Viscosity steady-state shear

The shear stress a is the force that a flowing liquid exerts on a surface, per unit area of that surface, in the direction parallel to the flow. The shear viscosity t] is then defined as 

After a steady shearing flow has been imposed on a fluid for a suitable period of time, the shear stress often (but not always) comes to a steady state, r(y), which depends on the imposed shear rate y. The ratio of the steady shear stress a to the shear rate y is then the steady-state shear viscosity t](y). 

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Steady state shear viscosity measurements indicate a power-law type relation for the variation of the shear viscosity with shear rate even in the lower shear rate range between 10"1 to 1 sec"l. The results at higher shear rates were questionable due to the slip between the sample and cone and plate fixtures. 

According to the theory of linear elastico-viscous behaviour (47) the steady-state shear viscosity t] and the steady-state shear compliance Je depend in the following way on the shear relaxation modulus G (t), where t is here the time of the relaxation experiment ... 

Lodge s expression for the steady-state shear viscosity reads (10) ... 

Fig. E3.5 Steady-state shear viscosity rj and first normal stress coefficient i, obtained from dynamic measurements versus shear rate for a low-density polyethylene melt, melt I. [H. M. Laun, Rheol. Ada, 17, 1 (1978).]...

J m(t — t )y 1](t, t) dt. Consider a fluid with a single relaxation time, 20, and modulus, Go, and with h(y) = e y. Calculate the steady-state shear viscosity function... 

For correlating extensional viscosity data, it is obvious to attempt the same method as was used for non-steady state shear viscosity. Thus, the ratio r)Jrjeo is presumed to be determined by two dimensionless groups i0 and f/i0. As e is constant (i.e. qe), the ratio of these groups is equal to the tensile deformation e. Therefore, t/t0 will likewise be a function of t/xa and . 

Cox and Merz illustrated empirically the equivalence of the co dependence of rj to the shear rate y dependence of the steady-state shear viscosity rj(y). This rule has been tested frequently, and found to be reliable for flexible-chain resins... 

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 ... 

Note that at the temperature 150 C, the transition from liquid-like to solid-like behavior occurs at frequency 1 sec, as indicated by the crossover of G and G roughly at this frequency. For this melt, a corresponding crossover in the steady-state shear-viscosity curve from a liquid-like plateau in to a solid-like shear thinning region occurs at a shear rate roughly in the vicinity of 1 sec (see Fig. 1-9). Thus, the crossover shear rate yc in steady shearing is about equal numerically to the crossover frequency coc... 

Even below c, the viscoelastic properties of a solution vary somewhat from those of a dilute solution, due to occasional coil overlaps (Ferry 1980). To obtain true dilute-solution properties from measurements made at a series of low, but finite, concentrations, one can subtract from those measurements any solvent contribution and then extrapolate the results to zero concentration. The intrinsic steady-state shear viscosity, for example, is one such extrapolated quantity it is defined as... 

Strands that terminate with a branch point at both of its ends can neither reptate nor completely retract. Relaxation of such strands presumably occurs by more complex, hierarchical processes discussed by McLeish (1988b). Here we simply note that the presence of branch points at both ends of a strand leads to much more strain hardening in extensional flows (Bishko et al. 1997 McLeish and Larson 1998). Low-density polyethylenes (LDPEs), which are highly branched, are well known for their extreme strain hardening behavior in extensional flows (Meissner 1972 Laun 1984) (see Fig. 3-39). The steady-state shear viscosity, as a function of shear rate, seems to be little affected by long-chain branching, however. 

Suppose there is a single relaxation time, so that m(/ — t ) = (G/t) exp[—(t — r )l. and suppose that h(y) = exp(—y). Calculate a formula for the steady-state shear viscosity as a function of shear rate. At high shear rate, what is the shear-thinning power-exponent p, with rj [Pg.188]

Since the steady-state shear viscosity is only slightly affected by the rods, the ratio of the extensional to the shear viscosity is of order 50 or so. 

Steady-state shear viscosity versus shear rate for PBLG solutions (molecular weight = 238,000) in m-cresol for several concentrations. The 38 wt% and 40 wt% samples show Region I behavior. (From Walker et al. 1995, with permission of the Journal of Rheology.)... 

Under steady shearing, these trapped disclinations should play the role of an anchoring condition, much like the role solid walls play in the flow properties of small-molecule nematics. A scaling analysis of this problem in Section 10.2.5 gives an equation, (10-28), for the steady-state shear viscosity for flow between surfaces with strong, homeotropic anchoring ... 

Fig. 7 Three-regime steady state shear viscosity for LCPs.

For determination of the steady state shear viscosity the Instron capillary viscometer model 3211 was used at 190 C. Six capillaries were used, three each of diameter d = 747 and 1273 ym. The length to diameter ratio In each series varied from L/d > 0.6 to 60. The standard Bagley and Bablnowltsch corrections as well as that for the pressure effects (45) were applied. The extrudate swell was determined on air-cooled extrudates, 5 cm in length. 

The process of macroscopic gelation that is sensed as a change in the rheological properties (a rapid increase in the steady-state shear viscosity) then involves the aggregation and crosslinking between these preformed gel particles to form an infinite network. 

As mentioned in Part 7.1, for polymer blends the relation between the steady-state shear viscosity and concentration can be quite complex. In the following discussion, the constant stress (not the constant rate) viscosity, corrected for the yield and time effects, will be considered. To illustrate flexibility of Equation 7.125 to describe (and thus to facilitate interpretation of the rheological results) r vs. < > dependence examples of computations are shown in Figures 7.24-7.31. 

PS/PMMA steady state shear viscosity Lyngaae-Jprgensen, 1983... 

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 6.30 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). 

HPMC chains adsorb onto the silica surface (9) and their adsorbed amounts are determined to be ca. 0.12g/g, irrespective of the silica concentration and HPMC. Figure 1 displays steady-state shear viscosities of the 5.0, 7.5, and 10.0 wt % silica suspensions dispersed in a 1.5g/100 mL HPMC solution, where the HPMC concentrations in the supernatants are 0.90, 0.60, and 0.30 g/100 mL in... 

Figure 1. Double-logarithmic plots of steady-state shear viscosities as a function of shear rate for 1,5 g/100 mL HPMC solution (open circle), 5.0 (filled triangle), 7.5 (filled circle), and 10.0 wt% (filled square) silica suspensions dispersed in a 1.5 g/100 mL HPMC solution.

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. 

Besides the relaxation time, the steady-state shear viscosity p is often used to characterize the mobility of polymers in the fluid phase as well. The change of shear viscosity with temperature reflects the viscous feature of the fluid. The most common fluids appear as the Arrhenius type (Arrhenius 1889),... 

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