White-Metzner equation

The behavior of a non-Newtonian viscoelastic fluid can be described by a constitutive equation which takes into account condition (1). Rheological behavior of the fluid is described by an equation derived from White-Metzner-Litvinov model and takes the following form 27,321 ... 

For each r, a constitutive equation must be selected. Dooley and Dietsche [5] evaluated the White-Metzner, the Phan-Thien Tanner-1, and the Giesekus models, given by,... 

Together with Eq. 3.3-17, Eq. 3.3-16 is the White-Metzner constitutive equation, which has been used frequently as a nonlinear viscoelastic model. Of course, for small deformations, X(i) = dx/dt, and the single Maxwell fluid equation (Eq. 3.3-9) is obtained. 

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]. 

Birefringence measurements are often performed to compare theoretical and experimental stress distributions in an abrupt contraction (see Section III-l). Such comparisons have been already published for the White-Metzner [30], K-BKZ [27, 31] and Wagner [32, 33] constitutive equations. Generally speaking. 

If one replaces /tr by any of the empiricisms for r] discussed in Section 2.1, then the constitutive equation, which is known as the White-Metzner (WM) model, can be written as... 

The constitutive equation used by Denn and co-workers (Denn et al., 1975 Fisher and Denn, 1976) was the White-Metzner model (see also Section 3.2.1 Eq. 3.42), which was thought to be applicable to high Deborah number processes such as melt spinning. The zz and rr components of the constitutive equation in cylindrical coordinates are... 

The vortex development of a low-density polyethylene in different flat dies under various proeessing conditions has been analyzed by the Finite Element Method enploying the modified White-Metzner model as constitutive equation. The theoretical results are conpared with the velocity distributions measured by Laser-Doppler Velocimetry (LDV). 

Non-isothermal viscoelastic steady two-dimensional finite element simulations were performed by solving the well-known mass, momentum and energy equations using the commercially available Compuplast software VEL 6.1. In this study, the modified White-Metzner (mWM) constitutive equation according to Barnes and Roberts [8] is employed. The non-isothermal mWM model is given by Eqs. (2) - (5). 

Often times concentrated polymeric solutions cannot be treated as Newtonian fluids, however, and this tends to offset the simplifications which result from the creeping flow approximation and the fact that the boundaries are well defined. The complex rheological behavior of polymeric solutions and melts requires that nonlinear constitutive equations, such as Eqs. (l)-(5), be used (White and Metzner, 1963) ... 

A. B. Metzner, J. L. White, and M. M. Denn, Constitutive Equation for Viscoelastic Fluids for Short Deformation Periods and for Rapidly Changing Flows Significance of the Deborah Number, AIChEJ. (12) 863,1966. 

Metzner A, White JL, Denn MM (1966) Constitutive equations for viscoelastic fluids for short deformation periods and for rapidly changing flows significance of the Deborah number. AIChE J 12 863-866... 

Subsequent work took into account the velocity and shear field adjustments at the end of a die (Nickell et al., 1974). White and Roman (1976) developed a modified formulation of Tanner s theory where the melt recovers from Poiseuille flow in a tube into a state of uniaxial tension. These theories have been validated by Vlachopoulos et al. (1972). However, the die swell theory was also studied using macroscopic mass and momentum balances given by Metzner and others (Metzner et al., 1961 Graessley et al., 1970), where the balance on the fluid between the capillary exit and flow downstream formed a relationship with the die swell. The first normal stress difference is shown in Equation [4.10]. 

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