Transient shear viscosity

as is often the case, the stress reaches a steady value only after a transient period of steady shearing starting from a state of rest, then the instantaneous stress j(y,t) during the transient start-up period, divided by the steady shear rate y, is the transient start-up 

The superscript indicates that the shear rate was increased from zero at time t = 0. This definition is only meaningful if the fluid at time r = 0 is in a well-defined state, usually a stress-free state, and if this starting state and the transient viscosity are reproducible from one run to the next. The superscript -p is sometimes omitted. Measurements of t) give information about rates of structural rearrangement within a deforming complex fluid. 

The creep test is a related way of obtaining time-dependent rheological information. In it, a constant shear stress, rather than a constant shear rate, is imposed on the material, and the shear rate is measured as a function of time until a steady shear rate is obtained. The creep test is especially useful for measuring the yield stress jy, since if the imposed stress is below cr the steady-state shear rate will be zero. 

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In Fig. 15.26 an example is given of Eq. (15.97) for a Maxwell element with G = 1000 N/m2 and t = 1 s. For small extensional rates of strain, the extensional viscosity is constant and equal to 3000 N s/m2. For higher extensional rates of strain, the viscosity increases. At the extensional rate of strain of 0.5 s-1 there is a transition to infinite extensional viscosities. The dotted line is the transient shear viscosity r)+(t) at low shear rates and equal to 14 of the transient extensional viscosity r)+ (f) at low extensional rates of strain. 

Figures 7 and 8 show the predictions of the Wagner model compared to experimental data for transient shear viscosity and first normal stress coefficient of LD. These have been calculated according to ...

FIGURE 4 Comparison of rheological model of Eqs. (47)-(49) with experiment for natural rubber, (a) Steady-state shear viscosity, (b) Transient shear viscosity at beginning of flow, (c) Stress, relaxation following now. 

FIGURE 6 Storage effects on stress transients at beginning of flow, (a) Transient shear viscosity of NR with 0.2 volume fraction N 326 black, T = 100°C. (b) Transient shear viscosity of SBR with 0.2 volume fraction N 326 black, T = 300°C. 

Figure 8.18 Sequential transient shear viscosity after various periods of rest for SBR. SBR + 0.2 phr N236 black, T = 100 °C.

The HDPE Tipelin and mLLDPE Exact 0201 (film blowing grades) have been chosen as the tested material. Linear viscoelastic properties (storage G and loss G" moduli), transient shear viscosity and transient first... 

Figure 5 Comparison between experimentally determined transient shear viscosity data (5a-5c)/first normal stress coefficient (5d-5f) and predictions of the XPP, PTT-XPP and mLeonov models for HDPE Tipelin at 180 °C.

The transient viscosity f] = T2i(t)/y0 diverges gradually without ever reaching steady shear flow conditions. This clarifies the type of singularity which the viscosity exhibits at the LST The steady shear viscosity is undefined at LST, since the infinitely long relaxation time of the critical gel would require an infinitely long start-up time. 

The trouble is that, under transient conditions, the shear recovery vs. preceding shear deformation can be much more sensitive to deviations from the strict behaviour of a second order fluid than the shear viscosity or the normal stress difference. A few entanglements between extraordinarily long chain molecules may be responsible for a maximum in the shear recovery. If this is the case, a shear recovery higher than the one... 

Surfactant solutions with rod-like micelles can have notable viscosities up to six times higher than the water viscosity [37]. This can be explained by the presence of entangled rodlike micelles (often also called worm-like micelles or thread-like micelles) which arrange themselves in a supramolecular transient network [38-41 ]. Such solutions often have elastic properties but they do not show a yield stress. This means that even high viscous solutions flow under the influence of very small shear stress. In this situation they show a zero shear viscosity which is given by ... 

In Fig. 15.27, the transient extensional viscosity of a low-density polyethylene, measured at 150 °C for various extensional rates of strain, is plotted against time (Munstedt and Laun, 1979). Qualitatively this figure resembles the results of the Lodge model for a Maxwell model in Fig. 15.26. For small extensional rates of strain (qe < 0.001 s ) 77+(f) is almost three times rj+ t). For qe > 0. 01 s 1 r/+ (f) increases fast, but not to infinite values, as is the case in the Lodge model. The drawn line was estimated by substitution of a spectrum of relaxation times of the polymer (calculated from the dynamic shear moduli, G and G") in Lodge s constitutive equation. The resulting viscosities are shown in Fig. 15.28 after a constant value at small extensional rates of strain the viscosity increases to a maximum value, followed by a decrease to values below the zero extension viscosity. 

