Generalized Newtonian constitutive models

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... 

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,... 

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... 

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... 

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. 

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... 

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]... 

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,... 

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... 

Our approach, based on an energetical description of turbulence and on a power law model for the constitutive equation of fluids, leads us to the formulation of a theory generalizing newtonian and laminar cases [9]. 

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. 

For purely viscous fluids, the rheological constitutive equation that relates the stresses x to the velocity gradients is the generalized Newtonian model [5,6,21] and is written as... 

The primary objective of modern science is to construct theories that faithfully describe the reality. Theories are nothing but models that are subject to perpetual experimental scrutiny and checks of internal consistency. When a particular theory is abandoned, it is for one of several reasons. Some theories, such as those of flogiston and caloric, are simply proven wrong. Others, such as the geocentric theory of the universe, are superseded by simpler descriptions of reality. Finally, some formalisms (such as the Newtonian mechanics) are found to possess only a limited validity or to constitute special cases of more general (unified) theories. 

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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