Viscous Flow Models

Strictly speaking, the viscosity rj, measured with shear deformation viscometers, should not be used to represent the elongational terms located on the diagonal of the stress and strain rate tensors. Elongational flows are briefly discussed later in this chapter. A rheologist s 

The Power Law Model. The power law model proposed by Ostwald [57] and de Waale [15] is a simple model that accurately represents the shear thinning region in the viscosity versus strain rate curve but neglects the Newtonian plateau present at small strain rates. The power law model can be written as follows  

The infinite viscosity at zero strain rates leads to an erroneous result when there is a region of zero shear rate, such as at the center of a tube. This results in a predicted velocity distribution that is flatter at the center than the experimental profile, as will be explained in more detail in Chapter 5. In computer simulation of polymer flows, this problem is often overcome by using a truncated model such as 

The Bird-Carreau-Yasuda Model. A model that fits the whole range of strain rates was developed by Bird and Carreau [7] and Yasuda [72] and contains five parameters  

The Bingham Fluid. The Bingham fluid is an empirical model that represents the rheological behavior of materials that exhibit a no flow region below certain yield stresses, tv, such as polymer emulsions and slurries. Since the material flows like a Newtonian liquid above the yield stress, the Bingham model can be represented by 

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In the viscous-flow model, oxidation rate and viscosity are related through kf. 

Our theoretical calculations indicated viscous flow to be dominant at high densities, as depicted the agreement of the theory in the inset in Figure 1. A viscous flow model may be used over length scales larger than the mean free path, which is largely satisfied for mesopores. To obtain the theoretical transport coefficient we solved die Navier Stokes equation... 

Finely Porous Model. In this model, solute and solvent permeate the membrane via pores which connect the high pressure and low pressure faces of the membrane. The finely porous model, which combines a viscous flow model eind a friction model (7, ), has been developed in detail and applied to RO data by Jonsson (9-12). The most recent work of Jonsson (12) treated several organic solutes including phenol and octanol, both of which exhibit solute preferential sorption. In his paper, Jonsson compared several models including that developed by Spiegler eind Kedem (13) (which is essentially an irreversible thermodynamics treatment), the finely porous model, the solution-diffusion Imperfection model (14), and a model developed by Pusch (15). Jonsson illustrated that the finely porous model is similar in form to the Spiegler-Kedem relationship. Both models fit the data equally well, although not with total accuracy. The Pusch model has a similar form and proves to be less accurate, while the solution-diffusion imperfection model is even less accurate. 

Figure 7. A viscous flow model for silicon dioxide film formation.

Viscous flow models have been applied to a variety of problems ranging from slow creep deformations and mud flows to more rapid fluid types of mass movement. Morgenstem (1967) described an early viscous flow analysis in which the soil shear resistance consisted of a velocity-dependent viscous component in addition to Coulomb friction. In his analysis Morgenstem (1967) derived an equation for determining the velocity required to change a slump into a turbidity current as shown in Figure 11.20. This expression is given as follows... 

Hirose, T, and Y. Murakami, Two-dimensional viscous flow model for power consumption in close-clearance agitators , J. Chem. Eng. Japan 19, 6, 568-574 (1986). 

Both creep and stress relaxation is modeled using computer simulation software based on simple spring (elastic deformation) and dashpot (viscous flow) models. Many polymers, when they approach the Tg, will exhibit viscoelastic behavior in which the physical characteristics are best described by considering the material as having both solid- and liquid-like properties. Viscoelasticity is an important property to be found in polymeric materials (see Viscoelasticity). 

In the absence of any concentration polarization, and Cfi are equal to Cg and respectively. The extent of concentration polarization and its effects on the solvent flux and solute transport for porous membranes and macrosolutes/proteins can be quite severe (see Section 6.3.3). This model is often termed the combined diffusion-viscous flow model (Merten, 1966), and it can be used in ultrafiltration (see Sections 6.3.3.2 and 7.2.1.3). The relations between this and other models, such as the finely porous model, are considered in Soltanieh and Gill (1981). 

