Newtonian fluids generalized

Theoretically the apparent viscosity of generalized Newtonian fluids can be found using a simple shear flow (i.e. steady state, one-dimensional, constant shear stress). The rate of deformation tensor in a simple shear flow is given as 

In addition to the apparent viscosity two other material parameters can be obtained using simple shear flow viscometry. These are primary and secondary nomial stress coefficients expressed, respectively, as 

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. 

Numerous examples of polymer flow models based on generalized Newtonian behaviour are found in non-Newtonian fluid mechanics literature. Using experimental evidence the time-independent generalized Newtonian fluids are divided into three groups. These are Bingham plastics, pseudoplastic fluids and dilatant fluids. 

Bingham plastics are fluids which remain rigid under the application of shear stresses less than a yield stress, Ty, but flow like a. simple Newtonian fluid once the applied shear exceeds this value. Different constitutive models representing this type of fluids were developed by Herschel and Bulkley (1926), Oldroyd (1947) and Casson (1959). 

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 difference between various models of the generalized Newtonian fluid is the expression used for the viscosity function fj y). 

A well-known relationship between the zero-shear-rate viscosity and molecular weight My, is that (Fox and Hory 1954) 

In comparing the Carreau and Cross models on a variety of commercial-grade polymer melts, Hieber and Chiang (1992) found that the Cross model provides a better overall fit for the shear-rate dependence. The Cross model has been widely used in injection molding modeling. 

The generahzed Newtonian fluid model has no memory, and no elasticity either it predicts a zero first normal stress difference Ni = Th — — 

---

Figure 1.2 Comparison of the rheological behaviour of Newtonian and typical generalized Newtonian fluids...

Typical rheograms representing the behaviour of various types of generalized Newtonian fluids are shown in Figure 1.2. 

MODELLING OF STEADY STATE STOKES FLOW OF A GENERALIZED NEWTONIAN FLUID... 

Similarly in the absence of body forces the Stokes flow equations for a generalized Newtonian fluid in a two-dimensional (r, 8) coordinate system are written as... 

Similarly the components of the equation of motion for an axisymmetric Stokes flow of a generalized Newtonian fluid are written as... 

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... 

The governing equations used in this case are identical to Equations (4.1) and (4.4) describing the creeping flow of an incompressible generalized Newtonian fluid. In the air-filled sections if the pressure exceeds a given threshold the equations should be switched to the following set describing a compressible flow... 

For a generalized Newtonian fluid the components of the extra stress and... 

For generalized Newtonian fluids the load vector (i.c. the right-hand side in Equation (5.31) is expressed as... 

All fluids for which the viscosity varies with shear rate are non-Newtonian fluids. For uou-Newtouiau fluids the viscosity, defined as the ratio of shear stress to shear rate, is often called the apparent viscosity to emphasize the distiuc tiou from Newtonian behavior. Purely viscous, time-independent fluids, for which the apparent viscosity may be expressed as a function of shear rate, are called generalized Newtonian fluids. 

The polymer behaves as a generalized Newtonian fluid, where viscosity is an arbitrary function of shear rate and temperature. 

These forms of the equation of motion are commonly called the Cauchy momentum equations. For generalized Newtonian fluids we can define the terms of the deviatoric stress tensor as a function of a generalized Newtonian viscosity, p, and the components of the rate of deformation tensor, as described in Table 5.3. 

Table 5.3 Stress Tensor Generalized Newtonian Fluid...

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]... 

The generalized Newtonian fluid models (GNF), which are widely used in polymer processing flow analysis, since they are capable of describing well the very strong shear rate dependence of melts. 

In solutions, the most important physical factors that influence the solubility of ingredients are type of fluid, mixing equipment, and mixing operations. Generalized Newtonian fluids are ideal fluids for which the ratio of the shear rate to the shear stress is constant at a particular time. Unfortunately, in practice, usually liquid dosage forms and their ingredients are non-Newtonian fluids in which the ratio of the shear rate to the shear stress varies. As a result, non-Newtonian fluids may not have a well-defined viscosity [32],... 

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. 

---