Generalized Newtonian constitutive equations

Material parameters defined by Equations (1.11) and (1.12) arise from anisotropy (i.e. direction dependency) of the microstructure of long-chain polymers subjected to liigh shear deformations. Generalized Newtonian constitutive equations cannot predict any normal stress acting along the direction perpendicular to the shearing surface in a viscometric flow. Thus the primary and secondary normal stress coefficients are only used in conjunction with viscoelastic constitutive models. 

The simplest generalized Newtonian constitutive equation is the power law model that assumes the viscosity has the following dependence on shear rate,... 

Using the generalized Newtonian constitutive equation, the deviatoric stress tensor is defined as... 

Equation 2.8-1 holds only for simple shearing flow, namely, when there is one velocity component changing in one (normal) spatial direction. The most general Newtonian constitutive equation that we can write for an arbitrary flow field takes the form ... 

Similar generalizations to multidimensional flow are necessary for non-Newtonian constitutive equations. 

All the non-Newtonian constitutive equations just given are simplifications of the most general time-independent constitutive equation for isotropic, incompressible non-Newtonian fluids that do not exhibit elasticity [4,5],... 

In this section, we combine the Cauchy equation and the Newtonian constitutive equation to obtain the Navier-Stokes equation of motion. First, however, we briefly reconsider the notion of pressure in a general, Newtonian fluid. 

Newtonian constitutive equation, (2 80), that the normal component of the surface force or stress acting on a fluid element at a point will generally have different values depending on the orientation of the surface. Nevertheless, it is often useful to have available a scalar quantity for a moving fluid that is analogous to static pressure in the sense that it is a measure of the local intensity of squeezing of a fluid element at the point of interest. Thus it is common practice to introduce a mechanical definition of pressure in a moving fluid as... 

The definition (2 85) is a purely mechanical definition of pressure for a moving fluid, and nothing is implied directly of the connection for a moving fluid between p and the ordinary static or thermodynamic pressure p. Although the connection between p and p can always be stated once the constitutive equation for T is given, one would not necessarily expect the relationship to be simple for all fluids because thermodynamics refers to equilibrium conditions, whereas the elements of a fluid in motion are clearly not in thermodynamic equilibrium. Applying the definition (2-85) to the general Newtonian constitutive model, (2-80), we find... 

The Newtonian constitutive equation is the simplest equation we can use for viscous liquids. It (and the inviscid fluid, which has negligible viscosity) is the basis of all of fluid mechanics. When faced with a new liquid flow problem, we should try the Newtonian model first. Any other will be more difficult. In general, the Newtonian constitutive equation accurately describes the rheological behavior of low molecular weight liquids and even high polymers at very slow rates of deformation. However, as we saw in the introduction to this chapter (Figures 2.1.2 and 2.1.3) viscosity can be a strong function of the rate of deformation for polymeric liquids, emulsions, and concentrated suspensions. 

The use of the shell balance cannot accommodate all the flows we find in polymer processing. In the next section we summarize the isothermal equations of change plus generalize the Newtonian and non-Newtonian constitutive equations to three dimensions. 

All of the described differential viscoelastic constitutive equations are implicit relations between the extra stress and the rate of deformation tensors. Therefore, unlike the generalized Newtonian flows, these equations cannot be used to eliminate the extra stress in the equation of motion and should be solved simultaneously with the governing flow equations. 

As already discussed, in general, polymer flow models consist of the equations of continuity, motion, constitutive and energy. The constitutive equation in generalized Newtonian models is incorporated into the equation of motion and only in the modelling of viscoelastic flows is a separate scheme for its solution reqixired. 

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. 

In generalized Newtonian fluids, before derivation of the final set of the working equations, the extra stress in the expanded equations should be replaced using the components of the rate of strain tensor (note that the viscosity should also be normalized as fj = rj/p). In contrast, in the modelling of viscoelastic fluids, stress components are found at a separate step through the solution of a constitutive equation. This allows the development of a robust Taylor Galerkin/ U-V-P scheme on the basis of the described procedure in which the stress components are all found at time level n. The final working equation of this scheme can be expressed as... 

Note 2 For a non-Newtonian liquid (see note 3), a form of the general constitutive equation which may be used is... 

We are now left to deal with the constitutive equation. For a generalized Newtonian fluid, we can write... 

Symmetry for the velocity profile will set <7 = 0. With the generalized Newtonian fluid constitutive equation, we get that... 

In this case, p is an arbitrary constant, chosen as the zero shear rate viscosity. The expression for the non-Newtonian viscosity is a constitutive equation for a generalized Newtonian fluid, like the power law or Ostwald-de-Waele model [6]... 

In the previous section we discussed the nature and some properties of the stress tensor t and the rate of strain tensor y. They are related to each other via a constitutive equation, namely, a generally empirical relationship between the two entities, which depends on the nature and constitution of the fluid being deformed. Clearly, imposing a given stress field on a body of water, on the one hand, and a body of molasses, on the other hand, will yield different rates of strain. The simplest form that these equations assume, as pointed out earlier, is a linear relationship representing a very important class of fluids called Newtonian fluids. 

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. 

Observing that the Newtonian structure is never observed in turbulent shear flows, but that persistently strained flows apparently develop an equilibrium structure, Bradshaw (B3) prefers to relate the Reynolds stresses directly to q A generalization of his constitutive equation is... 

In order to use these general momentum conservation equations to calculate the velocity field, it is necessary to express viscous stress terms in terms of the velocity field. The equations which relate the stress tensor to the motion of the continuous fluid are called constitutive equations or rheological equations of state. Although the governing momentum conservation equations are valid for all fluids, the constitutive equations, in general, vary from one fluid material to another and possibly also from one type of flow to another. Fluids, which follow Newton s law of viscosity (although it is referred to as a law, it is just an empirical proposition) are called Newtonian fluids. For such fluids, the viscous stress at a point is linearly dependent on the rates of strain (deformation) of the fluid. With this assumption, a general deformation law which relates stress tensor and velocity components can be written ... 

In fact, Equation 5.281 describes an interface as a two-dimensional Newtonian fluid. On the other hand, a number of non-Newtonian interfacial rheological models have been described in the literature. Tambe and Sharma modeled the hydrodynamics of thin liquid films bounded by viscoelastic interfaces, which obey a generalized Maxwell model for the interfacial stress tensor. These authors also presented a constitutive equation to describe the rheological properties of fluid interfaces containing colloidal particles. A new constitutive equation for the total stress was proposed by Horozov et al. ° and Danov et al. who applied a local approach to the interfacial dilatation of adsorption layers. 

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