General viscous model

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... 

Other models have been used (Bird et al., 1987, p. 228), but most studies have concentrated on the power law or the Cross (Carreau) models. Once one has chosen a numerical method, any of the general viscous models that depend on I ho can be used. 

While the viscous model for the evolution of protoplanetary disks has had some success in matching some of the general properties of protoplanetary disks, such as the observed mass accretion rates and effective temperatures, the exact source of the viscosity remains the subject of ongoing studies. Currently, the most popular candidates for driving the mass transport in protoplanetary disks are the magneto-rotational instability (MRI) and gravitational instability. A third candidate, shear instability, has also been proposed based on laboratory experiments of rotating fluids (Richard Zahn 1999), but questions remain as to whether these results can be extended to the scale of protoplanetary disks. 

A generalized kinetic model of cure is developed from the aspect of relaxation phenomena. The model not only can predict modulus and viscosity during the cure cycle under isothermal and non-isothermal cure conditions, but also takes into account filler effects on cure behavior. The increase of carbon black filler loading tends to accelerate the cure reaction and also broadens the relaxation spectrum. The presence of filler reduces the activation energy of viscous flow, but has little effect on the activation energy of the cure reaction. 

An important class of non-Newtonian fluids is formed by isotropic rheological stable media whose stress tensor [ry] is a continuous function of the shear rate tensor [e,j] and is independent of the other kinematic and dynamic variables. One can rigorously prove that the most general rheological model satisfying these conditions is the following nonlinear model of a viscous non-Newtonian Stokes medium [19] ... 

It is known that incompressible newtonian fluids at constant temperature can be characterized by two material constants the density p and the viscosity T. The characterization of a purely viscous nonnewtonian fluid using the power law model (or any of the so-called generalized newtonian models) is relatively straightforward. However, the experimental description of an incompressible viscoelastic nonnewtonian fluid is more complicated. Although the density can be measured, the appropriate expression for r poses considerable difficulty. Furthermore there is some uncertainty as to what other properties need to be measured. In general, for viscoelastic fluids it is known that the viscosity is not constant but depends on shear rate, that the normal stress differences are finite and depend on shear rate, and that the stress may also depend on the preshear history. To characterize a nonnewtonian fluid, it is necessary to measure the material functions (apparent viscosity, normal stress differences, etc.) in a relatively simple or standard flow. Standard flow patterns used in characterizing nonnewtonian fluids are the simple shear flow and shear-free flow. 

The creep of dam concrete includes two parts instantaneous elastic deformation and viscous deformation. A generalized Kelvin model consisting of two standard Kelvin model in series is used to describe the time-dependent deformation of the dam concrete as shown in Figure 1. 

If the viscous strain of Kelvin model is f at to t = to + Ar, and the stress remains constant during the Al time increment, the viscous strain increment of the generalized Kelvin model during At can be derived from equation (4) as... 

A general electrohydrodynamic model of a weakly conductive viscous jet accelerated by an external electric field was also derived, by considering inertial, hydrostatic, viscous, electric, and surface tension forces. Nonlinear rheologic constitutive equation for the jet radius was derived,... 

There are a number of additional physical phenomena, such as wall slip, electric effects and viscous energy dissipation, which may need to be taken into account. Generally applicable models are not available for some of these effects, particularly wall slip. 

A numerical study is carried out for illustrating the effect of in-stmcture damping models on the optimal distribution of dampers. Models used for the study are the classical Rayleigh model and the non-viscous model given by Equation (1) as it represents the most general damping model within the scope of a linear analysis (Woodhouse 1998). 

Fig. 8 Linear elastic and viscous modulus functions G co, T) and G"(a>, T) of gum EPDM2504, drawn using the G of a six elements generalized Maxwell model and the respective Cl, C2 parameters of a WLF type equation with 100 °C as reference temperature experimeutal data from a frequency-temperature sweep protocol at 1 deg. strain amplitude with a closed-cavity torsional harmonic rheometer are displayed for comparison with the calculated maps...

The most widely used form of the general viscous constitutive relation is the power law model... 

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 purely viscous fluids, the rheological constitutive equation that relates the stresses x to the velocity gradients is the generalized Newtonian model [5,6,21] and is written as... 

When the identified system description is parametric, e.g., in state-space form, no assumptions - other than that the structure s response can be described with the identified description - are needed for estimating the modal characteristics. The assumed damping model is most often general viscous damping, which contains proportional damping as a special case. This means that the approach can also be used when localized dampers are present and the mode shapes are complex. 

Zener body (Carcione et al. 1988), and (c) a generalized Maxwell model augmented with a viscous damper (Bielak et al. 2011)... 

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. 

A major complication, especially for separated flows, arises from the effect of slip. Slip occurs because the less dense and less viscous phase exhibits a lower resistance to flow, as well as expansion and acceleration of the gas phase as the pressure drops. The result is an increase in the local holdup of the more dense phase within the pipe (phase density, pm), as given by Eq. (15-11). A large number of expressions and correlations for the holdup or (equivalent) slip ratio have appeared in the literature, and the one deduced by Lockhart and Martinelli is shown in Fig. 15-7. Many of these slip models can be summarized in terms of a general equation of the form... 

Krt < k scalar spectral transport time scale defined in terms of the velocity spectrum (e.g., rst). 

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