A Higher Weissenberg Number Analysis of Die-swell Flow of Viscoelastic Fluids Using a Decoupled Finite Element Method Iwata, Shuichi * 1/Aragaki, Tsut

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1 A Higher Weissenberg Number Analysis of Die-swell Flow of Viscoelastic Fluids Using a Decoupled Finite Element Method Iwata, Shuichi * 1/Aragaki, Tsutomu * 1/Mori, Hideki * 1 Ishikawa, Satoshi * 1/Shin, Yusaku * 2 This paper describes a new decoupled finite element method, which has been developed based on previous work and applied successfully to the high Weissenberg number problem of die-swell flow simulation using a single relaxation mode Giesekus model. The method is a modified version of the 2 ~ 2-subelement/non-consistent streamline upwind (2 ~ 2 SU) scheme. To improve the convergence behavior, several additional techniques are incorporated : (1) unification of the number of Gaussian points to evaluate integrations with same accuracy for momentum equation and for constitutive equation, (2) as for discritization of momentum equation with the rearranged form substitution and for getting diagonal dominant matrices, treatment of velocity (ƒò) as known and treatment of velocity gradient ( ƒòv) (and its transpose) as unknown, both of which appear in the definition of upper convected derivative of the elastic part of extra-stress tensor ( ), and (3) stepwise reduction of the relaxation factors for velocity-stress fields and for free surface shape, respectively. The new method enables us to predict die-swell flow simulations at Weissenberg numbers over 400. Key words : Viscoelastic fluids/giesekus model/die-swell/decoupled finite element method/ High Weissenberg number problem Nagoya Institute of Technology Gokiso-cho, Showa-ku, Nagoya , Japan Mitsui Chemicals, Inc. 2-1 Tangodori, Minami-ku, Nagoya , Japan Seikei-Kakou Vol.12 No

2 Fig. 1 Boundary conditions and coordinates (a) (b) Fig. 2 (a) Relevant model parameters, shear viscosity and first normal stress difference, (b) Elongational viscosity

3 Fig. 3 A typical finite element mesh Fig. 4 Unification of number of Gaussian points Seikei-Kakou Vo1.12 No

4 Table 1 Stepwise reduction of relaxation factors, (a) wand (b) wh (a) (b) Table 2 Maximum Weissenberg numbers * ) Iwata et al''

5 Fig. 6 (a) Non-dimensional radial velocity along solid wall and liquid surface (a)velocity vector (b)stream line (c)radial velocity profile (d)main velocity profile Fig. 6 (b) Non-dimensional normal stress along solid wall and liquid surface (e)pressure (f)shear stress (g)normal stress difference Fig. 5 Predicted results at We = 400, a = 0. 5, s = 0. 1, i/o = 5000 Pa s, A =1 s Fig. 6 (c) Non-dimensional shear stress along solid wall and liquid surface Seikei-Kakou Vol. 12 No

6 Fig, 7 Convergence behavior ; (a) Surface radial velocity at a nodal point Q adjacent to die lip, (b) Maximum error of radial velocity u Table 3 Average iteration numbers (b)

7 (a)velocity vector (b)stream line (c)radial velocity profile (d)main velocity profile (e)pressure (f)shear stress (g)normal stress difference Fig. 8 Predicted results at We = 50, a = 0. 1, s = 0. 1, h o = 5000 Pa s, A =1 s 3) Baaijens, F.P.T.: J. Non-Newtonian Fluid Mech., 79, (1998) 5 ) Matallah, H,, Townsend, P. and Webster, M.F.:1 Non-Newtonian Fluid Mech., 75, (1998) Seikei-Kakou Vol. 12 No

Developement of Plastic Collocation Method Extension of Plastic Node Method by Yukio Ueda, Member Masahiko Fujikubo, Member Masahiro Miura, Member Sum

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