Weissenberg Definition

Dealy DM (2010) Weissenberg and Deborah numbers— their definition and use. Rheol Bull 79 14-18... 

The Weissenberg number compares the elastic forces to the viscous effects. It is usually used in steady flows. One can have a flow with a small Wi number and a large De number, and vice versa. Sometimes the characteristic time of the flow in the deflnition of the Deborah number has been taken to be the reciprocal of a characteristic shear rate of the flow in these cases, the Deborah number and the Weissenberg number have the same definition. Pipkin s diagram (see Fig. 3.9 in Tanner 2000) classifies shearing flow behavior in terms of De and Wi, and provides a useful guide for the choice of constitutive equations. 

There is a single dimensionless group, XVjL, which is known as the Weissenberg number, denoted by various authors as We or Wi. (We is more common, but it can lead to confusion with the Weber number, so Wi will be used here.) The shear rate in any viscometric flow is equal to a constant multiplied by V/L, so it readily follows that the ratio of the first normal stress difference to the shear stress is equal to twice that constant multiphed by Wi. Hence, Wi can be interpreted as the relative magnitude of elastic (normal) stresses to shear stresses in a viscometric flow. The ratio of the shear stress to the shear modulus, G, is sometimes known as the recoverable shear and is denoted Sr. Sr differs from Wi for a Maxwell fluid only by the constant that multiplies F jL to form the shear rate for a given flow. In fact, many authors define Wi as the product of the relaxation time and the shear rate, in which case Wi = Sr. It is important to keep the various definitions of Wi in mind when comparing results from different authors. 

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. 

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. 

The definition of the different regimes in terms of critical Weissenberg numbers can be refined intermediate regimes should be defined where the flow is strong enough but too short to reach stationary values for the orientation (Wio > 1) or the molecular stretch (Wig >1) (see Fig. 14.4). When the shear strain level is increased, the two transition points in the curve move to lower... 

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