Newtonian fluids shear rate/stress

Fig. 12.15 The ratio of entrance pressure drop to shear stress at the capillary wall versus Newtonian wall shear rate, T. . PP , PS O, LDPE +, HDPE , 2.5% polyisohutylene (PIB) in mineral oil x, 10% PIB in decalin A, NBS-OB oil. [Reprinted by permission from J. L. White, Critique on Flow Patterns in Polymer Fluids at the Entrance of a Die and Instabilities Leading to Extrudate Distortion, Appl. Polym. Symp., No. 20, 155 (1973).]...

Figure 7.30. Generic rheological behaviour of non-Newtonian fluids - shear stress versus shear rate...

In general, fluid velocity is given by the Navier-Stokes and continuity equations. For fluids that are Newtonian (shear stress linearly related to fluid shear rate) and incompressible, the Navier-Stokes equation is written as... 

Note that there are many (complex) fluids that do not exhibit a Newtonian plateau at low shear rate (stress) and whose shear viscosity function feeds the controversy on the existence of a yield stress. As noted by Barnes, when the flow is so slow than ages are necessary to detect it, at least one could consider that the yield stress is an engineering reality (H. Barnes. The yield stress—a review or "jtotvra pei" —everything flows /. Non-Newtonian Fluid Mech., 81,133-178,1999.)... 

Theoretically the apparent viscosity of generalized Newtonian fluids can be found using a simple shear flow (i.e. steady state, one-dimensional, constant shear stress). The rate of deformation tensor in a simple shear flow is given as... 

For some materials the linear constitutive relation of Newtonian fluids is not accurate. Either stress depends on strain in a more complex way, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known collectively as non-Newtonian. Many different types of behavior have been observed, ranging from fluids for which the viscosity in the Navier-Stokes equation is a simple function of the shear rate to the so-called viscoelastic fluids, for which the constitutive equation is so different that the normal stresses can cause the fluid to flow in a manner opposite to that predicted for a Newtonian fluid. 

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 Williamson equation is useful for modeling shear-thinning fluids over a wide range of shear rates (15). It makes provision for limiting low and high shear Newtonian viscosity behavior (eq. 3), where T is the absolute value of the shear stress and is the shear stress at which the viscosity is the mean of the viscosity limits TIq and, ie, at r = -H... 

Capillary viscometers are useful for measuring precise viscosities of a large number of fluids, ranging from dilute polymer solutions to polymer melts. Shear rates vary widely and depend on the instmments and the Hquid being studied. The shear rate at the capillary wall for a Newtonian fluid may be calculated from equation 18, where Q is the volumetric flow rate and r the radius of the capillary the shear stress at the wall is = r Ap/2L. 

Gla.ss Ca.pilla.ry Viscometers. The glass capillary viscometer is widely used to measure the viscosity of Newtonian fluids. The driving force is usually the hydrostatic head of the test Hquid. Kinematic viscosity is measured directly, and most of the viscometers are limited to low viscosity fluids, ca 0.4—16,000 mm /s. However, external pressure can be appHed to many glass viscometers to increase the range of measurement and enable the study of non-Newtonian behavior. Glass capillary viscometers are low shear stress instmments 1—15 Pa or 10—150 dyn/cm if operated by gravity only. The rate of shear can be as high as 20,000 based on a 200—800 s efflux time. 

In most rotational viscometers the rate of shear varies with the distance from a wall or the axis of rotation. However, in a cone—plate viscometer the rate of shear across the conical gap is essentially constant because the linear velocity and the gap between the cone and the plate both increase with increasing distance from the axis. No tedious correction calculations are required for non-Newtonian fluids. The relevant equations for viscosity, shear stress, and shear rate at small angles a of Newtonian fluids are equations 29, 30, and 31, respectively, where M is the torque, R the radius of the cone, v the linear velocity, and rthe distance from the axis. 

