Uniaxial elongational flow

Note 2 From the definition of general homogeneous flow (Definition 1.5) G X = GX = constant) in the particular case of steady uniaxial elongation flow... 

Part 2 presents a summary of the theoretical considerations and basic assumptions that lead to the model equations. Part 3 discusses some experimental aspects and focuses on the measmements in various shear and uniaxial elongational flow situations. Part 4 and 5 are devoted to the comparisons between experimental and predicted rheological functions. Problems encountered in the choice of correct sets of parameters or related to the use of each type of equation will be discussed in view of discrepancies between model and data. 

The mathematical form of the function can be derived simply from a fit of the experimental h(y) as obtained in step shear strain for example. However, the problem is further complicated if one now takes into account flows where the two invariants differ from each other as, for example, in uniaxial elongational flows where ... 

Provided that a suitable Y function is chosen, this is claimed to give a correct picture of many phenomena displayed in simple shear and uniaxial elongational flows. One should note that the original model of Phan-Thien and Tanner uses the non-af ne derivative together with nonlinear stress term. 

This section presents different results obtained using the Wagner equation with the form of the damping function of equation (33) together with the generalized invariant of equation (25). It is primarily devoted to comparisons between predictions and experimental results in shear and uniaxial elongational flows described in paragraph 3 for LD. 

Two different constitutive equations, namely the Wagner model and the Phan Thien Tanner model, both based on network theories, have been investigated as far as their response to simple shear flow and uniaxial elongational flow is concerned. This work was primarily devoted to the determination of representative sets of parameters, that enable a correct description of the experimental data for three polyethylenes, to be used in numerical calculation in complex flows. Additionally, advantages and problems related to the use of these equations have been reviewed. 

Table 7 gives a summary of qualitative performances and problems encountered for simple shear and uniaxial elongational flows, using the Wagner and the Phan Thien Tanner equations or more simple models as special cases of the former. Additional information may also be found in papers by Tanner [46, 64]. All equations presented hereafter can be cast in the form of a linear Maxwell model in the small strain limit and therefore are suitable for the description of results of the linear viscoelasticity in the terminal zone of polymer melts. 

M.H.Wagner, A constitutive analysis of uniaxial elongational flow data of a low density polyethylene melt, J. of Non-Newt. Fluid Mech. 1 (1978), 39-55. 

In a uniaxial elongational flow, the Trouton, or elongational viscosity of an anisometric particle suspension can be estimated by equation 59 as well with an intrinsic viscosity given by... 

Gabriel, G., and H. Miinstedt. 2003. Strain hardening of various polyolefins in uniaxial elongational flow. Journal of Rheology 47 619-630. 

All of the experiments just described are intended for thermoplastics. Only Cotton and Thiele [C19] have developed a uniaxial elongational flow instrument intended for elastomers. In this instrument an extruded rubber strand is placed over two puUeys. The two ends descend between a pair of knurled pulleys which draws the strand ends downward. 

Rotational convection and angular Brownian motion of clusters, as well as ather-mal nucleation resulting from transient free energy of the system are considered. One concludes that the cluster growth mechanism dominates the other mechanisms of oriented nucleation in the systems with transient chain deformation and orientation. Example computations illustrating transient effects in oriented nucleation are presented for the case of uniaxial elongational flow. 

Figure 4.3 shows the steady-state modulus, F , vs. the reduced elongation rate, 63T, computed from (4.17) for uniaxial elongational flow with fixed elongation rate, 63 = const. The modulus deviates from the modulus of the Gaussian system at the elongation rates e r > 0.5. 

Fig. 4.3. Reduced modulus at the steady-state limit, vs. reduced elongation rate, ear, computed from (4.17) for the non-linear and Gaussian systems subjected to uniaxial elongational flow and N = 100...

Fig. 4.4. Reduced chain elongation coefficient, Xs/X s, vs. reduced time, t/r, computed from (4.18), (4.19) for the uniaxial elongational flow with fixed elongation rates and A = 100. Steady-state levels of the elongation coefficients indicated...

The role of the transient effects in the molecular orientation, elastic free energy and, in consequence, in the kinetics of crystal nucleation is illustrated for the uniaxial elongational flow with fixed deformation rate, e r = const, which is less complex than the processes with time-dependent deformation rates. 

In the case of uniaxial elongational flow, the evolution equation for the distribution of cyUndrical cluster, n g,0,t), accounting for the flux of cluster growth, rotational diffusion and rotational flow convection reads... 

Fig. 4.11. Time evolution of the reduced critical cluster volume, g (6, t)/g 6. t = 0), vs. orientation angle, 0, between the initial state and the steady-state calculated for uniaxial elongational flow with fixed elongation rate, esr = 1. Af = 100...

Delaby, L, Ernst, B., Germain, Y, and Muller, R. (1994) Droplet deformation in polymer blends during uniaxial elongational flow - influence of viscosity ratio for large capillary numbers./. Rhed., 38 (6), 1705-1720. 

Delaby, I., Ernst, B., Froelich, D., and Muller, R. (1996) Droplet deformation in immiscible polymer blends during transient uniaxial elongational flow. Polym. Er. Sd.. 36 (12), 1627-1635. 

Oosterlinck, F., Mours, M., Laun, H.M., and Moldenaers, P. (2005) Morphology development of a polystyrene/ polymethylmethacrylate blend during start-up of uniaxial elongational flow. 

That is, the interfacial area increases exponentially with time. This is attributed to Erwin [22]. Equation (6.14) forms the basis of the view that uniaxial elongational flow is important to achieve good mixing. For a simple shear flow, the principal extension ratios A, An, and Am may be shown to be [28] ... 

Non-Newtonian Fluids in Microfluidics, Figure 2 Schematic diagram of two simple flow types, (a) Simple shear flow between parallel rigid surfaces with gap h. (b) Uniaxial elongational flow in a cylindrical fluid filament stretching along its axis z... 

Kobayashi, M., Takahashi, T., Takimoto, J., Koyama, K., Flow-induced whisker orientation and viscosity for molten composite systems in a uniaxial elongational flow-field, Polymer 36 (1995) 3927. 

Fig. 1 The orientability of a rigid rod in simple shear and uniaxial elongation flow fields. (From Ref. 32 with permission.)...

The morphology and properties of TLCPs can be significantly influenced within the spinline. In this processing regime, a nonisothermal uniaxial elongational flow field predominates. The morphology developed is maintained in the final extrudate due to solidification prior to take-up. The stretching flow results from differences between the take-up velocity and the... 

Figure 6.2 shows a particularly simple case uniaxial elongational flow, in which case the , coefficients take the values ... 

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