Rheological behavior Newtonian fluids

However, for non-Newtonian fluids, even though continuity equation and the equation of motion written as Equation 11.2 remain valid, the Newtonian constitutive equation is not correct and a different constitutive equation is needed. To find constitutive equations, experiments are performed on materials using standard flows described above. The functions of kinematic parameters that characterize the rheological behavior of fluids are called rheological material functions. Standardized material functions are shown in Table 11.1 [2-4]. 

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 flow of plastics is compared to that of water in Fig. 8-5 to show their different behaviors. The volume of a so-called Newtonian fluid, such as water, when pushed through an opening is directly proportional to the pressure applied (the straight dotted line), the flow rate of a non-Newtonian fluid such as plastics when pushed through an opening increases more rapidly than the applied pressure (the solid curved line). Different plastics generally have their own flow and rheological rates so that their non-Newtonian curves are different. 

Complex liquids seldom behave as classical Newtonian fluids thus, analysis of their behavior requires a thorough understanding of non-Newtonian rheology. The importance of this knowledge is illustrated by the following two examples ... 

Many materials are conveyed within a process facility by means of pumping and flow in a circular pipe. From a conceptual standpoint, such a flow offers an excellent opportunity for rheological measurement. In pipe flow, the velocity profile for a fluid that shows shear thinning behavior deviates dramatically from that found for a Newtonian fluid, which is characterized by a single shear viscosity. This is easily illustrated for a power-law fluid, which is a simple model for shear thinning [1]. The relationship between the shear stress, a, and the shear rate, y, of such a fluid is characterized by two parameters, a power-law exponent, n, and a constant, m, through... 

Hydraulic fracturing fluids are solutions of high-molecular-weight polymers whose rheological behavior is non-Newtonian. To describe the flow behavior of these fluids, it is customary to characterize the fluid by the Power Law parameters of Consistency Index (K) and Behavior Index (n). These parameters are obtained experimentally by subjecting the fluid to a series of different shear rates (y) and measuring the resultant shear stresses (t). The slope and Intercept of a log shear rate vs log shear stress plot yield the Behavior Index (n) and Consistency Index (Kv), respectively. Consistency Indices are corrected for the coaxial cylinder viscometers by ... 

Often times concentrated polymeric solutions cannot be treated as Newtonian fluids, however, and this tends to offset the simplifications which result from the creeping flow approximation and the fact that the boundaries are well defined. The complex rheological behavior of polymeric solutions and melts requires that nonlinear constitutive equations, such as Eqs. (l)-(5), be used (White and Metzner, 1963) ... 

The behavior of a non-Newtonian viscoelastic fluid can be described by a constitutive equation which takes into account condition (1). Rheological behavior of the fluid is described by an equation derived from White-Metzner-Litvinov model and takes the following form 27,321 ... 

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... 

Fig. 11.19 Viscosity of suspensions of spherical particles in Newtonian fluids, (a) Curve constructed by Bigg. [Reprinted by permission from D. M. Bigg, Rheological Behavior of Highly Filled Polymer Melts, Polym. Eng. Sci, 23, 206 (1983).] (b) Curves presented by Thomas (79).

Thus far we have given exclusive attention to the flow of purely viscous fluids. In practice the chemical engineer often encounters non-Newtonian fluids exhibiting elastic as well as viscous behavior. Such viscoelastic fluids can be extremely complex in their rheological response. The le vel of mathematical complexity associated with these types of fluids is much more sophisticated than that presented here. Within the limits of space allocated for this article, it is not feasible to attempt a summary of this very extensive field. The reader must seek information elsewhere. Here we shall content ourselves with fluids that do not exhibit elastic behavior. 

Figure 3.2 illustrates a classification of the rheological behavior of solids and fluids. Examples of different flow behaviors are shown in the lowest boxes. Figure 3.2 also illustrates the resulting shear stress as a function of the (shear-)deformation y or, for fluids, the shear rate y. The two most important material properties for our discussion in this chapter are the viscoelastic and the Newtonian fluid circled in the figure. 

Newtonian fluids display the simplest rheological behavior. They show a constant viscosity T] and there is a direct proportionality between the shear rate y and shear stress o ... 

There are three types of non-Newtonian fluids plastic, pseudoplastic, and dila-tant. Figure 4.39 shows the rheological behaviors of Newtonian and non-Newtonian fluids. A plastic fluid does not move until the shear stress exceeds a certain minimum value, known as the yield value (f), and is expressed mathematically ... 

I would also like to list some of the challenges that will provide the foundation for where the profession has to go (Fig. 2). This is not meant to be comprehensive, but to suggest some of what we should be doing. This wish list derives from work Bob Brown and I have done on modeling flows of polymer fluids. The first item has to do with the need to understand the effects of polymer structure and rheology on flow transitions in polymeric liquids and on polymer processing operations. In the past, we ve studied extensively the behavior of Newtonian fluids and how Newtonian flows evolve as, say, the Reynolds number is varied. We have tools available to... 

There are two general types of constitutive equations for fluids Newtonian and non-Newtonian. For Newtonian fluids, the relation between the stress tensor, t, and the rate of deformation tensor or the shear stress is linear. For non-Newtonian fluids the relation between the stress tensor and the rate of deformation tensor is nonlinear. The various Newtonian and non-Newtonian rheologies of fluids are shown in Figure 12.2. There are four types of behavior (1) Newtonian, (2) pseudo-plastic, (3) Bingham plastic, and (4) dilatent. The reasons for these different rheological behaviors will also be discussed in subsequent sections of this chapter. But first it is necessary to relate the stress tensor to the rate of deformation tensor. 

In some cases, an extrudable and injectable paste may consist of 65% vol. ceramic powder and 35% vol. polymeric binder. In others, an extrudable paste may consist of a highly loaded aqueous suspension of clay particles such that its rheology is plastic. Hie low shear (i.e., <100 sec ) viscosity of such a paste is between 2000 and 5000 poise at ambient temperature. Highly nonlinear stress strain curves are typical of ceramic pastes, as well as time dependent thixotropy. In many cases, pastes behave like visco-elastic fluids. This complex rheological behavior of ceramic pastes has made theoretical approadies to these problems difficult. For this reason, the discussion in this chapter is limited to Newtonian fluids where analytical solutions are possible, with obvious consequences as to accuracy of these equations for non-Newtonian ceramic pastes. 

In Eqs. (6) and (7) e represents the internal energy per unit mas, q the heat flux vector due to molecular transport, Sh the volumetric heat production rate, ta, the mass fraction of species i, Ji the mass flux vector of species i due to molecular transport, and 5, the net production rate of species i per unit volume. In many chemical engineering applications the viscous dissipation term (—t Vm) appearing in Eq. (6) can safely be neglected. For closure of the above set of equations, an equation of state for the density p and constitutive equations for the viscous stress tensor r, the heat flux vector q, and the mass flux vector 7, are required. In the absence of detailed knowledge on the true rheology of the fluid, Newtonian behavior is often assumed. Thus, for t the following expression is used ... 

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