| Abstract: | A seven parameter nonlinear viscoelastic constitutive equation was developed which accurately describes the behavior of polymer melts and solutions. The rheological parameters for a low density polyethylene (LDPE) and a high density polyethylene (HDPE) were estimated from steady viscometric data. The predicted steady shear and elongational flow, onset of steady shear and elongational flow, and small amplitude oscillatory flow material functions were calculated analytically and presented for the LDPE and the HDPE. The constitutive model does not contain spurious modes of instability for realistic values of the material parameters. The constitutive equation was used in a numerical simulation of the steady two-dimensional hydrodynamically developing flow of an incompressible viscoelastic fluid between parallel plates. The six governing equations were classified and found to be of mixed type, having both real and imaginary characteristics. The equation determining the characteristics for a fluid without a retardation term has six roots. Two of the roots are real, the characteristics following the streamlines. The four remaining roots depend on the solution, and cannot be determined explicitly. The roots can be real or complex permitting a change in type. The system of nonlinear partial equations was solved for the axial and transverse velocity components, the pressure, and the three non-zero stress components, by the method of orthogonal collocation on finite elements for the LDPE and the HDPE. For a Reynolds number of 10 and 100, the equations were found to change type with increasing Weissenberg number. The change in type was not found to cause the numerical breakdown of the calculations, but may be associated with it. Large spatial oscillations were always manifested as the limit point was reached. The solutions were obtained for Weissenberg numbers well beyond previous calculations, the results being very sensitive to the specified entrance stress boundary conditions. |
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