Fluids, non-Newtonian

We have defined a Newtonian fluid as one that has a constant viscosity at a given temperature and pressure irrespective of the way in which it is flowing or being moved. We can also say that the ratio of shear stress to shear rate in such a fluid would have a constant value at constant temperature and pressure. Effectively, all gases approximate to this Newtonian type of flow behavior and most low-viscosity liquids are also Newtonian. That a fluid is either compressible (like a gas) 

To decide whether a fluid is Newtonian or non-Newtonian, we need to know how it behaves in response to shear. 

As discussed in Section 2.4, Newtonian fluids are those which follow Newton s law, Eq. (3.5-1). 

If a fluid does not follow Eq. (3.5-1), it is a non-Newtonian fluid. Then a plot of t versus —dv/dr is not linear through the origin for these fluids. Non-Newtonian fluids can be divided into two broad categories on the basis of their shear stress/shear rate behavior those whose shear stress is indejjendent of time or duration of shear (time-independent) and those whose shear,stress is dependent on time or duration of shear (time-dependent). In addition to unusual shear-stress behavior, some non-Newtonian fluids also exhibit elastic (rubberlike) behavior which is a function of time and results in 

Pseudoplastic fluids. The majority of non-Newtonian fluids are in this category and include polymer solutions or melts, greases, starch suspensions, mayonnaise, biological fluids, detergent slurries, dispersion media in certain pharmaceuticals, and paints. The shape of the flow curve is shown in Fig. 3.5-1, and it generally can be represented by a power-law equation (sometimes called Ostwald-deWaele equation). 

For a Newtonian fluid, = 1. Solutions showing dilatancy are some corn flour-sugar solutions, wet beach sand, starch in water, potassium silicate in water, and some solutions containing high concentrations of powder in water. 

Thixotropic fluids. These fluids exhibit a reversible decrease in shear stress with time at a constant rate of shear. This shear stress approaches a limiting value that depends on the shear rate. Examples include some polymer solutions, shortening, some food materials, and paints. The theory for time-dependent fluids at present is still not completely developed. 

In most practical applications, polymer fluids do not behave like ideal Newtonian fluids. The occurrence of non-ideal viscoelastic behaviors of shear flow is often associated with a dimensionless number, called the Weissenberg number We (Weissenberg 1947 Dealy 2010), 

According to the relatimiship between the shear stress r and the shear rate y, non-Newtonian fluids can be classified into the following conventional types, as illustrated in Fig. 7.4. Curve d represents the dilatant fluid, whose viscosity t] (i.e. the slope of the curve) increases with the increase of shear rate /. Such a 

If the shear rates are constants, the non-Newtonian fluids can also be classified according to their viscosity dependence on time. This classification has been widely applied to describe the rheological characteristics of coatings. For the development of deformation, the time evolution corresponds to the effect of the increase of shear rate. Three typical cases occur with the time evolution the thixotropic fluids exhibit the decrease of viscosity, corresponding to pseudo-plastic fluids the rheopectic fluids exhibit the increase of viscosity, corresponding to dilatant fluids while the viscoelastic fluids exhibit partial recovery of the deformation of pseudo-plastic fluids after the removal of the stress. Since polymers can perform a large scale of elastic deformation, this character appears extremely significant. 

De Waele and Ostwald proposed a unified power-law equation, as 

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 pi which does not depend on the shear rate and, for streamline (laminar) flow, is equal to the ratio of the shear stress Ry) to the shear rate (dux/dy) as shown in equation 3.4, or  

The modulus sign is used because shear stresses within a fluid act in both the positive arid negative senses. Gases and simple low molecular weight liquids are all Newtonian, and viscosity may be treated as constant in any flow problem unless there are significant variations of temperature or pressure. 

Many fluids, including some that are encountered very widely both industi ially 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 jXa which is defined as the ratio of the shear stress to the shear rate at that point is given by  

When the apparent viscosity is a function of the shear rate, the behaviour is said to be shear-dependent when it is a function of the duration of shearing at a particular rate, it is referred to as time-dependent. Any shear-dependent fluid must to some extent be time-dependent because, if the shear rate is suddenly changed, the apparent viscosity does not alter instantaneously, but gradually moves towards its new value. In many cases, however, the time-scale for the flow process may be sufficiently long for the effects of time-dependence to be negligible. 

