Steady Shearing Flow

We consider here the flow vx—Ky, vy = 0, vz = 0, with k having a constant value. Then Eqs. (5.2) through (5.4) become  

The form of Eq. (5.1) suggests that for small shear rates a power series solution for xp(6, tj ) should be possible. Hence we try  

Since we have not assumed that ip is normalized, we may take xp0 = 1 with no loss of generality. Substitution of Eq. (6.4) into Eq. (5.1) shows that successive approximations xpk can be found from solving the system of differential equations =. (6,5) 

In obtaining the expressions for tpk, we have chosen to set all integration constants equal to zero. The choice of integration constants is completely arbitrary and affects only the normalization constant J. For our choice of integration constants, J = 4nL2. For other choices, tp and J will of course be different, but p/J will be the same. 

When the distribution function obtained in this way is substituted into Eqs. (6.1) and (6.2), and Eq. (5.6) is used, one obtains11  

Klingenberg et al. (1991a) find in their simulations a dependence of the dynamic yield stress on particle volume fraction 0 that is in qualitative agreement with experiment (see Fig. 8-7). Note that in both experiments and simulation, Oy is roughly linear in 0 for 0 0.30. 

For higher 0, ay levels off, or even has a maximum, evidently because chains are more efficient at momentum transfer when they are a single particle wide than when they are clumped into columns, as they tend to be when 0 is large (Klingenberg et al. 1991b). Note, also, however, that the magnitude of Oy from the simulations is much lower, by a factor or 50 or so, than the experimental values. At least part of this discrepancy can readily be attributed to the point-dipole approximation, which is only accurate when particles are widely spaced and when 1, conditions that aren t met. The magnitude of the errors made in the 

02 - i Sion ironi j. ncin. r nys. 7 o i ou, opyrigni 1991, American Institute of Physics.) 

By comparison with the numerical calculations of Kirkwood and Flock (40) and Stewart and S0rensen (72) Eq. (6.7), including the term in (2k) is good to within 1 % up to Ak = 0.46. 

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Kaye, A., Lodge, A. S. and Vale, D. G., 1968. Determination of normal stress difference in steady shear flow. Rheol. Acta 7, 368-379. 

Steady shear flow properties are sensitive indicators of the approaching gel point for the liquid near LST, p < pc. The zero shear viscosity rj0 and equilibrium modulus Ge grow with power laws [16]... 

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. 

Near LST, the relaxation times become very long, and steady shear flow cannot be reached in the relatively short transient experiment. Large strains are the consequence for most reported data. 

It became clear in the early development of the tube model that it provided a means of calculating the response of entangled polymers to large deformations as well as small ones [2]. Some predictions, especially in steady shear flow, lead to strange anomaUes as we shall see, but others met with surprising success. In particular the same step-strain experiment used to determine G(t) directly in shear is straightforward to extend to large shear strains y. In many cases of such experiments on polymer melts both Hnear and branched, monodisperse and polydisperse,the experimental strain-dependent relaxation function G(t,Y) may be written... 

When considering a sohd-liquid interface, we begin with the simplest case of steady shear flow parallel to the surface, with D /Dt = 0 and v = V . Equation (1) reduces to Newtonian viscous flow,... 

Actual calculations of w,E) for isotropic and nematic solutions will be described in Sects. 8 and 9 respectively. Furthermore, derived approximately for isotropic solutions in a steady shear flow in Sect. 8, but it will be neglected for nematic solutions in Sect. 9. 

The viscous properties of HIPEs and high gas fraction foams have also been studied extensively, using a two dimensional, monodisperse, hexagonal cell model. Khan and Armstrong [52] showed that, under steady shear flow (i.e. beyond the yield point of the system), the foam viscosity was inversely proportional to shear rate. At high rates of shear, a constant viscosity value was approached. Gas fraction, <)>, was assumed to be very close to unity. 

Quantitative evidence regarding chain entanglements comes from three principal sources, each solidly based in continuum mechanics linear viscoelastic properties, the non-linear properties associated with steady shearing flows, and the equilibrium moduli of crosslinked networks. Data on the effects of molecular structure are most extensive in the case of linear viscoelasticity. The phenomena attributed to chain entanglement are very prominent here, and the linear viscoelastic properties lend themselves most readily to molecular modeling since the configuration of the system is displaced for equilibrium only slightly by the measurement. 

