Newtonian constitutive equation

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... 

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

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 ... 

Equations 2.8-2 and 2.8-3 are coordinate-independent compact tensorial forms of the Newtonian constitutive equation. In any particular coordinate system these equations break up into nine (six independent) scalar equations. Table 2.3 lists these equations in rectangular, cylindrical and spherical coordinate systems. 

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... 

So far, we have simply stated the Cauchy equation of motion and the Newtonian constitutive equations as a set of nine independent equations involving u, T, and p. It is evident in this case, however, that the constitutive equation, (2-80), for the stress [or equivalently (2-86)] can be substituted directly into the Cauchy equation to provide a set of equations that involve only u and p (orp). These combined equations take the form... 

Finally, it was stated previously that fluids that satisfy the Newtonian constitutive equation for the stress are often also well approximated by the Fourier constitutive equation, (2-67), for the heat flux vector. Combining (2-67) with the thermal energy, (2-52), we obtain. 

The above viscosity relation does not yet take into account the non-Newtonian feature of the resist solution. When the rotational speed is high, the fluid is sheared to a large extent, inducing appreciable shear thinning of the material. Hence, approximation of the fluid behavior by a Newtonian constitutive equation leads to inaccurate predictions. The Newtonian model not only predicts too thick a film, but also gives too weak a film thickness dependence on spinner speed, as will be shown below. 

These observations pose an interesting question because the results reported in Refs. (Santamaria Holek, 2005 2009 2001) were obtained under the assumption of local equilibrium in phase space, that is, at the mesoscale. It seems that for systems far from equilibrium, as those reported in (Sarman, 1992), the validity of the fundamental h5q)othesis of linear nonequilibrium thermodynamics can be assumed at the mesoscale. After a reduction of the description to the physical space, this non-Newtonian dependedence of the transport coefficients on the shear rate appears. This point will be discussed more thoroughly in the following sections when analyzing the formulation of non-Newtonian constitutive equations. 

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 film-blowing process is used industrially to manufacture plastic films that are biaxially oriented. Many attempts have been made to predict and model this complex but important process, which continues to mystify rheologists and polymer processing engineers worldwide. A constitutive equation, able to predict well the polymer melt in all forms of deformation, is required to model the process, together with the standard conservation equations of continuity, momentum, and energy. Pearson and Petrie [125,126] were the first to predict the forces within the blown film by the use of the thin-shell approximation, force balances, and the Newtonian constitutive equation. The use of the thin-shell approximation and force balances is standard in any attempt to model the film-blowing process, and it has been used in the vast majority of subsequent studies. 

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. 

The dimensions of the mold are H= 30.48 cm and L = 20.0 cm. The applied pressure differential is 6.9 X 10 Pa. Calculate the wall thickness distribution and the time to fill the mold using first the Newtonian constitutive equation and then the PTT model. Compare the predictions of the two models. 

However, for non-Newtonian fluids, even though continuity equation and the equation of motion written as Equation 11.2 remain valid, the Newtonian constitutive equation is not correct and a different constitutive equation is needed. To find constitutive equations, experiments are performed on materials using standard flows described above. The functions of kinematic parameters that characterize the rheological behavior of fluids are called rheological material functions. Standardized material functions are shown in Table 11.1 [2-4]. 

The only way to test the validity of a constitutive model is by using experiments and measnring material functions. The ways in which structured fluids fail to follow the Newtonian constitutive equation vary enormously. Two typical examples are polymeric liquids and dispersed systems. 

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