Constitutive equations generalized Newtonian fluid

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... 

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 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. 

The choice of constitutive equations depends on the particular problems investigated. If the flow phenomena are dominated by the shear-rate dependent viscosity, it makes sense to use inelastic, or generalized Newtonian fluids, for which the extra stress tensor is proportional to the rate of deformation in the form... 

The time-temperature superposition principle can be incorporated into isothermal constitutive equations (including the generalized Newtonian fluid model) to solve non-isothermal flow problems. If we define... 

When the viscosity is a function of shear rate, then the relationship between shear stress and shear rate is given by equation (2.9). Since its form is similar to equation (2.36) except for Ae shear rate dependent viscosity, the equation is said to represent a Generalized Newtonian fluid. In such a fluid, the presence of normal stresses defined by equations (2.10) and (2.11) is considered to be negligible for a specific flow situation. In effect, equation (2.5b) represents the constitutive equation for a Generalized Newtonian fluid. The hypothesis of a Generalized Newtonian fluid differs from the simple Newtonian case by the assumption that the functional relationship between the stress tensor and the kinematic variable need not be only linear. It holds, however, the suggestion that only the kinematic variable of the first order can influence the state of stress in the fluid and no attempt is to be made to describe the normal stresses in it. 

Although there have been numerous numerical simulations of EHL contacts lubricated with non-Newtonian liquids, few have assumed for the lubricant shear response the ordinary shear-thinning that can be observed in rheometers. One of the authors has developed a numerical scheme [12] for the calculation of central film thickness that can utilize any generalized Newtonian fluid model for the nonlinear shear response and any general pressure-viscosity response. It is an extremely simple Grubin style inlet zone analysis that, instead of utilizing the Reynolds equation, integrates the constitutive relation across the thickness of the film. The pressure boundary condition that we use for the Hertz boundary has been criticized [13] however, the film thickness result is... 

The simplest non-Newtonian constitutive equation is the so-called generalized Newtonian fluid ... 

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. 

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 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. 

Fluids for which this constitutive equation is an adequate model are known as Newtonian fluids. We have shown that the Newtonian fluid model is the most general form that is linear and instantaneous in E and isotropic. If the fluid is also incompressible,... 

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... 

Given the apparent arbitrariness of the assumptions in a purely continuum-mechanics-based theory and the desire to obtain results that apply to at least some real fluids, there has been a historical tendency to either relax the Newtonian fluid assumptions one at a time (for example, to seek a constitutive equation that allows quadratic as well as linear dependence on strain rate, but to retain the other assumptions) or to make assumptions of such generality that they must apply to some real materials (for example, we might suppose that stress is a functional over past times of the strain rate, but without specifying any particular form). The former approach tends to produce very specific and reasonable-appearing constitutive models that, unfortunately, do not appear to correspond to any real fluids. The best-known example is the so-called Stokesian fluid. If it is assumed that the stress is a nonlinear function of the strain rate E, but otherwise satisfies the Newtonian fluid assumptions of isotropy and dependence on E only at the same point and at the same moment in time, it can be shown (see, e.g., Leigh29) that the most general form allowed for the constitutive model is... 

Newtonian fluids are a subclass of purely viscous fluids. Purely viscous nonnewtonian fluids can be divided into two categories (1) shear-thinning fluids, and (2) shear-thickening fluids. Such fluids can be described by a constitutive equation of the general form... 

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