Shear stress, Newtonian fluids rheology

Define rheology, shear force, shear stress, shear rate, Newtonian fluid, dynamic viscosity, centi-poise, kinematic viscosity, centistokes, viscometry, and viscometer. 

The transport properties of polymeric materials which distinguish them most from other materials are their flow properties or rheological behavior. There are many differences between the flow properties of a polymeric fluid and typical low molecular weight fluids such as water, benzene, sulfuric acid, and other fluids, which we classify as Newtonian. Newtonian fluids can be characterized by a single flow property called viscosity (/r) and their density (p). Polymeric fluids, on the other hand, exhibit a viscosity function that depends on shear rate or shear stress, time-dependent rheological properties, viscoelastic behavior such as elastic recoil (memory), additional normal stresses in shear flow, and an extensional viscosity that is not simply related to the shear viscosity, to name a few differences. 

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 branch of science which is concerned with the flow of both simple (Newtonian) and complex (non-Newtonian) fluids is known as rheology. The flow characteristics are represented by a rheogram, which is a plot of shear stress against rate of shear, and normally consists of a collection of experimentally determined points through which a curve may be drawn. If an equation can be fitted to the curve, it facilitates calculation of the behaviour of the fluid. It must be borne in mind, however, that such equations are approximations to the actual behaviour of the fluid and should not be used outside the range of conditions (particularly shear rates) for which they were determined. 

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... 

Rheology concerns the study of the deformation and flow of soft materials when they respond to external stress or strain. If the ratio of its shear stress and shear rate is a straight line, the material is termed Newtonian otherwise, it is termed non-Newtonian (Figure 4.3.2(a)). As the slope of the curve is the viscosity rj, a shear-thinning fluid exhibits a reduced viscosity as the shear stress increases, whereas a shear-... 

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 ... 

Polymer rheology can respond nonllnearly to shear rates, as shown in Fig. 3.4. As discussed above, a Newtonian material has a linear relationship between shear stress and shear rate, and the slope of the response Is the shear viscosity. Many polymers at very low shear rates approach a Newtonian response. As the shear rate is increased most commercial polymers have a decrease in the rate of stress increase. That is, the extension of the shear stress function tends to have a lower local slope as the shear rate is increased. This Is an example of a pseudoplastic material, also known as a shear-thinning material. Pseudoplastic materials show a decrease in shear viscosity as the shear rate increases. Dilatant materials Increase in shear viscosity as the shear rate increases. Finally, a Bingham plastic requires an initial shear stress, to, before it will flow, and then it reacts to shear rate in the same manner as a Newtonian polymer. It thus appears as an elastic material until it begins to flow and then responds like a viscous fluid. All of these viscous responses may be observed when dealing with commercial and experimental polymers. 

Perhaps the most important and striking features of high internal phase emulsions are their rheological properties. Their viscosities are high, relative to the bulk liquid phases, and they are characterised by a yield stress, which is the shear stress required to induce flow. At stress values below the yield stress, HIPEs behave as viscoelastic solids above the yield stress, they are shear-thinning liquids, i.e. the viscosity varies inversely with shear rate. In other words, HIPEs (and high gas-fraction foams) behave as non-Newtonian fluids. 

We can distinguish between two types of stresses on an interface a shear stress and a dilatational stress. In a shear stress experiment, the interfacial area is kept constant and a shear is imposed on the interface. The resistance is characterized by a shear viscosity, similar to the Newtonian viscosity of fluids. In a dilatational stress experiment, an interface is expanded (dilated) without shear. This resistance is characterized by a dilatational viscosity. In an actual dynamic situation, the total stress is a sum of these stresses, and both these viscosities represent the total flow resistance afforded by the interface to an applied stress. There are a number of instruments to study interfacial rheology and most of them are described in Ref. [1]. The most recent instrumentation is the controlled drop tensiometer. 

Index Entries Com stover rheological measurement shear stress shear rate non-Newtonian fluids Power Law parameters. 

To avoid the apparent complications with absolute rheologic measurement techniques, a number of investigators (4,5). have used relative measurement systems to make rheologic measurements. The major difference between the relative and absolute measurement techniques is that the fluid mechanics in the relative systems are complex. The constitutive equations needed to find the fundamental rheologic variables cannot be readily solved. Relative measurement systems require the use of Newtonian and non-Newtonian calibrations fluids with known properties to relate torque and rotational speed to the shear rate and shear stress (6). 

The impeller method is a technique commonly used to determine rheologic properties of fluids subject to particle settling. The impeller method utilizes a viscometer along with Newtonian and non-Newtonian calibration fluids to obtain constants that relate shear stresses and shear rates to experimentally measured values of torque and rotational speed. Newtonian calibration fluids are used to determine the impeller constant, c, and non-Newtonian calibration fluids are used to calculate the shear rate constant, k. These constants are then used to aid in the determination of rheologic properties of a selected non-Newtonian fluid, such as wet grains. 

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 ... 

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. 

All of these rheological expressions (equations 13.16, 13.17, and 13.18) can be used to analyze the flow under the doctor blade in tape casting. Using the momentum balance equation 13.14 and one of the preceding equations for the shear stress, the differential equation which governs the velocity V,., can be determined. For Newtonian fluids, the solution is given by... 

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. 

The stress in viscoelastic liquids at steady-state conditions is defined, in simple shear flow, by the shear rate and two normal stress differences. Chapter 13 reviews the evolution of both the normal stress differences and the viscosity with increasing shear rate for different geometries. Semiquantitative approaches are used in which the critical shear rate at which the viscosity starts to drop in non-Newtonian fluids is estimated. The effects of shear rate, concentration, and temperature on die swell are qualitatively analyzed, and some basic aspects of the elongational flow are discussed. This process is useful to understand, at least qualitatively, the rheological fundamentals of polymer processing. 

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