All fluids for which the viscosity varies with shear rate are non-Newtonian fluids. For uou-Newtouiau fluids the viscosity, defined as the ratio of shear stress to shear rate, is often called the apparent viscosity to emphasize the distiuc tiou from Newtonian behavior. Purely viscous, time-independent fluids, for which the apparent viscosity may be expressed as a function of shear rate, are called generalized Newtonian fluids. 

In a perfectly viscous (Newtonian) fluid the shear stress, t is directly proportional to the rate of strain (dy/dt or y) and the relationship may be written as... 

Now for a Newtonian fluid, the shear stress, Xy, is related to the viscosity, t], and the shear rate, y, by the equation... 

In a fluid under stress, the ratio of the shear stress, r. to the rate of strain, y, is called the shear viscosity, rj, and is analogous to the modulus of a solid. In an ideal (Newtonian) fluid the viscosity is a material constant. However, for plastics the viscosity varies depending on the stress, strain rate, temperature etc. A typical relationship between shear stress and shear rate for a plastic is shown in Fig. 5.1. 

Since it is recognised that the fluid is Non-Newtonian, this is often referred to as the apparent shear rate to differentiate it from the true shear rate. If the pressure drop, P, across the die is also measured then the shear stress, r, may be calculated from... 

For Newtonian fluids the dynamic viscosity is constant (Equation 2-57), for power-law fluids the dynamic viscosity varies with shear rate (Equation 2-58), and for Bingham plastic fluids flow occurs only after some minimum shear stress, called the yield stress, is imposed (Equation 2-59). 

Figure 4-185. Shear stress-shear rate diagram, (a) Newtonian fluid, (b) Bingham plastic fluid, (c) Power iaw fiuid. (d) Herschei-Buckiey fiuid.

A Newtonian fluid is one in which, provided that the temperature and pressure remain constant, the shear rate increases linearly with shear stress over a wide range of shear rates. As the shear stress tends to retard the fluid near the centre of the pipe and accelerate the slow moving fluid towards the walls, at any radius within the pipe it is acting simultaneously in a negative direction on the fast moving fluid and in the positive direction on the slow moving fluid. In strict terms equation 3.3 should be written with the incorporation... 

In the previous sections of this chapter, the calculation of frictional losses associated with the flow of simple Newtonian fluids has been discussed. A Newtonian fluid at a given temperature and pressure has a constant viscosity /r which does not depend on the shear rate and, for streamline (laminar) flow, is equal to the ratio of the shear stress (R-,) to the shear rate (d t/dy) as shown in equation 3.4, or ... 

Many fluids, including some that are encountered very widely both industrially and domestically, exhibit non-Newtonian behaviour and their apparent viscosities may depend on the rate at which they are sheared and on their previous shear history. At any position and time in the fluid, the apparent viscosity pa which is defined as the ratio of the shear stress to the shear rate at that point is given by ... 

A further important property which may be shown by a non-Newtonian fluid is elasticity-which causes the fluid to try to regain its former condition as soon as the stress is removed. Again, the material is showing some of the characteristics of both a solid and a liquid. An ideal (Newtonian) liquid is one in which the stress is proportional to the rate of shear (or rate of strain). On the other hand, for an ideal solid (obeying Hooke s Law) the stress is proportional to the strain. A fluid showing elastic behaviour is termed viscoelastic or elastoviseous. 

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. 

In this section, consideration will be given to the equilibrium relationships between shear stress and shear rate for fluids exhibiting non-Newtonian behaviour. Whenever the shear stress or the shear rate is altered, the fluid will gradually move towards its new equilibrium state and, for the present, the period of adjustment between the two equilibrium states will be ignored. 

Pigure 3.24. Shear stress-shear rate behaviour of Newtonian and non-Newtonian fluids plotted using linear... 

The relation between shear stress and shear rate for the Newtonian fluid is defined by a single parameter /z, the viscosity of the fluid. No single parameter model will describe non-Newtonian behaviour and models involving two or even more parameters only approximate to the characteristics of real fluids, and can be used only over a limited range of shear rates. 

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