Oldroyd constitutive equation

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. 

Lumley, J. L. Applicability of Oldroyd constitutive equation to flow of dilute polymer solutions. Phys. Fluids 14 (1971) 2282. 

The Oldroyd-type differential constitutive equations for incompressible viscoelastic fluids can in general can be written as (Oldroyd, 1950)... 

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... 

Many industrially important fluids cannot be described in simple terms. Viscoelastic fluids are prominent offenders. These fluids exhibit memory, flowing when subjected to a stress, but recovering part of their deformation when the stress is removed. Polymer melts and flour dough are typical examples. Both the shear stresses and the normal stresses depend on the history of the fluid. Even the simplest constitutive equations are complex, as exemplified by the Oldroyd expression for shear stress at low shear rates ... 

Schmidt et al. (102) carried out a detailed experimental study of PET blow molding with a well-instrumented machine and compared the results with theoretical predictions using FEM and an Oldroyd B constitutive equation. They measured and calculated internal gas pressure, coupled it with the thermomechanical inflation and performed experiments and computations with free parison inflation. 

The three constant Oldroyd model is a nonlinear constitutive equation of the differential corrotational type, such as the Zaremba-Fromm-Dewitt (ZFD) fluid (Eq. 3.3-11). [For details, see R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Second Edition, Vol. 1, Wiley, New York, 1987, Table 7.3-2.]... 

Equations (9.6) and (9.7) make up the simplest set of constitutive equations for dilute polymer solutions, which, after excluding the internal variables j, can be written in the form of a differential equation that has the form of the two-constant contra-variant equation investigated by Oldroyd (1950) (Section 8.6). 

The constitutive equations are the Oldroyd-B model and a modified Oldroyd-B model in which the viscosity depends on the rate of strain. In [79], Laure et al. study the spectral stability of the plane Poiseuille flow of two viscoelastic fluids obeying an Oldroyd-B law in two configurations the first one is the two layer Poiseuille flow in the second case the same fluid occupies the symmetric upper and lower layers, surrounding the central fluid. (See Figure 9.)... 

In the story of numerical flow simulation, the ability to predict observed and significant viscoelastic flow phenomena of polymer melts and solutions in an abrupt contraction has been unsuccessful for many years, in relation to the incomplete rheological characterization of materials, especially in elongation. The numerical treatments have often been confined to flow of elastic fluids with constant viscosity, described by differential constitutive equations as the Upper Convected Maxwell and Oldroyd-B models. Fortunately, the recent possibility to use real elastic fluids with constant viscosity, the so-called Boger fluids [10], has narrowed the gap between experimental observation and numerical prediction [11]. 

It should be pointed out that the improvement of convergence might also be related to realistic preditions of shear and elongational viscosities by the Phan-Thien Tanner model, when compared to the Upper Convected Maxwell, Oldroyd-B and White-Metzner models. Satisfactory munerical results were also obtained with multi-mode integral constitutive equations using a spectnun of relaxation times [7, 17, 20-27], such as the K-BKZ model in the form introduced by Papanastasiou et al. [19]. 

The streamline patterns are quite identical for both constitutive equations. However, the vortex is more pronounced for the multimode Phan-Thien Tanner model, whereas the swelling is greater for the generalized Oldroyd-B model... 

Equations (3-32)-(3-34) are equivalent to the so-called Oldroyd-B equation. The Oldroyd-B equation is a simple, but qualitatively useful, constitutive equation for dilute solutions of macromolecules (see Section 3.6.2). Refinements to the simple elastic dumbbell model, such as the effects of the nonlinearity of the force-extension relationship at high extensions, are discussed in Section 3.6.2.2.I. 

Helical flow involves the steady laminar motion of a fluid in an annulus, when one (or both) of two coaxial cylinders is rotated and an axial pressure gradient is simultaneously imposed (Fig. 8). It occurs in mechanical equipment like deep well drilling and in production lines for extrusion of artificial casings [16]. Theoretical studies on the helical flow of general fluids have been done by Rivlin [17] and Coleman and Noll [18]. The work of Tanner [19,20] and Savins and Wallick [21] on helical flows emphasizes the Oldroyd type of constitutive equation. Dierckes and Schowalter [22] as well... 

Here r is given by the UCM equation 4.3.7 (or equivalently by the Lodge equation, eq.4.3.18), and is usually just a Newtonian term 2r7jD, where rj, is the solvent viscosity. The combination of these two terms is the Oldroyd-B constitutive equation (Oldroyd, 1950 see Exercise 4.6.4). Figure 4.3.4 compares the storage mod-... 

To address the shortcomings of the predictions associated with Eq. (3.9), Oldroyd (1958) has proposed a constitutive equation of the following form ... 

Small-amplitude oscillatory analysis can readily be applied to any nonlinear constitutive equation. For instance, applying Eq. (3.79) to the Oldroyd three-constant model, Eq. (3.21), we obtain... 

Figure 3.11 gives plots of n /irjo versus A.jC that are predicted from two constitutive equations (1) the upper convected Maxwell model, and (2) the Oldroyd three-constant model. It is seen in Figure 3.11 that both models predict values of increasing very rapidly without bound as e increases, in contrast to the experimental results given in Figure 3.10. As a matter of fact, all the expressions summarized in Table 3.3 predict similar elongational behavior, which is considered to be physically unrealistic.

If, as shown in Table 1, simple postulates are made for the anisotropic tensors (case I is due to Giesekus ), then a variety of constitutive equations can be obtained, including some which are special cases of the Oldroyd model. 

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