Non-Newtonian equations

Application of the weighted residual method to the solution of incompressible non-Newtonian equations of continuity and motion can be based on a variety of different schemes. Tn what follows general outlines and the formulation of the working equations of these schemes are explained. In these formulations Cauchy s equation of motion, which includes the extra stress derivatives (Equation (1.4)), is used to preseiwe the generality of the derivations. However, velocity and pressure are the only field unknowns which are obtainable from the solution of the equations of continuity and motion. The extra stress in Cauchy s equation of motion is either substituted in terms of velocity gradients or calculated via a viscoelastic constitutive equation in a separate step. 

Equation (47) has been derived previously by the rheologists from the so-ealled power law G = hr" which assumes a constant n at any chosen applied pressure (k being a constant). This derivation is not exact since n varies with the position of the fluid inside the capillary. It is unity at the center of the capillary where the flow is Newtonian and becomes greater than one toward the wall when the flow is non-Newtonian. Equation (47) can now be derived without this unrealistic assumption. 

In this chapter we will look at a number of situations where, if we know the (Newtonian) viscosity of a liquid, we are able to predict something about its flow in certain simple geometries. First we will look at flows where the viscosity alone is important, and then we will see what happens when we have to take the hquid s density into accoimt, so that secondary flows and eventually turbulence are seen in high-speed flows. Lastly, we will consider some flow situations where surface tension as well as viscosity is important. Learning to use these equations is a very useful precursor to looking at non-Newtonian equations. They can also be used to obtain an order of magnitude estimate of many comphcated flow situations. 

Colloidal dispersions often display non-Newtonian behaviour, where the proportionality in equation (02.6.2) does not hold. This is particularly important for concentrated dispersions, which tend to be used in practice. Equation (02.6.2) can be used to define an apparent viscosity, happ, at a given shear rate. If q pp decreases witli increasing shear rate, tire dispersion is called shear tliinning (pseudoplastic) if it increases, tliis is known as shear tliickening (dilatant). The latter behaviour is typical of concentrated suspensions. If a finite shear stress has to be applied before tire suspension begins to flow, tliis is known as tire yield stress. The apparent viscosity may also change as a function of time, upon application of a fixed shear rate, related to tire fonnation or breakup of particle networks. Thixotropic dispersions show a decrease in q, pp with time, whereas an increase witli time is called rheopexy. 

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... 

In this chapter the general equations of laminar, non-Newtonian, non-isothermal, incompressible flow, commonly used to model polymer processing operations, are presented. Throughout this chapter, for the simplicity of presentation, vector notations are used and all of the equations are given in a fixed (stationary or Eulerian) coordinate system. 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

Phan-Thien, N. and Tanner, R.T., 1977. A new constitutive equation derived from network theory, Non-Newtonian Fluid Mech. 2, 353-365. 

U-V-P schemes belong to the general category of mixed finite element techniques (Zienkiewicz and Taylor, 1994). In these techniques both velocity and pressure in the governing equations of incompressible flow are regarded as primitive variables and are discretized as unknowns. The method is named after its most commonly used two-dimensional Cartesian version in which U, V and P represent velocity components and pressure, respectively. To describe this scheme we consider the governing equations of incompressible non-Newtonian flow (Equations (1.1) and (1.4), Chapter 1) expressed as... 

Level of enforcement of the incompressibility condition depends on the magnitude of the penalty parameter. If this parameter is chosen to be excessively large then the working equations of the scheme will be dominated by the incompressibility constraint and may become singular. On the other hand, if the selected penalty parameter is too small then the mass conservation will not be assured. In non-Newtonian flow problems, where shear-dependent viscosity varies locally, to enforce the continuity at the right level it is necessary to maintain a balance between the viscosity and the penalty parameter. To achieve this the penalty parameter should be related to the viscosity as A = Xorj (Nakazawa et al, 1982) where Ao is a large dimensionless parameter and tj is the local viscosity. The recommended value for Ao in typical polymer flow problems is about 10. ... 

