Viscous fluid models

Sampling of physical model of process. At this stage is accepted any idealization, the schematization, etc. For example, instead of an imperfect gas is accepted any of classical models model of perfect fluid (gas) or viscous fluid model. Within the limits of the set model, it is possible to inject the assumption of external affecting in addition. 

The Emerman model described in the previous section is hardly applicable to the carbon black-filled CCM as the black particles have sizes of hundreds angstrom and such a composite, compared with the molding channel size, may be considered as a homogeneous viscous fluid. Therefore, the polymer structure, crystallinity and orientation play an important role for such small particles. The above-given example of manufacture of the CCM demonstrates the importance of these factors being considered during processing of a composite material to and article with the desired electrical properties. 

In contrast with the one-dimensional model, the two-dimensional model allows to determine the actual parameter distribution in flow fields of the working fluid and its vapor. It also allows one to calculate the drag and heat transfer coefficients by the solution of a fundamental system of equations, which describes the flow of viscous fluid in a heated capillary. 

As noted before, thin film lubrication (TFL) is a transition lubrication state between the elastohydrodynamic lubrication (EHL) and the boundary lubrication (BL). It is widely accepted that in addition to piezo-viscous effect and solid elastic deformation, EHL is featured with viscous fluid films and it is based upon a continuum mechanism. Boundary lubrication, however, featured with adsorption films, is either due to physisorption or chemisorption, and it is based on surface physical/chemical properties [14]. It will be of great importance to bridge the gap between EHL and BL regarding the work mechanism and study methods, by considering TFL as a specihc lubrication state. In TFL modeling, the microstructure of the fluids and the surface effects are two major factors to be taken into consideration. 

Sundaraj, U., Dori, Y., and Macosko, C. W., Sheet formation in immiscible polymer blends model experiments on an initial blend morphology. Polymer 36,1957-1968 (1995). Swanson, P. D., and Ottino, J. M., A comparative computational and experimental study of chaotic mixing of viscous fluids, J. Fluid Mech. 213, 227-249 (1990). 

Here the time derivative of the strain is represented by Newton s dot. This is the response of a purely viscous fluid. Now suppose we consider a combination of these models. The two simplest arrangements that we can visualise is the models in series or parallel. When they are placed in series we have a Maxwell model and in parallel we have a Kelvin (or sometimes a Kelvin-Voigt) model. 

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. 

When a tube or pipe is long enough and the fluid is not very viscous, then the dispersion or tanks-in-series model can be used to represent the flow in these vessels. For a viscous fluid, one has laminar flow with its characteristic parabolic velocity profile. Also, because of the high viscosity there is but slight radial diffusion between faster and slower fluid elements. In the extreme we have the pure convection model. This assumes that each element of fluid slides past its neighbor with no interaction by molecular diffusion. Thus the spread in residence times is caused only by velocity variations. This flow is shown in Fig. 15.1. This chapter deals with this model. 

Boussinesq (B4) proposed that the lack of internal circulation in bubbles and drops is due to an interfacial monolayer which acts as a viscous membrane. A constitutive equation involving two parameters, surface shear viscosity and surface dilational viscosity, in addition to surface tension, was proposed for the interface. This model, commonly called the Newtonian surface fluid model (W2), has been extended by Scriven (S3). Boussinesq obtained an exact solution to the creeping flow equations, analogous to the Hadamard-Rybczinski result but with surface viscosity included. The resulting terminal velocity is... 

Note that if j = 1, (9.12) is formally identical with the classical expression (9.7) the classical multiple oscillator model, which will be discussed in Section 9.2, is even more closely analogous to (9.12). However, the interpretations of the terms in the quantum and classical expressions are quite different. Classically, o30 is the resonance frequency of the simple harmonic oscillator quantum mechanically 03 is the energy difference (divided by h) between the initial or ground state / and excited state j. Classically, y is a damping factor such as that caused by drag on an object moving in a viscous fluid quantum mechanically, y/... 

