General viscous fluid

A large number of models that depend on rate of deformation have been developed, but they all arise logically fi om the general viscous fluid. The general viscous model can be derived by a process very similar to the derivation of the general elastic solid in Section 1.6. Here we propose that stress depends only on the rate of deformation 

Expanding the function in a power series gives T = folf + /iD -1- fil3 +  

Note that D° = I and for an incompressible fluid f = —p. Again we can evoke the Cayley Hamilton theorem, eq. 1.6.2. Thus 

This constitutive equation is also known as a Reiner-Rivlin fluid. The Newtonian fluid is simply a special case with t , (Iho, 11 ho) = I. a constant, and r 2 = 0. 

The term gives rise to normal stresses in steady shear flow, but unfortunately they are not even in qualitative agreement with experimental observations. This can Ik readily seen by noting 

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Since the r], term gives qualitatively the wrong result, it is usually discarded. Therefore, the general viscous fluid reduces to... 

These models fit the shear rate dependence of viscosity very well and are very usefid to engineers. They form the backbone of polymer processing flow analyses. If the problem is to predict pressure drop versus steady flow rate in channels of relatively constant cross section, or torque versus steady rotation rate, the general viscous fluid gives excellent results. We need to be sure that we pick a model that describes our particular material over the rates and stresses of concern, however. With numerical methods, the multiple parameter models are readily solved. 

If shear thinning is the main phenomenon to be described, the simplest model is the general viscous fluid. Section 2.4. It has no time dependence, nor can it predict any normal stresses or extensional thickening (however, recaU eq. 2.4.24). Nevertheless, it should generally be the next step after a Newtonian solution to a complex process flow. The power law. Cross or Carreau-type models are available on all large-scale fluid mechanics computation codes. As discussed in Section 2.7, they accurately predict pressure drops in flow through channels, forces on rollers and blades, and torques on mixing blades. 

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 ). 

Flow in tubular reactors can be laminar, as with viscous fluids in small-diameter tubes, and greatly deviate from ideal plug-flow behavior, or turbulent, as with gases, and consequently closer to the ideal (Fig. 2). Turbulent flow generally is preferred to laminar flow, because mixing and heat transfer... 

All fluids for which the viscosity varies with shear rate are non-Newtonian fluids. For uou-Newtouiau fluids the viscosity, defined as the ratio of shear stress to shear rate, is often called the apparent viscosity to emphasize the distiuc tiou from Newtonian behavior. Purely viscous, time-independent fluids, for which the apparent viscosity may be expressed as a function of shear rate, are called generalized Newtonian fluids. 

For steady-state laminar flow of any time-independent viscous fluid, at average velocity V in a pipe of diameter D, the Rabinowitsch-Mooney relations give a general relationship for the shear rate at the pipe wall. 

When a pump performance is defined for water, the corrected performance for a viscous fluid can be developed using Figure 3-56 or 3-57. In order to develop the curves for viscosity conditions of 100 SSU or 1,000 SSU as shown in Figure 3-58, the following general procedure is used [17]. 

The plate heat exchanger can also be used for evaporation of highly viscous fluids when the evaporation occurs in plate or the liquid flashes after leaving the plate. Applications generally have been restricted to the soap and food industries. The advantage of these units is their ability to concentrate viscous fluids of up to 50 poise. 

Mechanical and chemical methods for qualitative and quantitative measurement of polymer structure, properties, and their respective processes during interrelation with their environment on a microscopic scale exist. Bosch et al. [83] briefly discuss these techniques and point out that most conventional techniques are destructive because they require sampling, may lack accuracy, and are generally not suited for in situ testing. However, the process of polymerization, that is, the creation of a rigid structure from the initial viscous fluid, is associated with changes in the microenvironment on a molecular scale and can be observed with free-volume probes [83, 84]. 

In general positive displacement pumps have limited flow capacity but are capable of relatively high pressures. Thus these pumps operate at essentially constant flow rate, with variable head. They are appropriate for high pressure requirements, very viscous fluids, and applications that require a precisely controlled or metered flow rate. 

While the value for N is useful for comparing the relative efficiencies of different columns, the HETP is useful in assessing the varying efficiency of the same column under different conditions. The value decreases as the efficiency of the column increases, a characteristic that is generally better for small particle supports and less viscous fluids. In order to assess column efficiency independently of the... 

The diffusion process in general may be viewed as the model for specific well-defined transport problems. In particle diffusion, one is concerned with the transport of particles through systems of particles in a direction perpendicular to surfaces of constant concentration in a viscous fluid flow, with the transport of momentum by particles in a direction perpendicular to the flow and in electrical conductivity, with the transport of charges by particles in a direction perpendicular to equal-potential surfaces. 

In extrusion, a fluid material, generally rendered fluid through heating, is forced through a shaping device. Since there is a need for quickness and because the preshaped material is quite viscous, extrusion requires high pressure to drive or force the melt through a die. The melts can be extruded as pipes, sheets, films, or poured into molds. 

For an orthotropic particle in steady translation through an unbounded viscous fluid, the total drag is given by Eq. (4-5). In principle, it is possible to follow a development similar to that given in Section IT.B.l for axisymmetric particles, to deduce the general behavior of orthotropic bodies in free fall. This is of limited interest, since no analytic results are available for the principal resistances of orthotropic particles which are not bodies of revolution. General conclusions from the analysis were given in TLA. 

Thus, according to Eq. (40) heat transfer coefficients to viscous fluids are higher than those to less viscous materials, in contradiction to the behavior usually observed. Although the experimental data upon which Eq. (40) was based appear to be precise, it may be concluded that the correlation itself is not generally useful. 

General comments about flow of viscous fluids... 

Generally speaking viscous fluid flow is not irrotational. Nevertheless, in regions of irrotational flow there is a great simplification of the acceleration vector. Referring back to Eqs. 2.55 and 2.56, note that for irrotational flow... 

As discussed in Section 2.6, vorticity is a measure of the angular rotation rate of a fluid. Generally speaking, vorticity is produced by forces that cause rotation of the flow. Most often, those forces are caused by viscous shearing action. As viscous fluid flows over solid walls, for example, the shearing forces caused by a no-slip condition at the wall is an important source of vorticity. The following analysis shows how vorticity is transported throughout a flow field by convective and viscous phenomena. 

Rheodynamics of non-linear viscous fluids flowing in circular channels with moving walls is described most comprehensively in 1S-34). With respect to the above conclusion (see sect 2.2.1) that the high elasticity of a melt influences insignificantly flow rate parameters of a flow, the combined shear is discussed in 24128-30,341 on the basis of a general approach to the analysis of viscosimetric flows developed by B. Colleman and W. Noll. 

Merzhanov Stolin (Ref 17) examined hydro-dynamic heating of a viscous fluid undergoing Cuette-type flow. After developing the general equations, they obtained physical analogies between Cuette flow and the well-known hydrodynamic heating that occurs in a Newtonian fluid with an exponential dependence of vis-... 

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