Extensional viscosity time dependent

The extensional viscosity of the two fluids was determined using a punch that splits the liquid thread with a total length of 1 mm to 6.5 mm (method see [15]). The resulting time-dependent thread diameter and the calculated extensional viscosities are shown in Fig. 3.22. 

Left Time-dependent thread diameter between two plates. Right Calculated extensional viscosities as a function of time... 

The exponential decrease in the thread diameter and the highly time-dependent increase in extensional viscosity is clearly visible in the case of the PEO solution. However, the silicone oil displays a linear drop in the diameter of the thread (Newtonian fluid) and no increase of the extensional viscosity over time the extensional viscosity corresponds to approximately three times the shear viscosity. 

The instantaneous extensional stress W(e, t) is the force F(e, t) along the cylinder axis required to pull the cylinder ends apart, divided by the instantaneous cross-sectional area A s, t) of the cylinder thus a(e, r) = F e, t)/A(e, t). The time-dependent extensional viscosity, rj(e, t), is then ct( , t)/e. If this viscosity reaches a time-independent value within the duration of the experiment, that value is called the steady-state extensional viscosity, r ( ). 

Figure 1-14 compares the time-dependent extensional viscosity to the time-dependent shear viscosity, after onset of flow, at various shear and extension rates, for the same molten polyethylene described in Figs. 1-9 to 9-11 (Meissner 1972). This figure shows that the behavior of the extensional viscosity can be very different from that of the shear viscosity the former increases while the latter decreases with increasing strain rate at a fixed time after inception of steady flow. Thus, while the shear viscosity is shear thinning, the extensional viscosity is extension thickening. 

The temporary network model predicts many qualitative features of viscoelastic stresses, including a positive first normal stress difference in shear, gradual stress relaxation after cessation of flow, and elastic recovery of strain after removal of stress. It predicts that the time-dependent extensional viscosity rj rises steeply whenever the elongation rate, s, exceeds 1/2ti, where x is the longest relaxation time. This prediction is accurate for some melts, namely ones with multiple long side branches (see Fig. 3-10). (For melts composed of unbranched molecules, the rise in rj is much less dramatic, as shown in Fig. 3-39.) However, even for branched melts, the temporary network model is unrealistic in that it predicts that rj rises to infinity, whereas the data must level eventually off. A hint of this leveling off can be seen in the data of Fig. 3-10. A more realistic version of the temporary network model... 

As discussed in Section 12.3.3, unusual time- and shear-rate-dependencies have been reported for some wormy micellar solutions at dilute concentrations—for example, 1-5 mTAB/NaSal. At higher concentrations, 7-250 mM, of a similar surfactant, tetrade-cyltrimethyammonium bromide in NaSal, the extensional viscosity increases with in-creasing extension rate until a maximum is reached, and extension thinning then follows Thomme and Warr 1994). Prud homme and Warr interpret the maximum as the critical... 

The deformation of dispersed drops in immiscible polymer blends with the viscosity ratio X = 0.005-13 during extensional flow was studied by Delaby et al. [1994, 1995]. In the latter paper, the time-dependent drop deformation during a start-up flow at constant deformation rate was derived. The model is restricted to small drop deformations. 

If modified coefficients, Tjg and Deg, are used, as suggested in Refs. [1, 2] on the basis of the FENE dumbbell model. Fig. lb is obtained. The onset behaviour is described by the onset Deborah number, De o e,0 " with g 0 critical elongation rate of the porous media flow and T = relaxation time of the polymer solution, whilst the maximum value of the attainable increase of the extensional viscosity in normalized form only depends on the... 

Since the direction of the elongated particles does not coincide with the axial direction as a consequence of the spiraling of the streamlines as well as of the perpendicular shear, we find that extensional viscosity can act in an anisotropic way in the plain perpendicular to the axial movement. As long as the molecules are stretched the viscosity used in eq. (2.1) therefore becomes time dependent and its value increases with wall distance.. It can easily be seen from the integration (2.9) of the vorticity equation (2.3) that... 

Elongational Viscosity n (extensional viscosity, Trouton viscosity) The viscosity that characterizes an element undergoing Elongational Flow (above). It is equal to the tensile stress divided by the rate of elongation and for polymers it depends on the rate, but may increase with rate, unlike the usual reduction of shear viscosity with rate. Tensile viscosities are apt to be many times larger than shear viscosities for the same resin, temperature, and deformation rate. Values in the range of 10 -10 Pa s have been reported. For Newtonian liquids, the elongation viscosity is three times the shear viscosity (at the same temperature). 

In an analoguous manner the time dependent extensional viscosity , defined as... 

Figure 7.19 shows the time dependent viscosities derived from Eqs. (7.143) and (7.154) for both simple shear and extensional flow. For simplicity a single exponential relaxation with a relaxation time r is assumed for The... 

Fig. 7.19. Time dependent viscosities for shear and extensional flow, and as predicted by Lodge s equation of state. Calculations are performed for different Hencky strain rates en, assuming a single exponential relaxation modulus G(t) exp — t/r...

The approaches above can describe only steady state, time-independent viscosities. In Chapter 4 we will show that for time-dependent viscoelastic models, like Maxwell s, extensional thickening arises naturally. 

Although in a shear flow the viscosity of a polymeric fluid usually decreases with increasing deformation rate, in an extensional flow the viscosity frequently increases with increasing extension rate that is, the fluid is extensional thickening (recall Figure 2.1.3). Figure 4.2.5 shows the time-dependent uniaxial extensional viscosity... 

Figure 9.21 shows the time-dependent viscosities derived from Eqs. (9.178) and (9.189) for both simple shear and extensional flow. For simplicity a single exponential relaxation with a relaxation time t is assumed for G t"). The dotted line represents the time-dependent viscosity for simple shear, which is independent of 7. A qualitatively different result is found for the extensional flow. As we can see, the time-dependent extensional viscosity ff t) increases with ch and for en > 0.5t a strain hardening arises. 

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