A tubular reactor model that may apply to viscous fluids such as polymers has a radial distribution of linear velocities represented by... 

With the above information, it becomes possible to combine viscous characteristics with elastic characteristics to describe the viscoelasticity of polymeric materials.86-90 The two simplest ways of combining these features are shown in Figure 2.49, where a spring having a modulus G models the elastic response. The viscous response is modelled by what is called a dashpot. It consists of a piston moving in a cylinder containing a viscous fluid of viscosity r. If a downward force is applied to the cylinder, more fluid flows into it, whereas an upward force causes some of the fluid to flow out. The flow is retarded because of the high viscosity and this element thus models the retarded movement and flow of polymer chains. 

Case 2 Continuous deformation vertically averaged strain. The second and third models take a radically different viewpoint of continental deformation single faults are not important but rather add up to appear continuous over long lengthscales. These models use continuum mechanics to describe the lithosphere as a viscous fluid and consider the ratio of stresses arising from buoyancy forces following crustal thickening and horizontal plate... 

In saturated porous media viscous fluid flow is slow. This can be observed in reality as well as in standard experiments. Therefore, dynamic effects will be neglected in the model (x" = o). Furthermore, it will be postulated that the local temperatures of all constituents are equal and that the motions of solid Xs> ice Xb and gel water Xp are the same, i.e., 0 = 0 and xs = Xi = Xp- The distance and response time for movement from gel to ice are negligible. Experiments have shown that the motion occurs in situ, compare Stockhausen Setzer [3],... 

Following Gaskell s work, a great deal of effort was invested by numerous researchers in the field to improve on his model. Most of this effort, however, basically concentrated on solving the Gaskell model with more realistic, constitutive equations and attempts to account for nonisothermal effects. In the original Gaskell model, a purely viscous (nonelastic and time-independent) fluid model is assumed, with specific... 

When a spherical particle exists in a stagnant, suspending gas, its velocity can be predicted from viscous fluid theory for the transfer of momentum to the particle. Perhaps no other result has had such wide application to aerosol mechanics as Stokes (1851) theory for the motion of a solid particle in a stagnant medium. The model estimates that the drag force 2) acting on the sphere is... 

Very important among the screw models are the 1-dimensional models applying the dimensionless parameters introduced by Pawlowski [2] for highly viscous fluids with a constant viscosity. In this case, there are linear relationships for the pressure and power characteristics as a function of throughput. The dimensionless representation, see Fig. 1.7, often used in this book is thus especially important... 

In order to illustrate the specific material properties of polymers, we compare a viscous fluid (silicone oil) with a viscoelastic shear thinning fluid (aqueous polyethylene oxide solution). These fluids are used as model fluids in order to show the flow behavior limits for polymer melts, which corresponds to the behavior of a viscous fluid at very low shear rates and to the behavior of a shear thinning fluid at very high shear rates. 

The advantages and disadvantages of 3-dimensional have already been outlined in Section 6.4. Figure 6.14 gives an overview of the model depths and high-lights their current possibilities. Because currently it is possible to precisely model viscous fluids only in filled... 

Visco-elastic models have been developed for the nonlinear mechanical properties of fluids and solids. For a viscous fluid in simple shear flow, the shear stress, r y y), is a function of the effective viscosity, rj(-y) and the shear rate, y, as follows ... 

For viscoelastic fluids, the formalism of a viscous fluid and an elastic solid are mixed [31]. The equations for the effective viscosity, dynamic viscosity, and the creep compliance are given in Table 12.4 for a viscous fluid, an elastic solid, and a visco-elastic solid and fluid. For the viscoelastic fluid model the dynamic viscosity, >j (tu), and the elastic contribution, G (ti)), are plotted as a function of (w) in Figure 12.31. With one relaxation time, X, the breaks in the two curves occur at co. 

---