementary model for (he fibrous filter and is the geometry of interest for deposition on pipes, wires, and other such objects in an air flow (Chapter 3). The flow patterns at low and high Reynolds numbers differ significantly, and thi.s affects impaction efficiencies. For Re > 100. the velocity distribution outside the velocity boundary layer can be approximated by inviscid flow theory. This approximates the velocity distribution best over the front end of the cylinder which controls the impaction efficiency. The components of the velocity in the direction of the mainstream flow, x, and normal to the main flow, y, are... 

The reaction engineering aspects of these polymerizations are similar. Good heat transfer to a comparatively inviscid phase makes them suitable for vinyl addition polymerizations. Free-radical catalysis is mostly used, but cationic catalysis is used for nonaqueous dispersion polymerization (e.g., of isobutene). High conversions are generally possible, and the resulting polymer, either as a latex or as beads is directly suitable for some applications (e.g., paints, gel permeation chromatography beads, expanded polystyrene). Suspension polymerizations are run in the batch model. Continuous emulsion polymerization is common. 

Our model about the porous matrix includes, as a particular case, previous void theories which does not anticipate any size effects in torsion on the other hand the virtual inertia considered in our immiscible mixture allows the presence of an inviscid drag term depending on changes of the radius of the material elements, other than on the relative velocities, a term which is usually absent in previous theories (see Giovine (1990)). 

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