Viscous number

The outlined scheme is shown to yield stable solutions for non-zero Weissenberg number flows in a number of benchmark problems (Swarbric and Nassehi, 1992b). However, the extension of this scheme to more complex problems may involve modifications such as increasing of elemental subdivisions for stress calculations from 3 x 3 to 9 x 9 and/or the discretization of the stress field by biquadratic rather than bi-linear sub-elements. It should also be noted that satisfaction of the BB condition in viscoelastic flow simulations that use mixed formulations is not as clear as the case of purely viscous regimes. 

The majority of polymer flow processes are characterized as low Reynolds number Stokes (i.e. creeping) flow regimes. Therefore in the formulation of finite element models for polymeric flow systems the inertia terms in the equation of motion are usually neglected. In addition, highly viscous polymer flow systems are, in general, dominated by stress and pressure variations and in comparison the body forces acting upon them are small and can be safely ignored. 

Reynolds dumber. One important fluid consideration in meter selection is whether the flow is laminar or turbulent in nature. This can be deterrnined by calculating the pipe Reynolds number, Ke, a dimensionless number which represents the ratio of inertial to viscous forces within the flow. Because... 

La.mina.r Flow Elements. Each of the previously discussed differential-pressure meters exhibits a square root relationship between differential pressure and flow there is one type that does not. Laminar flow meters use a series of capillary tubes, roUed metal, or sintered elements to divide the flow conduit into innumerable small passages. These passages are made small enough that the Reynolds number in each is kept below 2000 for all operating conditions. Under these conditions, the pressure drop is a measure of the viscous drag and is linear with flow rate as shown by the PoiseuiHe equation for capilary flow ... 

Flow Past Bodies. A fluid moving past a surface of a soHd exerts a drag force on the soHd. This force is usually manifested as a drop in pressure in the fluid. Locally, at the surface, the pressure loss stems from the stresses exerted by the fluid on the surface and the equal and opposite stresses exerted by the surface on the fluid. Both shear stresses and normal stresses can contribute their relative importance depends on the shape of the body and the relationship of fluid inertia to the viscous stresses, commonly expressed as a dimensionless number called the Reynolds number (R ), EHp/]1. The character of the flow affects the drag as well as the heat and mass transfer to the surface. Flows around bodies and their associated pressure changes are important. 

For blending of low viscosity Hquids to practical homogeneity, turnover of three tank volumes is normally adequate. For viscous Hquids or for a higher degree of Hquid homogeneity, a greater number of turnovers is needed. 

Static mixing of immiscible Hquids can provide exceUent enhancement of the interphase area for increasing mass-transfer rate. The drop size distribution is relatively narrow compared to agitated tanks. Three forces are known to influence the formation of drops in a static mixer shear stress, surface tension, and viscous stress in the dispersed phase. Dimensional analysis shows that the drop size of the dispersed phase is controUed by the Weber number. The average drop size, in a Kenics mixer is a function of Weber number We = df /a, and the ratio of dispersed to continuous-phase viscosities (Eig. 32). 

For laminar flow (Re < 2000), generally found only in circuits handling heavy oils or other viscous fluids, / = 16/Re. For turbulent flow, the friction factor is dependent on the relative roughness of the pipe and on the Reynolds number. An approximation of the Fanning friction factor for turbulent flow in smooth pipes, reasonably good up to Re = 150,000, is given by / = (0.079)/(4i e ). 

Viscous Drag. The velocity, v, with which a particle can move through a Hquid in response to an external force is limited by the viscosity, Tj, of the Hquid. At low velocity or creeping flow (77 < 1), the viscous drag force is /drag — SirTf- Dv. The Reynolds number (R ) is deterrnined from... 

A parameter used to characterize ER fluids is the Mason number, Af, which describes the ratio of viscous to electrical forces, and is given by equation 14, where S is the solvent dielectric constant T q, the solvent viscosity 7, the strain or shear rate P, the effective polarizabiUty of the particles and E, the electric field (117). 

Piston Cylinder (Extrusion). Pressure-driven piston cylinder capillary viscometers, ie, extmsion rheometers (Fig. 25), are used primarily to measure the melt viscosity of polymers and other viscous materials (21,47,49,50). A reservoir is connected to a capillary tube, and molten polymer or another material is extmded through the capillary by means of a piston to which a constant force is appHed. Viscosity can be determined from the volumetric flow rate and the pressure drop along the capillary. The basic method and test conditions for a number of thermoplastics are described in ASTM D1238. Melt viscoelasticity can influence the results (160). 

Based on such analyses, the Reynolds and Weber numbers are considered the most important dimensionless groups describing the spray characteristics. The Reynolds number. Re, represents the ratio of inertial forces to viscous drag forces. 

The behavior of colloidal suspensions is controlled by iaterparticle forces, the range of which rarely extends more than a particle diameter (see Colloids). Consequentiy suspensions tend to behave like viscous Hquids except at very high particle concentrations when the particles are forced iato close proximity. Because many coating solutions consist of complex mixtures of polymer and coUoidal material, a thorough characterization of the bulk rheology requires a number of different measurements. 

The final factor influencing the stabiHty of these three-phase emulsions is probably the most important one. Small changes in emulsifier concentration lead to drastic changes in the amounts of the three phases. As an example, consider the points A to C in Figure 16. At point A, with 2% emulsifier, 49% water, and 49% aqueous phase, 50% oil and 50% aqueous phase are the only phases present. At point B the emulsifier concentration has been increased to 4%. Now the oil phase constitutes 47% of the total and the aqueous phase is reduced to 29% the remaining 24% is a Hquid crystalline phase. The importance of these numbers is best perceived by a calculation of thickness of the protective layer of the emulsifier (point A) and of the Hquid crystal (point B). The added surfactant, which at 2% would add a protective film of only 0.07 p.m to emulsion droplets of 5 p.m if all of it were adsorbed, has now been transformed to 24% of a viscous phase. This phase would form a very viscous film 0.85 p.m thick. The protective coating is more than 10 times thicker than one from the surfactant alone because the thick viscous film contains only 7% emulsifier the rest is 75% water and 18% oil. At point C, the aqueous phase has now disappeared, and the entire emulsion consists of 42.3% oil and 57.5% Hquid crystalline phase. The stabilizing phase is now the principal part of the emulsion. 

This postulate imposes an idealization, and is the basis for all subsequent property relations for PVT systems. The PVT system sei ves as a satisfactoiy model in an enormous number of practical applications. In accepting this model one assumes that the effects of fields (e.g., elec tric, magnetic, or gravitational) are negligible and that surface and viscous-shear effects are unimportant. 

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