Some materials have the characteristics of both solids and liquids. For instance, tooth paste behaves as a solid in the tube, but when the tube is squeezed the paste flows as a plug. The essential characteristic of such a material is that it will not flow until a certain critical shear stress, known as the yield stress is exceeded. Thus, it behaves as a solid at low shear stresses and as a fluid at high shear stress. It is a further example of a shear-thinning fluid, with an infinite apparent viscosity at stress values below the yield value, and a falling finite value as the stress is progressively increased beyond this point. 

Pseudoplastic liquids have a t — y plot that is concave downward. The simplest mathematical representation of such relations is a power law 

Since n is less than unity, the apparent viscosity decreases with the deformation rate. Examples of such materials are some polymeric solutions or melts such as rubbers, cellulose acetate and napalm suspensions such as paints, mayonnaise, paper pulp, or detergent slurries and dilute suspensions of inert solids. Pseudoplastic properties of wallpaper paste account for good spreading and adhesion, 

Dilutant liquids have rheological behavior essentially opposite those of pseudoplastics insofar as viscosity behavior is concerned. The r — y plots are concave upward and the power law applies 

Bingham plastics require a finite amount of shear stress before deformation begins, then the deformation rate is linear. Mathematically, 

Generalized Bingham or yield-power law fluids are represented by the equation 

The Bingham fluid has a linear shear stress-shear strain relationship, but a critical shear stress is required to initiate the flow. The shear stress-shear rate curve of a Bingham fluid does not pass through the origin, and it can be expressed as  

Theoretically, in shear-thickening and shear-thinning flows, there is not a characteristic viscosity since t] = T/y is not a constant. Therefore, the term apparent viscosity (t]J is introduced and it is deflned as the shear stress divided by shear rate  

To mathematically describe the flow behaviors of shear-thickening and shearthinning fluids, a power law equation can be used  

Based on the definition of viseosily, the power law equation ean be rearranged to obtain  

According to Equation 8.12, the log rf - log y curves of Newtonian, shear-thickening, and shear-thiiming fluids also are straight lines and the slopes are determined by the n-1 values. 

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Non-Newtonian flow processes play a key role in many types of polymer engineering operations. Hence, formulation of mathematical models for these processes can be based on the equations of non-Newtonian fluid mechanics. The general equations of non-Newtonian fluid mechanics provide expressions in terms of velocity, pressure, stress, rate of strain and temperature in a flow domain. These equations are derived on the basis of physical laws and... 

Numerous examples of polymer flow models based on generalized Newtonian behaviour are found in non-Newtonian fluid mechanics literature. Using experimental evidence the time-independent generalized Newtonian fluids are divided into three groups. These are Bingham plastics, pseudoplastic fluids and dilatant fluids. 

Herschel, W.H. and Bulkley, R., 1927. See Rudraiah, N, and Kaloni, P.N. 1990. Flow of non-Newtonian fluids. In Encyclopaedia of Fluid Mechanics, Vol. 9, Chapter 1, Gulf Publishers, Houston. 

Johnson, M. W. and Segalman, D., 1977. A model for viscoelastic fluid behaviour which allows non-affine deformation. J. Non-Newtonian Fluid Mech. 2, 255-270. 

Pearson,. I.R.A., 1994. Report on University of Wales Institute of Non-Newtonian Fluid Mechanics Mini Symposium on Continuum and Microstructural Modelling in Computational Rheology. /. Non-Newtonian Fluid Mech. 55, 203 -205. 

Townsend, P. and Webster, M. I- ., 1987. An algorithm for the three dimensional transient simulation of non-Newtonian fluid flow. In Pande, G. N. and Middleton, J. (eds). Transient Dynamic Analysis and Constitutive Laws for Engineering Materials Vul. 2, T12, Nijhoff-Holland, Swansea, pp. 1-11. 

