Weissenberg number problem

As pointed out in Section II-3, numerical modelling of viscoelastic flows leads to numerical difficulties related to the mixed character (elliptic - hypeiholic) of the constitutive equation, to the propagation of "stress singularities" and to the so-called "High Weissenberg Number" problem. 

Keunings R (1986) On the High Weissenberg number problem. J Non-Newtonian Fluid Mech 20 209-226... 

There is a good but dated tutorial review, including a discussion of the high Weissenberg number problem, in... 

To simulate viscoelastic two-phase flow problems, the VOF method has been extended to capture the rheological properties of the Oldroyd-B fluid. To alleviate the High Weissenberg Number Problem in the simulation of viscoelastic flow, stabilization approaches have been adapted and implemented in FS3D. The simulation results show that the viscoelastic effect is reflected in the oscillation process during the collision, and the elasticity restrains the deformation of the collision complex. 

In the simulation of viscoelastic flow, a significant numerical problem, the so-called High Weissenberg Number Problem (HWNP), often occurs with loss of convergence of numerical algorithms. In order to alleviate the problem, we have, as the first attempt, implemented the conformation tensor Positive Definiteness Preserving Scheme (PDFS) by Stewart et al. [28], and then adapted and implemented the Log-Conformation Representation (LCR) approach by Fattal and Kupferman [7] in the viscoelastic two-phase flow solver in FS3D. 

Keunings, R. (1986) On the high weissenberg number problem. J. Non-Newtonian Fluid Mech., 20, 209-226. 

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 outlined scheme is shown to yield stable solutions for non-zero Weissenberg number flows in a number of benchmark problems (Swarbric and Nassehi, 1992b). However, the extension of this scheme to more complex problems may involve modifications such as increasing of elemental subdivisions for stress calculations from 3 x 3 to 9 x 9 and/or the discretization of the stress field by biquadratic rather than bi-linear sub-elements. It should also be noted that satisfaction of the BB condition in viscoelastic flow simulations that use mixed formulations is not as clear as the case of purely viscous regimes. 

Examples of CFD applications involving non-Newtonian flow can be found, for example, in papers by Keunings and Crochet (1984), Van Kemenade and Deville (1994), and Mompean and Deville (1996). Van Kemenade and Deville used a spectral FEM and experienced severe numerical problems at high values of the Weissenberg number. In a later study Mompean and Deville (1996) could surmount these numerical difficulties by using a semi-implicit finite volume method. 

Many studies have been devoted to the Taylor-Couette problem (flow between two concentric cylinders with radii R and R2, Ri < R2, of infinite length, and rotating with angular velocities fij and 02 repectively). For instance Zielinska and Demay [74] consider the general Maxwell models with —1 <0 < 1. They show that the axisymmetric steady flow (the Couette flow) does not exist for 2dl values of parameters where the steady state exists moreover all models, except for a very close to —1, predict stabilization of the Couette flow in the spectral sense, for small enough values of the Weissenberg number. (See also [55].)... 

The EVSS-G method introduced by Brown et al. uses the velocity gradient as an additional unknown ([7]). In order to come back to primitive variable, Guenette and Fortin ([20]) have introduced a (U, p, o, D) method where no explicit change of variable is performed in the constitutive equation. Hence this method is easier to implement. The elements used by these authors are continuous for velocity, discontinuous Pj for and pressure and continuous Qi for G and D. This method was tested on the 4 1 contraction and the stick-slip problem. This method seems robust and no limiting Weissenberg number was reached when using the PTT model for the stick-slip problem. 

Problem 3.8 Numerically solve Eq. (3-78), the differential approximation to the Doi-Edwards equation for entangled linear melts, in a steady-state shearing flow. Plot the dimensionless shear stress ayijG against Weissenberg number W/ = )>r for Wi between 0.1 and 100. 

It should be noted that for some problems, more than one characteristic time for the flow can be identified [3]. Thus, a second dimensionless group, the Weissenberg number, Wi, is sometimes used in polymeric fluid dynamics. It can be defined as... 

The situation is even more complex when viscous dissipation and the release of the crystallization enthalpy are taken into account. For pressure-driven flows, where the shear rate varies from the core to the wall of the flow channel, the flow becomes nonisothermal and a locally increased temperature can reduce the flow effects, since the characteristic relaxation times, and thus the Weissenberg number, will decrease. This might lead to cases where a stronger flow has less effect on the formation of oriented crystalline structures. These cases can only be analyzed by using a numerical model, as for example a finite element code, that includes all physical aspects of the problem [11,72-77]. 

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