From the solute flux expression (3.4.91c) in the combined diffusion-viscous flow model, one can develop an expression for S,n,e or Rn e- Consider relation (3.4.93b) ... 

A more recent model for the preexponential factor including viscous flow across the solid-liquid interface is [14]... 

This determines the total flux at the li/nic of viscous flow. Equations (5.18 and (5.19) therefore describe the limiting form of the dusty gas model for high pressure or large pore diameters -- the limit of bulk diffusion control and viscous flow,... 

Our approach in this chapter is to alternate between experimental results and theoretical models to acquire familiarity with both the phenomena and the theories proposed to explain them. We shall consider a model for viscous flow due to Eyring which is based on the migration of vacancies or holes in the liquid. A theory developed by Debye will give a first view of the molecular weight dependence of viscosity an equation derived by Bueche will extend that view. Finally, a model for the snakelike wiggling of a polymer chain through an array of other molecules, due to deGennes, Doi, and Edwards, will be taken up. 

One simple rheological model that is often used to describe the behavior of foams is that of a Bingham plastic. This appHes for flows over length scales sufficiently large that the foam can be reasonably considered as a continuous medium. The Bingham plastic model combines the properties of a yield stress like that of a soHd with the viscous flow of a Hquid. In simple Newtonian fluids, the shear stress T is proportional to the strain rate y, with the constant of proportionaHty being the fluid viscosity. In Bingham plastics, by contrast, the relation between stress and strain rate is r = where is... 

The study of flow and elasticity dates to antiquity. Practical rheology existed for centuries before Hooke and Newton proposed the basic laws of elastic response and simple viscous flow, respectively, in the seventeenth century. Further advances in understanding came in the mid-nineteenth century with models for viscous flow in round tubes. The introduction of the first practical rotational viscometer by Couette in 1890 (1,2) was another milestone. 

A real material whose behaviour can be modelled in this way initially undergoes irreversible deformation as the stress is applied. This eventually ceases, and the material then behaves effectively as an elastic solid. Release of the stress will cause a rapid return to a less strained state, corresponding to the spring component of the response, but part of the deformation, arising due to viscous flow in the dashpot will not disappear. 

Caustic Waterflooding. In caustic waterflooding, the interfacial rheologic properties of a model crude oil-water system were studied in the presence of sodium hydroxide. The interfacial viscosity, the non-Newtonian flow behavior, and the activation energy of viscous flow were determined as a function of shear rate, alkali concentration, and aging time. The interfacial viscosity drastically... 

In practice, it is often possible with stirred-tank reactors to come close to the idealized mixed-flow model, providing the fluid phase is not too viscous. For homogenous reactions, such reactors should be avoided for some types of parallel reaction systems (see Figure 5.6) and for all systems in which byproduct formation is via series reactions. 

Figures 5 and h show how the shape of the creep curve is modified by changes in the constants of the model. The values of the constants are given in Table I. Curve I is the same as shown in Figure 4, curve II shows onlv a small amount of viscous creep, and in curve 111, viscous flow is a prominent part of the total creep. The same data were used in Figures 5 and 6, but notice the dramatic, change in the shapes of the curves when a linear time scale is replaced by a logarithmic time scale. In the model, most of the recoverable creep occurs "Within about one decade of the retardation time.

In the impact process that involves large temperature differences (AT) between the surface and the droplet, such as the ones considered in this study (e.g., AT=300 500 C), the value for Res is about 0.5 1.0. Thus, the inertial force of the vapor flow would be of the same order of magnitude as the viscous force, and cannot be neglected in Eq. (47) for the vapor-flow model. 

The second term on the right-hand side of Eqs. (145) and (146) contains the viscous-stress models ag and asm. Even for laminar flow, suitable forms for these models are difficult to determine a priori. Typical models used in CFD introduce an effective viscosity pea for each phase, and describe the viscous stresses as follows. 

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