Luo, X. L, and Tanner, R. L, 1989. A decoupled finite element streamline-upwind scheme for viscoelastic flow problems. J. Non-Newtonian Fluid Mech. 31, 143-162. 

Papaiiastasiou, T. C., Scriven, L. E. and Macoski, C. W., 1987. A finite element method for liquid with memory. J. Non-Newtonian Fluid Mech 22, 271-288. 

Using the described algorithm the flow domain inside the cone-and-plate viscometer is simulated. Tn Figure 5.17 the predicted velocity field in the (r, z) plane (secondary flow regime) established inside a bi-conical rheometer for a non-Newtonian fluid is shown. 

Hiebcr, C, A. and Shen, S.F., 1980. A finite element/finite difference simulation of the injection-moulding filling process. J. Non-Newtonian Fluid Mech. 7, 1-32. 

Olagunju, D.O. and Cook, L. P., 1993. Secondary flows in cone and plate flow of an Oldroyd-B fluid. J. Non-Newtonian Fluid Mech. 46, 29-47. 

Petera, J. and Nassehi, V., 1995. Use of the finite element modelling technique for the improvement of viscometry results obtained by cone-and-plate rheometers. J. Non-Newtonian Fluid Mech. 58, 1-24. 

Note that convected derivatives of the stress (and rate of strain) tensors appearing in the rheological relationships derived for non-Newtonian fluids will have different forms depending on whether covariant or contravariant components of these tensors are used. For example, the convected time derivatives of covariant and contravariant stress tensors are expressed as... 

External and internal loop air-lifts and bubble column reactors containing a range of coalescing and non-Newtonian fluids, have been studied (52,53). It was shown that there are distinct differences in the characteristics of external and internal loop reactors (54). Overall, in this type of equipment... 

Non-Newtonian Fluids Die Swell and Melt Fracture. Eor many fluids the Newtonian constitutive relation involving only a single, constant viscosity is inappHcable. Either stress depends in a more complex way on strain, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known coUectively as non-Newtonian and are usually subdivided further on the basis of behavior in simple shear flow. 

Pseudoplastic fluids are the most commonly encountered non-Newtonian fluids. Examples are polymeric solutions, some polymer melts, and suspensions of paper pulps. In simple shear flow, the constitutive relation for such fluids is... 

Heat Exchangers Using Non-Newtonian Fluids. Most fluids used in the chemical, pharmaceutical, food, and biomedical industries can be classified as non-Newtonian, ie, the viscosity varies with shear rate at a given temperature. In contrast, Newtonian fluids such as water, air, and glycerin have constant viscosities at a given temperature. Examples of non-Newtonian fluids include molten polymer, aqueous polymer solutions, slurries, coal—water mixture, tomato ketchup, soup, mayonnaise, purees, suspension of small particles, blood, etc. Because non-Newtonian fluids ate nonlinear in nature, these ate seldom amenable to analysis by classical mathematical techniques. 

A significant heat-transfer enhancement can be obtained when a nonckcular tube is used together with a non-Newtonian fluid. This heat-transfer enhancement is attributed to both the secondary flow at the corner of the nonckcular tube (23,24) and to the temperature-dependent non-Newtonian viscosity (25). Using an aqueous solution of polyacrjiamide the laminar heat transfer can be increased by about 300% in a rectangular duct over the value of water (23). 

A knowledge of the viscous and thermal properties of non-Newtonian fluids is essential before the results of the analyses can be used for practical design purposes. Because of the nonlinear nature, the prediction of these properties from kinetic theories is as of this writing in its infancy. Eor the purpose of design and performance calculations, physical properties of non-Newtonian fluids must be measured. 

For non-Newtonian fluids the correlations in Figure 35 can be used with generally acceptable accuracy when the process fluid viscosity is replaced by the apparent viscosity. For non-Newtonian fluids having power law behavior, the apparent viscosity can be obtained from shear rate estimated by... 

Occasionally, piping systems are designed to carry multiphase fluids (combinations of gases, Hquids, and soflds), or non-Newtonian fluids. Sizing piping for such systems is beyond the scope of this article. PubHcations covering multiphase flow (20) and non-Newtonian flow (21) are available. 

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