In polymer nanocomposites (PNCs), where polymeradsorption induced interparticle bridges are formed through polymer-mediated interparticle interactions, the influence of particles on polymer dynamics is considerably higher than the colloidal limits where the interparticle distances are larger than the size of polymers [32], To investigate the steady shear rheological properties of polymeric systems from macromolecular structure and relaxation behavior standpoint, it is indispensable to investigate the zero-shear viscosity ([[eta].sup.0.sub.P]) and first normal

stress difference of polymer matrices in linear viscoelastic region through Rouse model [33]:

Thirdly, the principal

stress difference of adding polypropylene fiber cementsoil was increasing with the increase of polypropylene fiber content under the same confining pressure.

Another rheological quantity of interest, which will be investigated, is the second normal

stress difference ([N.sub.2]), which is the difference between the normal stresses generated in the flow gradient and neutral directions (10-12), When considering plastic extrusion factors such as the frictional force along the steel wall, channel geometry, flow direction, extrusion velocity, and material type, among others, will affect the direction and magnitude of the second normal

stress difference (12), Co-extrusion with large discrepancies in [N.sub.2] between the fluids is the main cause of the appearance and development of elastic instabilities in non-axisymmetric channels, such as the ones treated herein (7),

The first normal

stress difference of the three polymers was measured in a parallel plates rheometer ARES, from Rheometric Scientific, with plate diameter of 25 mm.

In general, the normal stress effect during extrusion of a polymer melt is characterized by the normal

stress difference, including the first normal

stress difference ([N.sub.1]) and the second normal

stress difference ([N.sub.2]).

Measurements of the steady state shear viscosity [eta]([gamma]), where [gamma] is the shear rate and first normal

stress difference [N.sub.1] ([gamma]) = -([[tau].sub.11] - [[tau].sub.22] where 1 is the direction of the fluid velocity, 2 is the direction of velocity variation, [[tau].sub.11] the normal stress along the direction of the fluid velocity, and r22 the normal stress along the direction of velocity variation (44), (45) were made in a controlled strain ARES rheometer, from Rheometric Scientific, using a 25 mm parallel-plate fixture, gap of 1 mm, at 250[degrees]C, under nitrogen atmosphere; the Rabinowitsch correction on the shear stress [[tau].sub.12] ([[tau].sub.12] = [[tau].sub.measured] * ((3 + n)/4), where n = power law index) was also done.

The experiments were conducted in channels with circular and noncircular shapes for three different elastic response resins (polyethylene, polystyrene, and polycarbonata Observations served to explain that as a material such as polystyrene develops down a noncircular channel, stresses build up in the geometry which causes imbalances in the second normal

stress difference. These observations are unlike those seen in radial symmetric channels and it is this imbalance that generates secondary flows (12).

This finding follows from the fact that the first normal

stress difference is responsible for the die swelling in the r direction, whereas the second normal

stress difference is responsible for the swelling in the [theta] direction, and the former is probably an order of magnitude larger than the latter.

The change of the birefringence sign corresponded to the change of the

stress difference during cooling, as observed in Fig.

In the case of isothermal flow, a profile of principal

stress difference from the surface to the core shows a V-shape, so molecular orientation is directly related to fountain flow (9).

Kamal and Mutel (10) solved numerically the Fokker-Planck equation and the shear stress and obtained the normal

stress difference of suspensions.

It is expected that the Weissenberg number, which is the ratio of the first normal

stress difference and the shear stress, will be very high near the entrance of the channel.