Constitutive equations White-Metzner model

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]. 

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). 

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. 

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... 

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). 

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