Dimensionless shear rate

Fig. 10. Local values of the dimensionless shear rate in the model eddy [86]...

Fig. 4.2.2 Dimensionless shear rate and viscosity as a function of radius for a power-law fluid under the conditions shown in Figure 4.2.1. For a highly shear thinning material, the shear rate is large near the wall and close to zero near the center. The viscosity can vary by several orders of magnitude in the pipe.

For a fixed coefficient of friction p. and a fixed dimensionless shear rate K, the measurement of the power consumption per unit volume of the moist powder mass is proportional to the cohesive stress Uc- Thus, if the granulating liquid is added to the powder mass at a constant rate, the power consumption profile describes in a first approximation the cohesive stress (Tc as a function of the relative saturation S of the void space between the particles (Fig. 5). 

Fig. 8.1. Dynamic viscosity t] (a>) and steady state viscosity f](y) for undiluted narrow distribution polystyrenes. The data are plotted in reduced form to facilitate comparison. The dimensionless shear rate or frequency is t]0Mwy/gRT >r r/ M co/gRT. [See Eq.(8.3)]. The dynamic viscosities are for Mw = 215000 (O) and Mw = 581000 ( ) at 160° C (312). The steady shear viscosity is for Mw = 411000 (A) at 176° C (313). The shapes in the onset region are similar for the three curves, but the apparent limiting slope for the dynamic...

Fig. 8.11. Dimensionless shear rate / 0 locating the onset of shear rate dependence for viscosity in narrow distribution polystyrene systems. Symbols are O for solutions at 30° C in n-butyl benzene (155), 9 f°r solutions at 25° C in arochlor (177), and 4 for undiluted polymers at 159° and 183° C (324). Values for the intrinsic viscosity (cM=0) lie in the range /i0 = 1-2, varying somewhat with solvent-polymer interaction and molecular weight...

Fig. 8.12. Dimensionless shear rate /30 locating the onset of shear dependence in the viscosity for narrow distribution poly(a-methyl styrene) systems. Symbols are (198, 199) O M = 3.3 x 106, 6 M=1.82x 106,O-M= 1.19 x 106, and 9 M = 0.444 x 106. Values for intrinsic viscosity (cM=0) are similar to those for polystyrene (see caption of Fig. 8.11)...

Fig. 8.13. Dimensionless shear rate /30 locating the onset of shear rate dependence in the viscosity in narrow distribution systems of linear polymers vs cM/qM. Symbols for data on additional polymers are A for undiluted 1,4 polybutadiene (322), for undiluted poly(dimethyl siloxane) (323), and O for solutions of polyvinyl acetate in diethyl phthalate (195). The dotted lines indicate the ranges of for the intrinsic viscosity...

The solution of this set of equations gives the non-isothermal induction period x (8) as a function of non-isothermal shear rate 8 for different values of the parameters P and Fig. 2.32 shows the results of calculations for a wide range of dimensionless shear rates from 0.01 up to 100. The parameter P is equal to 0.03 and % varies from 0 to 1. At high shear rates, the decrease in the induction period is proportional to 8 1. This means that a 100-fold increase in shear rate results in an almost 100-fold reduction in the induction period, which could well be catastrophic for material processing if the process rate is increased. The influence of the parameter on x (8) is significant only for high shear rates. 

Figure 2.32. Predicted dependences of a dimensionless induction period in a non-isothermal process on dimensionless shear rate at different values of the parameter i 0 (curve 1) 0.6 (curve 2) 1.2 (curve 3) 1.8 (curve 4).

Let us discuss the experimental data on the curing of phenolic resins, which are shown in Fig. 2.33. In order to make theoretical calculations, the following characteristics of the resin derived from the experimental data were used 114,118 at To = 393K, the "static" value of to corresponding to the limit of very low shear rates, equals 240 s oo = 0.2 MPa Cp = 2 kJ/(kg K) p = lxlOr kg/m3 U = 41.8 kJ/mol. Using these values, the dimensionless shear rate can be expressed as... 

Eq. (2.85) is also applicable in this case, but the dimensionless shear rate must be replaced with another dimensionless parameter 5, which characterizes the non-isothermal effect. This is expressed as... 

SHEARING FLOW. Figure 3-14 shows the shear-rate-dependent viscosities of polystyrenes of various molecular weights in a couple of low-viscosity solvents, decalin and toluene (Noda et al. 1968 see also Kotaka et al. 1966). Plotted is the intrinsic relative viscosity, [r] l[r] o, against a dimensionless shear rate,... 

Figure 3.14 Curves of intrinsic relative shear viscosity versus dimensionless shear rate for dilute solutions of poly(a-methylstyrene) with molecular weights of (1) 690,000, (2) 1,240,000, (3) 1,460,000, (4) 1,820,000, (5) 7,500,000, and (6) 13,600,000 in toluene (a good solvent), and (7) 13,600,000 in decalin (a theta solvent). (Reprinted with permission from Noda et al.. Journal of Physical Chemistry 72 2890 Copyright 1968, American Chemical Society.)...