It is evident that application of Green s theorem cannot eliminate second-order derivatives of the shape functions in the set of working equations of the least-sc[uares scheme. Therefore, direct application of these equations should, in general, be in conjunction with C continuous Hermite elements (Petera and Nassehi, 1993 Petera and Pittman, 1994). However, various techniques are available that make the use of elements in these schemes possible. For example, Bell and Surana (1994) developed a method in which the flow model equations are cast into a set of auxiliary first-order differentia] equations. They used this approach to construct a least-sciuares scheme for non-Newtonian flow equations based on equal-order C° continuous, p-version hierarchical elements. 

Incorporation of viscosity variations in non-elastic generalized Newtonian flow models is based on using empirical rheological relationships such as the power law or Carreau equation, described in Chapter 1. In these relationships fluid viscosity is given as a function of shear rate and material parameters. Therefore in the application of finite element schemes to non-Newtonian flow, shear rate at the elemental level should be calculated and used to update the fluid viscosity. The shear rale is defined as the second invariant of the rate of deformation tensor as (Bird et at.., 1977)... 

For the straight line in Fig. 2.5 where m = 1.0, this equation expresses direct proportionality between and y, the condition of Newtonian behavior. In the non-Newtonian region where m < 1, Eq. (2.11) may describe the data over an order of magnitude or so. Next we consider the relationship between the constant K and viscosity. If Eq. (2.11) is solved for K and the resulting expression multiplied and divided by 7 ", we obtain... 

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. 

Polymer melts are frequendy non-Newtonian. In this case the earlier expression given for the shear rate at the capillary wall does not hold. A correction factor (3n + 1)/4n, called the Rabinowitsch correction, must be appHed in such a way that equation 21 appHes, where 7 is the tme shear rate at the wall and nis 2l power law factor (eq. 22) determined from the slope of a log—log plot of the tme shear stress at the wad, T, vs 7. For a Newtonian hquid, n = 1. A tme apparent viscosity, Tj, can be calculated from equation 23. 

In addition to non-Newtonian flow, the main correction necessary for concentric cylinder measurements is that on account of end effects. Because the inner cylinder is not infinitely long, there is drag on the ends as well as on the face of the cylinder. The correction appears as an addition, to the length, b. The correction is best deterrnined by measuring the angular velocity and torque at several values of b, that is, at various depths of immersion. The data are plotted as M/Q vs b, and extrapolation is made to a value of at M/H = 0. The quantity (/i + h ) is substituted for b in the various equations. 

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. 

Non-Newtonian Flow For isothermal laminar flow of time-independent non-Newtonian hquids, integration of the Cauchy momentum equations yields the fully developed velocity profile and flow rate-pressure drop relations. For the Bingham plastic flmd described by Eq. (6-3), in a pipe of diameter D and a pressure drop per unit length AP/L, the flow rate is given by... 

If the material is Non-Newtonian then the true shear rates should be equated /2n-t-l 6Qa pn4-l 4Q,... 

For non-Newtonian drilling fluids, the concept of an effective viscosity can be used to replace the dynamic viscosity in Equation 4-89. 

Since non-Newtonian flow is typical for polymer melts, the discussion of a filler s role must explicitly take into account this fundamental fact. Here, spoken above, the total flow curve includes the field of yield stress (the field of creeping flow at x < Y may not be taken into account in the majority of applications). Therefore the total equation for the dependence of efficient viscosity on concentration must take into account the indicated effects. 

According to the structure of this equation the quantity cp indicates the influence of the filler on yield stress, and t r on Newtonian (more exactly, quasi-Newtonian due to yield stress) viscosity. Both these dependences Y(cp) andr r(cp) were discussed above. Non-Newtonian behavior of the dispersion medium in (10) is reflected through characteristic time of relaxation X, i.e. in the absence of a filler the flow curve of a melt is described by the formula ... 

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