Reduced shear stress

Another alternative is to correlate data in terms of a reduced shear stress, —-, in... 

With this quantity one obtains a simple relation between reduced first normal stress difference (pu — p2J)jv k T and reduced shear stress p2ilv k T. For this purpose, velocity gradient q is eliminated from eqs. (3.37) and (3.38). This relation reads ... 

According to eq. (3.38) the left-hand side of eq. (3.41) can also be considered as twice the reduced stored free energy in steady shear flow. Using for this quantity the character FR and for the reduced shear stress the character ft, one obtains instead of eq. (3.41) ... 

This treatment will not be discussed in further detail, as a knowledge of distinct relaxation times is not required for the present purpose. Interest is focussed on the relation between reduced first normal stress difference and reduced shear stress, as expressed by eq. (3.41). The simplest way to evaluate this equation for a polydisperse polymer has been given by Peterlin (76). This procedure has extensively been used by Daum (32, 73) in his experimental investigations. 

As pointed out in the paragraph after eq. (3.43), the dimensionless quantity / can be interpreted as a reduced shear stress or also, when the measured ("Newtonian ) zero shear viscosity r]y of the solution is used in eq. (3.43), as a reduced shear rate. 

Fig. 3.2. Reduced stored free energy Fa vs. reduced shear stress / for a 0.57 wt. per cent solution of Taps. No. 5 in methyl (4-bromo-phenyl) carbinol at various temperatures, viz. ( ) 18°, ( ) 30°, (a) 40° and (A) 50° C (32)...

For these measurements, temperature has been varied between 55 and 110° C. In this temperature range, the solvent viscosity changes by a factor three 4.7 to 1.5 cps). It is very improbable that a noticeable internal friction factor would change just by the same factor. Moreover, as has already been pointed out at the end of Section 5.2.2, the curves obtained by plotting cot2 c vs reduced shear stress fjN are practically coinciding for dilute solutions of cellulose tricarbanilate fractions with M S 500,000 and for anionic polystyrenes. So one can conclude that the internal friction of the thermodynamically stiff molecules of cellulose tricarbanilate must be rather low. 

Figure 3.35 Steady-state values of the reduced shear stress <712/ and first normal stress difference N / as functions of dimensionless shear rate y Zr predicted by the equations of a constraint-release reptation theory (see Problem 3.10) for Xd/Zr — (a) 50, (b) 150, and (c) 500, where Zd is the reptation time and Zr is the Rouse retraction time. See also Marracci and lanniruberto (1997). (From Larson et al. 1998, with permission.)...

Krieger (1972) has argued that the shear-rate-dependent suspension viscosity t (0, y) is more appropriate than the solvent viscosity t]s to use in defining a dimensionless shear rate. Since r 4>, y)y is the shear stress a, Krieger s suggested dimensionless group is really a reduced shear stress ... 

Figure 6.4 Relative viscosity rjr = rj/rjg verses reduced shear stress for latices of poly(methylmethacrylate) spheres of radii 85, 141, 204, and 310 nm, sterically stabilized by adsorbed triblock copolymer, poly(dimethylsiloxane)-polystyrene-poly(dimethylsiloxane), in silicone fluids of viscosity 7.98 cP and 44.1 cP at 30°C. The line is the fit from Eq. (6-14a). (From Choi and Krieger 1986, reprinted with permission from Academic Press.)...

Figure 6.30 Relative viscosity versus reduced shear stress for aqueous suspensions of charged polystyrene spheres (a = 110 nm) at a concentration of (p = 0.40 at HCl concentrations of 0 (V), 1.88 x 10 " (T), 1.88 X10 ( ), and 0.0188 (0) tfrom Krieger and Eguiluz 19761. The solid lines are calculated using the same equations as in Fig. 6-29. (From Buscall 1991, reproduced with permission of the Royal Society of Chemistry.)...

From Fig. 6-4, shear thinning begins at a reduced shear stress of around 0.01, where the relative viscosity rjr is around 10. By definition of r]r, we have... 

Figure 11.14 Reduced shear stress (transient stress divided by steady-state stress) plotted...

Figure 12.18 (a) The solid line is the reduced shear stress oj Go as a function of reduced shear rate yx predicted by Eqs. (12-34)-(12-36). The long-dashed line is a hypothetical modification obtained by considering high-ftequency Rouse modes. The short-dashed line is the stress predicted by flow stratification, (b) Comparison of the theory with flow-stratification to experimental data of Rehage and Hoffiuann (1991). (From Spenley et al. 1993, reprinted with permission from the American Physical Society.)... 

Brownian motion can be taken into account by plotting against the reduced shear stress given as... 

Figure 10 shows the variation of the relative viscosity with the counterion molarity at different reduced shear stress values for monodisperse polystyrene latex having a diameter of 0.192 xm at dispersed-phase volume fraction = 0.509. Clearly, is a function of the electrolyte concentration in addition to the reduced shear stress. 

Figure 9. Relative viscosities vs. reduced shear stress for 50% monodispersions of polystyrene spheres of various sizes in different media. Points are taken from Figure 8. (Reproduced with permission from reference 31. Copyright 1972...

Dilatant Flows Krieger and Choi [1984] smdied the viscosity behavior of sterically stabilized PMMA spheres in silicone oil. In high viscosity oils, thixotropy and yield stress was observed. The former was well described by Eq 7.41. The magnimde of Oy was found to depend on ( ), the oil viscosity, and temperature. In most systems, the lower Newtonian plateau was observed for the reduced shear stress value = Oj d / RT > 3 (d is the... 

The reduced shear stress is recognized to be essentially our Peclet number defined by Eq. (9.2.10). As can be seen from Fig. 9.2.3, as the Peclet number is increased, the viscosity reaches a stationary value and in this limit the suspension behaves as a Newtonian fluid. In the opposite limit, as the Peclet number tends to zero, the relative viscosity approaches a higher stationary value. The transition is seen to take place in the neighborhood of a Peclet number close to unity, consistent with our earlier discussion. 

Figure 9.2.3 Relative viscosity as a function of reduced shear stress for monodisperse colloidal polystyrene spheres of different sizes in different fluids (benzyl alcohol, meta-cresol, and water) at 50% volume fraction (after Krieger 1972).

Figure 9-22. Dependence of the reduced intrinsic viscosity Ma/lvi] on the reduced shear stress = (W 7i(A r7)/Z )021 of polystyrene with molar mass = 7.1 X 10 (0,D, A), 3.2 X 10 ...

A thicker bond line can reduce shear stress concentration by spreading the strain over a larger dimension, resulting in less strain on the adhesive. An increase in bond line gap from 0.0025 to 0.10 cm can decrease the stress ratio from 18.2 to 3.06. ... 

At higher temperatures, the inhibitor reacts with the polyester, thus forming branched structures causing the stress stress to increase. The excess of inhibitor acts as a diluting compound, reducing shear stress. 

Indicates a departure from a pure liquid state. A blend of the base oil with a Vl-improver, (Figure 4) (11.5% W) - modelling widely used, vlsco-statlc oils - retains the shear thickening characteristics at extremely high stress, however with a reduced shear stress sensitivity. 

---