Newtonian stress

Problems of forced convection diffusion in non-Newtonian flow have to this author s knowledge not yet been attacked. The equations needed for solving such problems are given in this article. The equation of motion in terms of the stress tensor [Eq. (25)] can be used to describe non-Newtonian flow provided that a suitable form for the stress tensor is used examples of two non-Newtonian stress tensors are given in Eqs. (28a) and (28b). 

Additional operations may be found on pp. xxiii to xxvi of Ref. (Hll). Most of these relations may be found in cylindrical, spherical, and other coordinate systems in standard reference works. Several of them do not, however, seem to be tabulated for handy reference these operations are given here in cylindrical and spherical coordinates. Expressions for the Newtonian stress tensor in terms of the velocity gradients and the coefficient of viscosity may be found in Ref. (G7, pp. 103-105). 

Baldwin PR, Helfand E (1990) Dilute solutions in steady shear flow Non-Newtonian stress. Phys Rev A 41 6772-6785... 

Newtonian liquids show a stress-strain relationship represented by a straight line (Figure 8.6a). In Figure 8.6b there are viscosities of two Newtonian fluids at different shear rates. These two materials, represented by 1 and 2, may have the same apparent viscosity when there is no direct proportionality between shear stress and the rate of shear. The flow behavior for non-Newtonian stress-strain relationship... 

The two expansion coefficients ijs and rjv are called the shear and bulk viscosities, respectively. The shear viscosity is all that is required to describe our gedanken experiment. The bulk viscosity describes the viscous or dissipative part of the response to a compression. This linear constitutive relation is called the Newtonian stress tensor. A fluid correctly described by this form is called a Newtonian fluid.16... 

Viscoelastic effects alter the steady drop shape from being ellipsoidal to drop shapes with more blunt ends. The results reflect a balance between the direct tensile stress contribution of the viscoelastic fluid to the normal stress balance and modifications of the viscous (i.e., Newtonian) stress and pressure due to viscoelastic changes in the flow Hsu and Leal 2009... 

A modified k - e model for determination of turbulent non-newtonian stresses has been used to provide velocity profiles in Poiseuille-Couette flow. Generalized functions are found for the non-newtonian turbulent viscosity coefficients and G > which lead us to formulation of the generalized... 

Equations (195) and (197) form a linear system for and r 7- Solving these equations as before (139)-(142), and inserting the result into (194), we obtain a Newtonian stress with unchanged values for the viscosities by choosing such that... 

Most authors do have an extra f term inside the stress divergence and give a Newtonian stress expression in terms of the macroscopic velocities. If we rigorously smooth the microscopic Newtonian stress for the case of constant density and constant viscosity, we get... 

Flow behaviour of polymer melts is still difficult to predict in detail. Here, we only mention two aspects. The viscosity of a polymer melt decreases with increasing shear rate. This phenomenon is called shear thinning [48]. Another particularity of the flow of non-Newtonian liquids is the appearance of stress nonnal to the shear direction [48]. This type of stress is responsible for the expansion of a polymer melt at the exit of a tube that it was forced tlirough. Shear thinning and nonnal stress are both due to the change of the chain confonnation under large shear. On the one hand, the compressed coil cross section leads to a smaller viscosity. On the other hand, when the stress is released, as for example at the exit of a tube, the coils fold back to their isotropic confonnation and, thus, give rise to the lateral expansion of the melt. 

Colloidal dispersions often display non-Newtonian behaviour, where the proportionality in equation (02.6.2) does not hold. This is particularly important for concentrated dispersions, which tend to be used in practice. Equation (02.6.2) can be used to define an apparent viscosity, happ, at a given shear rate. If q pp decreases witli increasing shear rate, tire dispersion is called shear tliinning (pseudoplastic) if it increases, tliis is known as shear tliickening (dilatant). The latter behaviour is typical of concentrated suspensions. If a finite shear stress has to be applied before tire suspension begins to flow, tliis is known as tire yield stress. The apparent viscosity may also change as a function of time, upon application of a fixed shear rate, related to tire fonnation or breakup of particle networks. Thixotropic dispersions show a decrease in q, pp with time, whereas an increase witli time is called rheopexy. 

Non-Newtonian flow processes play a key role in many types of polymer engineering operations. Hence, formulation of mathematical models for these processes can be based on the equations of non-Newtonian fluid mechanics. The general equations of non-Newtonian fluid mechanics provide expressions in terms of velocity, pressure, stress, rate of strain and temperature in a flow domain. These equations are derived on the basis of physical laws and... 

In the simplest case of Newtonian fluids (linear Stokesian fluids) the extra stress tensor is expressed, using a constant fluid viscosity p, as... 

Equations (1.6) and (1.7) are used to formulate explicit relationships between the extra stress components and the velocity gradients. Using these relationships the extra stress, t, can be eliminated from the governing equations. This is the basis for the derivation of the well-known Navier-Stokes equations which represent the Newtonian flow (Aris, 1989). 

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... 

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. 

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). 

A frequently used example of Oldroyd-type constitutive equations is the Oldroyd-B model. The Oldroyd-B model can be thought of as a description of the constitutive behaviour of a fluid made by the dissolution of a (UCM) fluid in a Newtonian solvent . Here, the parameter A, called the retardation time is de.fined as A = A (r s/(ri + s), where 7]s is the viscosity of the solvent. Hence the extra stress tensor in the Oldroyd-B model is made up of Maxwell and solvent contributions. The Oldroyd-B constitutive equation is written as... 

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. 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

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. 

For simplicity, we define T - and T (A iooTe/At). As explained by Luo and Tanner (1989), the decoupled method requires a suitable variable transfonna-tion in the governing equations (3.20) and (3.21). This is to ensure that the discrete momentum equations always contain the real viscous term required to recover the Newtonian velocity-pressure formulation when Ws approaches zero. This is achieved by decomposing the extra stress T 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... 

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

Note that convected derivatives of the stress (and rate of strain) tensors appearing in the rheological relationships derived for non-Newtonian fluids will have different forms depending on whether covariant or contravariant components of these tensors are used. For example, the convected time derivatives of covariant and contravariant stress tensors are expressed as... 

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