In steady shear flows, only the shear-rate dependence of viscosity is well documented in terms of molecular structure effects. Molecular theories are more... 

The response of simple fluids to certain classes of deformation history can be analyzed. That is, a limited number of material functions can be identified which contain all the information necessary to describe the behavior of a substance in any member of that class of deformations. Examples are the viscometric or steady shear flows which require, at most, three independent functions of the shear rate (79), and linear viscoelastic behavior (80,81) which requires only a single function, in this case a relaxation function. The functions themselves must be determined experimentally for each substance. 

For steady shear flow, the shear rate y is constant for all past time. Since deformation history now depends only on the parameter y, the stress components become functions of y alone ... 

In steady shear flow, the viscosity is independent of shear rate — for all y. This property alone represents a serious qualitative failure of the conventional bead-spring models. The normal stress functions are (108) ... 

Williams begins with Fixman s equation (220) for the stress contributed by intermolecular forces in flexible chain systems. The theory assumes that the polymer concentration is high enough that intermolecular interactions control the stress. The shear stress contributed by polymer molecules in steady shear flow is expressed in the form... 

Such behavior is qualitatively understandable in terms of partial disentanglement in steady shear flow. In highly entangled systems (cM>gM )Je0 is of the form (Section 5) ... 

Values of p22 — P33 = N2 appear to be negative and approximately 10-30% of Nj in magnitude (82). The conventional bead-spring models yield N2=0. Indeed, N2 in steady shear flow is identically zero for all free draining models, regardless of the force-distance law in the connectors (102a). Thus, finite extensibility and, by inference at least, internal viscosity do not in themselves provide non-zero N2 values. Bird and Warner (354) have recently analyzed the rigid dumbbell model with intramolecular hydrodynamic interaction, the latter represented by the Oseen approximation. In this case N2 turns out to be non-zero but positive. 

Studies have been made of the stresses produced in several non-steady flow histories. These include the buildup to steady state of a and pu — p22 at the onset of steady shearing flow (355-35 ) relaxation of stresses from their steady state values when the flow is suddenly stopped (356-360) stress relaxation after suddenly imposed large deformations (361) recoil behavior when the shear stress is suddenly removed after a steady state in the non-linear region has been reached (362) and parallel or transverse oscillations superimposed on steady shearing flow (363-367). Experimental problems caused by the inertia and compliance of the experimental apparatus are much more severe than in steady state measurements (368,369). Quantitative interpretations must therefore still be somewhat tentative. Nevertheless, the pattern of behavior emerging is suggestive with respect to possible molecular flow mechanisms. 

Stress Development at the Onset of Steady Shearing Flow (355-35 )... 

Small Oscillations Superimposed on Steady Shearing Flow (363-367)... 

The dynamic moduli for infinitesimal superimposed deformations parallel and transverse to the flow direction in steady shearing flow should be unaffected by flow if the shear rate is sufficiently small. According to the theory of simple fluids, the superimposed dynamic moduli for shearing flows in the non-Newtonian region must change in order to conform with the relations (370,372 ... 

Fractional reduction in entanglement density due to steady shear flow g(0) = E/Eo (Part 8). 

Fractional reduction in energy dissipation rate per molecule due to dis-entanglement in steady shear flow (Part 8). 

Graessley,W.W., Segal,L. Flow behavior of polystyrene systems in steady shearing flow. Macromolecules 2,49-57 (1969). 

Woods,M.E., Krieger,I.M. Rheological studies on dispersions of uniform colloidal spheres. I. Aqueous dispersions in steady shear flow. J. Colloid Sci. 34,91-99 (1970). 

Booij.H.C. Influence of superimposed steady shear flow on the dynamic properties of non-Newtonian fluids. Rheol. Acta 5,215-221 (1966). 

Simmons,J.M. Dynamic modulus of polyisobutylene solutions in superposed steady shear flow. Rheol. Acta. 7,184-188 (1968). 

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