Ns = 9 / Figure 3.20 Dependence of the first normal stress coefficient on dimensionless shear rate as pre-... 

Figure 3.33 The solid curve is the dimensionless shear stress versus dimensionless shear rate yrd predicted by the Doi-Edwards constitutive equation, Eq. (3-71). The dashed curve adds a speculated contribution to the stress from Rouse modes. (From Doi and Edwards 1979, reproduced by permission of The Royal Society of Chemistry.)...

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.)...

In the above, X is the chain stretch, which is greater than unity when the flow is fast enough (i.e., y T, > 1) that the retraction process is not complete, and the chain s primitive path therefore becomes stretched. This magnifies the stress, as shown by the multiplier X in the equation for the stress tensor a, Eq. (3-78d). The tensor Q is defined as Q/5, where Q is defined by Eq. (3-70). Convective constraint release is responsible for the last terms in Eqns. (A3-29a) and (A3-29c) these cause the orientation relaxation time r to be shorter than the reptation time Zti and reduce the chain stretch X. Derive the predicted dependence of the dimensionless shear stress On/G and the first normal stress difference M/G on the dimensionless shear rate y for rd/r, = 50 and compare your results with those plotted in Fig. 3-35. 

One can then define a dimensionless shear rate, or Peclet number, as... 

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 11.17 Dimensionless shear stress and first normal stress difference as functions of dimensionless shear rate predicted by the Doi with the closure approximation for the fourth moment, (uuun) — (uu) (uu) and U — 2U = 6.

Figure 11.23—Comparison of theo-retical and experimental first and second normal stress differences N and N2. The theoretical results (a) were calculated from the Smoluchowski equation (11-3) using the Onsager potential with U = 10.67, the minimum value for a fully nematic state, y/ >r is the dimensionless shear rate (or Deborah number), where Dr is the rotary diffusivity of a hypothetical isotropic fluid at the same concentration. Only the molecular-elastic contribution to the stress tensor was considered. The experimental results (b) are for 12.5% (by weight) PBLG (molecular weight = 238,000) in w-cresol. (Reprinted with permission from Magda et al., Macromolecules 24 4460. Copyright 1991, American Chemical Society.)...

Figure 9 The variation number average degree of polymerization with time obtained from multiparticle Brownian dynamics is shown for rigid (left) and flexible (right) molecules at different dimensionless shear rates (/). The shear induced enhancement of polymerization rate is much larger for the rigid molecules (Agarwal and BChakhar [57]).

We can estimate the error for the simplified rheometer geometry that we have just analyzed. The dimensionless shear rate can be calculated from the solution, (4—45), with the functions tij(y) given by (4-55), (4-59), and (4-61). If we assume that the shear stress will be measured at the bottom wall, y = 0, then we want to evaluate the shear rate at the same location. Thus... 

From the Eqs. (8.96) there follow the basic dimensionless parameters influencing the system s evolution the Peclet number Pe = ya /Dsp = (>n/xa y/kT that gives the ratio of the hydrodynamic force of shear flow to the thermodynamic Brownian force dimensionless shear rate y = (>nfia yl p, equal to the ratio of the hydrodynamic force of shear flow to the force of non-hydrodynamic interaction and the volume concentration of particles W. 

Figure 18.6 Viscosity of particle dispersions the solid lines correspond to the shear rate limits calculated from the Krieger-Dougheity equation (see text) using the values of and dimensionless shear rate or Peclet number (Pe) given in the inset table, of Chou et aL [76] on acrylic latex particles (O) 7S0 nm spheres, ( ) 960 run spheres, (A) multilob particles...

However, calculation of the form birefringence is tedious (see refe 66 and 70, and also Section 5.5). Here we will give a simple approximate treatment. Under weak shear flow eqn (4.115), the deformation of the structure factor g(ifc) is proportional to the dimensionless shear rate x/F, and will be written as... 

The data of Mewis et al. (39) and d Haene (40) for poly(methyl methacrylate) spheres stabilized by poly( 12-hydroxy stearic acid) and dispersi in decalin correlate reasonably well with results for hard spheres for low to moderate volume fractions, although the critical stress is somewhat smaller. For highly concentrated dispersions, however, packing constraints cause some interpenetration of the layers at rest and viscous forces at high shear rates drive the particles even closer together. Consequently, the effective layer thickness decreases with increasing 0 and Pe, the dimensionless shear rate. 

Dimensional analysis thus reduces the task of characterizing the dependency by three orders of magnitude. Further reductions are possible in practical situations. For example, at moderate shear rates, the steady-state relative viscosity of a dispersion of neutrally buoyant spheres depends on only the dimensionless concentration (volume fraction) and the Peclet Number Pe, which is a dimensionless shear rate ... 

---