Oldroyd B model

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... 

In this section the discretization of upper-convected Maxwell and Oldroyd-B models by a modified version of the Luo and Tanner scheme is outlined. This scheme uses the subdivision of elements suggested by Marchal and Crochet (1987) to generate smooth stress fields (Swarbrick and Nassehi, 1992a). 

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.)... 

When We = 0, the Oldroyd-B model (26) reduces to a three-field version of the Stokes problem. For e < 1, this problem is stable under condition (27). It was proven in [106] that, in the case of the Maxwell-type problem (where = 1), one has to add a second inf sup condition to obtain stability ... 

J. Baranger and D. Sandri, Some remarks on the discontinuous Galerkin method for the finite element approximation of the Oldroyd-B model, submitted. 

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]. 

Two equations have been selected for the viscoelastic extra-stress component a generalized Oldroyd-B model (GOB) and a multimode Phan-Thien Tanner model (mPTT). The veilues of the corresponding parameters are given in sub-section 3-2... 

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... 

The velocity aloi the symmetry axis is significantly different an overshoot at the contraction as well as an undershoot downstream of the die exit are observed with the generalized Oldroyd-B model, but only a smooth overshoot is indicated by the multimode Phan-Thien Tanner model. The final value of the velocity after swelling is more important for the PTT, which is consistent with the lower value of swelling observed in Fig 20. 

O) generalized Oldroyd-B model ( ) multimode Phan-Thien Temner ffloaei. 

At low shear rate (3.3 s ), experimental and computed free surfaces are compared in Fig. 26. The final extrudate swell values are equivalent, but the experimental shape of the free surface is different from the computed ones. In addition, a slight overswelling at the die exit is observed for the (JOB model, which is observed neither experimentally, nor using a classic Oldroyd-B model. 

As indicated above, such an agreement is perhaps expected. On the other hand, it is remarkable that a rather complex phenomenological theory postulated for an LC continuum can be reconciled with an even more complex molecular theory built on the concept of intermolecular potential. Perhaps the only other such happy instance is the agreement between the continuum Oldroyd-B model for viscoelastic liquids and the molecular model based on a dilute suspension of linear Hookean dumbbells in a Newtonian solvent. ... 

The VOF-code has been extended to simulate viscoelastic flows. In the current investigation, the Oldroyd-B model is employed which can be derived either from the dilute solution theory with bead-spring dumbbell model or from the stress-... 

To simulate the viscoelastic flow, the Oldroyd-B model has been implemented in the VOF-code. Stabilization approaches, such as the Positive Definiteness Preserving Scheme and the Log-Conformation Representation approach have been adapted and implemented in the code to stabilize the simulations at high Weissenberg numbers. The collision of viscoelastic droplets behaves as an oscillation process. The amplitude of the oscillation increases and the oscillation frequency decreases when the Deborah number becomes larger. The phenomenon can be explained with the dilute solution theory with Hookean dumbbell models. An increase of the fluid relaxation time yields a decrease of the stiffness of the spring in the dumbbell and restrains the deformation of the droplets. In addition, with larger the viscosity ratio the collision process is more similar to the Newtonian one since the fluid has less portion of polymers. 

Show that the Oldroyd-B model as Oldroyd first presented it,... 

An example of (a) is the convected Jeffreys model or Oldroyd B model " ... 

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