Extra stress

A constitutive equation is a relation between the extra stress (t) and the rate of deformation that a fluid experiences as it flows. Therefore, theoretically, the constitutive equation of a fluid characterises its macroscopic deformation behaviour under different flow conditions. It is reasonable to assume that the macroscopic behaviour of a fluid mainly depends on its microscopic structure. However, it is extremely difficult, if not impossible, to establish exact quantitative... 

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). 

Common experimental evidence shows that the viscosity of polymers varies as they flow. Under certain conditions however, elastic effects in a polymeric flow can be neglected. In these situations the extra stress is again expressed, explicitly, in terms of the rate of deformation as... 

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. 

Normally, the extra stress in the equation of motion is substituted in terms of velocity gradients and hence this equation includes second order derivatives of... 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

The elastic part of the extra stress for upper-eonvected Maxwell (UCM) and Oldroyd-B fluids is written as (see Chapter 1)... 

After the substitution of Cauchy stress via Equation (3.20) and the viscous part of the extra stress in terms of rate of defonnation, the equation of motion is written as... 

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... 

Figure 3.1 Subdivided element for the discretization of viscoelastic extra-stress...

The inverse of the Cauchy-Green tensor, Cf, is called the Finger strain tensor. Physically the single-integral constitutive models define the viscoelastic extra stress Tv for a fluid particle as a time integral of the defonnation history, i.e. 

Therefore the viscoelastic extra stress acting on a fluid particle is found via an integral in terms of velocities and velocity gradients evalua ted upstream along the streamline passing through its current position. This expression is used by Papanastasiou et al. (1987) to develop a finite element scheme for viscoelastic flow modelling. 

The (CEF) model (see Chapter 1) provides a simple means for obtaining useful results for steady-state viscometric flow of polymeric fluids (Tanner, 1985). In this approach the extra stress in the equation of motion is replaced by explicit relationships in terms of rate of strain components. For example, assuming a zero second normal stress difference for veiy slow flow regimes such relationships arc written as (Mitsoulis et at., 1985)... 

The extra stress is proportional to the derivatives of velocity components and consequently the order of velocity derivatives in terms arising from... 

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... 

A similar approximation should be applied to the components of the equation of motion and the significant terms (with respect to ) consistent with the expanded constitutive equation identified. This analy.sis shows that only FI and A appear in the zero-order terms and hence should be evaluated up to the second order. Furthermore, all of the remaining terms in Equation (5.29), except for S, appear only in second-order terms of the approximate equations of motion and only their leading zero-order terms need to be evaluated to preserve the consistency of the governing equations. The term E, which only appears in the higlier-order terms of the expanded equations of motion, can be evaluated approximately using only the viscous terms. Therefore the final set of the extra stress components used in conjunction with the components of the equation of motion are... 

The fermentation in these big vessels (max 3—5 brews) is normally performed pressureless, but several investigations have been made on fermentation under pressure and using somewhat higher temperatures in order to reduce the time needed still further. Some brewers agree with the statement that no yeast yet known can tolerate this extra stress and at the same time ferment a beet of equivalent excellent quaUty, but new selected yeast strains (genetic engineering) in the future may be better suited to these conditions. 

To see what is going on physically, it is easier to return to our first condition. At low stress, if we make a little neck, the material in the neck will work-harden and will be able to carry the extra stress it has to stand because of its smaller area load will therefore be continuous, and the material will be stable. At high stress, the rate of workhardening is less as the true stress-true strain curve shows i.e. the slope of the o/e curve is less. Eventually, we reach a point at which, when we make a neck, the workhardening is only just enough to stand the extra stress. This is the point of necking, with... 

At still higher true stress, do/de, the rate of work-hardening decreases further, becoming insufficient to maintain stability - the extra stress in the neck can no longer be accommodated by the work-hardening produced by making the neck, and the neck grows faster and faster, until final fracture takes place. 

The extra stress terms result from six additional stresses, three normal stresses, and three shear stresses ... 

In equation 1.94, (Tyx)v is the viscous shear stress due to the mean velocity gradient dvjdy and pv yv x is the extra shear stress due to the velocity fluctuations v x and v y. These extra stress components arising from the velocity fluctuations are known as Reynolds stresses. (Note that if the positive sign convention for stresses were used, the sign of the Reynolds stress would be negative in equation 1.94.)... 

In ISO 3384, the specified accuracy of force measurement is 1% and the compression must be maintained to within 0.01 mm. If a small extra stress or strain is added when force is measured, this must be less than IN or 0.05 mm. The compression plates must be flat to within 0.01mm with surface finish not worse than 0.4 pm Ra and, for ring test pieces, have a central hole to allow the circulation of liquid. The stiffness of the plates must be such that they bend by less than 0.01 mm under load. 

It has also been suggested that flow might occur at lower stresses than those predicted above by movement of material within the individual layers (Chu and Barnett, 1995). This has been observed in pearlitic structures made up of alternating layers of ferrite and cementite, and observations in other multilayer systems suggest that that deformation might occur in this way (Gil-Sevillano, 1979). Two cases have been identified the first where only the movement of a pre-existing dislocation loop is required, the second where the activation of a dislocation source within the layer is needed. Gil-Sevillano (1979) showed that the extra stress, Atm, required to move a dislocation half-loop in a layer of width 1 is... 

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