This section draws heavily from two good books Colloidal Dispersions by Russel, Seville, and Schowalter [31] and Colloidal Hydrodynamics by Van de Ven [32] and a review paper by Jeffiey and Acrivos [33]. Concentrated suspensions exhibit rheological behavior which are time dependent. Time dependent rheological behavior is called thixotropy. This is because a particular shear rate creates a dynamic structure that is different than the structure of a suspension at rest. If a particular shear rate is imposed for a long period of time, a steady state stress can be measured, as shown in Figure 12.10 [34]. The time constant for structure reorganization is several times the shear rate, y, in flow reversal experiments [34] and depends on the volume fraction of solids. The viscosities discussed in Sections 12.42.2 to 12.42.9 are always the steady shear viscosity and not the transient ones. 

The slip parameter can be easily determined from various experiments in shear situations by some fit of the steady state shear viscosity and primary normal stress coefficient. Analytic expressions are easily derived in steady state and transient flows in the form ... 

In transient shear flows starting from an isotropic distribution of fiber orientations, considerably higher viscosities will be initially observed, until the fibers become oriented. In Bibbo s experiments, t]r for isotropically oriented fibers is around 3.5 for v = 75. These viscosities can also be predicted reasonably well by semidilute theory and by simulations (Mackaplow and Shaqfeh 1996). Figure 6-25 shows the shear stress as a function of strain for a polyamide 6 melt with 30% by weight glass fibers of various aspect ratios, where the fibers were initially oriented in the flow-gradient direction. Notice the occurrence of a stress overshoot (presumably due to polymer viscoelasticity), followed by a decrease in viscosity, as the fibers are reoriented into the flow direction. 

Figure 12 shows the transient shear modulus g(f,y) of the ISHSM from (27c), which determines the viscosity via ri = It is the time derivative of the... 

Transient shear flows involve examining the shear stress and viscosity response to a time-dependent shear. The stress build up at the start of steady flow (<7+) and at the cessation of steady flow (a ) and the stress decay (ff(0) after a dynamic instantaneous impulse of deformation strain (y) can be used to characterize transient rheological behaviour. 

FIGURE 6.19 (Upper panel) Steady-state shear viscosity versus shear rate (soUd symbols), dynamic viscosity versus frequency (open symbols), and transient viscosity calculated from Eq. (6.65) versus the inverse of the time of shearing (solid line). (Lower panel) Dynamic storage and loss modulus master curve for the same entangled polybutadiene solution (Roland and Robertson, 2006). 

FIGURE 6.20 Maximum in transient viscosity after initiation of shear flow varsus the duration of a quiescent interval between shearing. The growth of the overshoot peak is caused by reentanglement of the solution. The inset shows representative transient shear stress data (Roland and Robertson, 2006). 

FIGURE 6.30 Dynamic viscosity (squares), steady-state shear viscosity (circles), and the transient viscosity calculated using Eq. (6.65) (solid lines) for a linear (Mw = 389 kg/mol) and a highly branched (Mq/ = 1080 kg/mol) with 21 branches per chain and a branch Mq = 52.7 kg polyisobutylene (Robertson et al., 2002). 

Despite the fact that here one has the typical composition of a microemulsion, i.e., surfactant-water-oil, one does not find a low viscosity microemulsion but instead a highly viscous system. The addition of water results in the formation of flexible cylindrical reverse micelles that form a transient network of entangled micelles and has been characterized by means of dynamic shear viscosity measurements [73,74]. Light scattering experiments on systems with cyclohexane as the oil have demonstrated that a water-induced micellar growth occurs and that these systems may be described analogously to semidilute polymer solutions [75-77]. 

The morphology may affect the rheological properties under shear and extension in different manners. If the dispersed phase is rigid but deformable, it more effectively contributes to the rheological properhes of the blend. In Section 8.3.2, the transient extensional viscosity was measured at a lower temperature than the melting temperature of the dispersed phase. Rigid fibrils enhance extensional viscosity even with a small amount of the dispersed phase (1 wt%). Nevertheless, the morphological effect under shear flow is not... 

The fibre-windup technique [Padmanabhan et al., 1996] can provide transient extensional-viscosity data using modified rotational shear rheometers. One end of the sample is clamped while the other end is wound around a drum at a constant rotational speed to achieve a given extension rate, with the rheometer s torque transducers being used to obtain the extensional viscosity. The technique is claimed to provide valuable extensional viscosity data for high viscosity liqmds. 

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