Planar contraction

L. M. Quinzani, R. C. Armstrong, and R. A. Brown, Birefringence and laser-Dop-pler velocimetry studies of viscoelastic flow through a planar contraction, J. Non-Newt. Fluid Mech., 52,1 (1994). 

Quinzani LM, Armstrong RC and Brown RA (1994) Birefringence and laser-Dop-pler velocimetry (LDV) studies of a viscoelastic fluid through a planar contraction. J Non-Newtonian Fluid Mech 52 1-36. 

Yoo, J. Y, and Na, Y., A numerical study of the planar contraction flow of a viscoelastic fluid using the SIMPLER algorithm. J. Non-Newtonian Fluid Mech. 39,89 (1991). 

By applying force balances over a cone or wedge-shaped element, Cogswell (1972) developed relationships between pressure drop and stress. For a sudden planar contraction (6 = 90° in Figure 3-29), the apparent planar extensional viscosity is (Padmanabhan and Bhattacharya, 1993) ... 

Fig. 8.7 /Cl defined by Eq. (14) for various planar contractions for a manifold the curve for / e = 0 Is indistinguishable from that for Re = ],...

The core of the computationally crucial part of CONNFFESSIT calculations has been described in great detail in Sections III and IV. The remaining elements in a CONNFFESSIT calculation are relatively straightforward. Two recent references describe in detail specific applications to the journal bearing problem (Laso et al. 1997) and to the planar contraction flow (Laso 1998) as well as the implementation on parallel and vector machines. 

Nigen S, Walters K (2002) Viscoelastic contractions flows comparison of axisymmetry and planar configurations. J Non-Newtonian Fluid Mech 102 343-359 13. Chiba K, Sakatani T, Nakamura K (1990) Anomalous flow patterns in viscoelastic entry flow through a planar contraction. J Non-Newtonian Fluid Mechn 36 193-203 Townsend P, Walters K (1994) Expansion flows of non-Newtonian liquids. Chemical Eng Sci 49 749-763 Hawa T, Rusak Z (2001) The dynamics of a laminar flow in a symmetric channel with a sudden expansion. J Fluid Mech 436 283-320... 

The polymer is a branched polyethylene melt with = 1.55 x 10 and Mw/M = 11.9, flowing at 170 °C in a 3.3 1 planar contraction. The hnear viscoelastic properties and the nonlinear parameters for a four-mode PTT equation are shown in Table 10.1. Different values of e were used for each mode, but with a constant value of f = 0.08. These parameters provide a reasonable fit to the transient and steady-state shear and extensional data, although the nonlinear parameters for the two longest relaxation times cause small oscillations in startup of simple shear that are not observed experimentally using parameters that ehmi-nate the shear oscillations causes the calculated extensional stresses to be too low, and the contraction flow results are sensitive to the extensional stresses. The mean relaxation time was 1.74 s, the average velocity in the downstream channel was 7.47 mm/s, and the downstream channel half-width (the characteristic length) was 0.775 mm. The Weissenberg number based on downstream channel properties was therefore 16.8. 

The sink flow analysis, which assumes a purely extensional flow (i.e., no shear component), was presented by Metzner and Metzner (1970) to evaluate the extensional viscosity from orifice Apen measurements. For an axisymmetric contraction, the flow into the orifice is analogous to a point sink for a planar contraction flow, the analogy is with a line sink (Batchelor, 1967). In the case of axisymmetric contraction (Figure 7.8.1), the use of spherical coordinates and continuity gives the velocity components... 

For the planar contraction, the cylindrical coordinate system will be convenient, and the velocity and extension rate equations respectively are given by... 

Bishko, G. B., Harlen, O. G., McLeish, T. C. B., Nicholson, T. M. Numerical simulation of the transient flow of branched polymer melts through a planar contraction using the pom-pom model. /. Non-Newt. Fluid Mech. (1999) 82, pp. 255-273... 

W.M.H. Verbeeten, G.W.M. Peters, F.P.T. Baaijens. Numerical simulation of the planar contraction flow for a polyethylene melt using the XPP model. J. Non-Newtonian Fluid Mech., 2004(117) 73-84. 

---