Weissenberg numbers

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. 

With a new software program it is possible to measure the Texture Constant" of pectins. This Texture Constant K is calculated by the ratio of the maximum force during the time interval of the measurement and the measured area below the force-time curve. The resulting constants K correlate well with the dynamic Weissenberg number of oscillating measurements carried through with the same pectin gels. 

The dynamic Weissenberg number W can be calculated from data obtained by the strain frequency sweep measurement. It s the ratio of the elastic to the viscous shares in the measured gel and leads to an objective description of the sensoric properties, representing the basis for the standardization of pectins. 

As an example of such applications, consider the dynamics of a hexible polymer under a shear how [88]. A shear how may be imposed by using Lees-Edwards boundary conditions to produce a steady shear how y = u/Ly, where Ly is the length of the system along v and u is the magnitude of the velocities of the boundary planes along the x-direction. An important parameter in these studies is the Weissenberg number, Wi = Ti y, the product of the longest... 

Fig. 21a-d. Flow from startup into a 4 1 contraction computed for the pom-pom model with fixed total and cross-bar molecular weight and Weissenberg number (dimensionless deformation rate) of 3[(a) and (b)] and 8 [(c) and (d)]. (a) and (c) show (colour coded) the level of dimensionless stretch of the pom-pom cross-bar, X. (b) and (c) show the respective streamlines. Note the spur of preoriented material joining the wall to the funnel and the reduction of the corner vortex at high flow rates when extension thinning sets in. (Computations courtesy of Dr. T. M. Nicholson)... 

Elastic behavior of liquids is characterized mainly by the ratio of first differences in normal stress, Ai, to the shear stress, t. This ratio, the Weissenberg number Wi = Ai/r, is usually represented as a function of the rate of shear y. 

By keeping the Weissenberg number Wi (a pure material number) and the Hedstrom number He constant, measurements are performed and presented in a dimensionless frame ... 

For a correct dimensional-analytical representation of the viscoelastic behavior of a fluid, the ratio of normal stress to shear stress is used. The so-called Weissenberg number is defined as... 

In the case of non-Newtonian mixtures, which also exert pseudoplastic and viscoelastic behavior, the pi-space is widened by the Weissenberg number, Wi. In addition, it has to be decided which effective viscosity, peffi has to enter the Reynolds number ... 

These test data are correlated satisfactorily by the Weissenberg number (here ned as Wi = /.n) and the term ( x0, i/Mo.2) which accounts for the pumping irection, Fig 23. ji represents the starting viscosity of the lower, more viscous liqui ( )> an the upper, less viscous (2) liquid. Upflow z= 0.059, downflow z= 0.17. 

Boger (1987) has reviewed viscoelastic flows through contractions and has pointed out that three-dimensional and time-dependence characteristics have to be taken into account for flows at high Weissenberg number We defined by ... 

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. 

The parameters in these equations are the Reynolds number Re = pULfTf (U and L are a typical velocity and a t rpical length of the flow, and rj = r),+t]p is the total viscosity of the liquid), the Weissenberg number We = Xl/fL, and the retardation parameter e = fip/fi. Obviously, 0< <1 =1 corresponds to Maxwell-type fluids, and 0 < e < 1 corresponds to Jeffreys-type fluids. Observe the change of notation in equation (7), where /3(Vv,r) denotes now adl the nonlinear terms in Vv and t other than the term (v V)r. f denotes some given body forces. 

Let V be a steady solution of the Navier-Stokes equations (with prescribed body forces and zero boundary velocity). Note that v is not assumed to be small . Then, there exists a steady solution (Vt,Tj,pj) of any Jeffreys model with a sufficiently small Weissenberg number We, and with a sufficiently small retardation parameter e, such that (ve,rj) is close to (v,0) and close 0. (See [26].)... 

In a recent paper [70] Renardy has investigated the nonlinear stability of flows of Jeffreys-type fluids at low Weissenberg numbers. More precisely, assuming the existence of a steady flow (v, r), he proves that this flow is linearly and Liapunov stable provided the spectrum of the linearized operator lies entirely in the open plane 3 A < 0 and that the following quantity is sufficiently small... 

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].)... 

M. Renardy, Nonlinear stability of flows of Jeffreys fluids at low Weissenberg numbers. Arch. Rat. Mech. Anal. 132 (1995) 37-48. 

This method is highly stable and results have been obtained at large values of the Weissenberg number for various flows. Figure 2 from [28] show the evolution of velocity and T y stress... 

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. 

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