Low shear limit

The Cross equation assumes that a shear-thinning fluid has high and low shear-limiting viscosity (16) (eq. 4), where a and n are constants. 

This model contains four rheological parameters the low shear limiting viscosity (rj0), the high shear limiting viscosity a time constant (X),... 

If r]oo <3C ( /. j/o), the Carreau model reduces to a three-parameter model ( 0,k, and p) that is equivalent to a power law model with a low shear limiting viscosity, also known as the Ellis model ... 

There are three important points about Equation (3.47). Firstly the viscosity is the low shear limiting value, rj(0), indicating that we may expect some thinning as the deformation rate is increased. The reason is that a uniform distribution was used (ensured by significant Brownian motion, i.e. Pe < 1) and this microstructure will change at high rates of deformation. Secondly there is a difference between the result for shear and that for extension. Thirdly the equation is only accurate up to cp < 0.1 as terms of order 3 become increasingly important. If we write the equation in the form often used for polymer solutions we have for Equation (3.47 a) ... 

Figure 3.12 The low shear limiting viscosity for unimodal, bimodal and trimodal size distributions calculated from Equation (3.55 )...

The effective hard sphere diameter may be used to estimate the excluded volume of the particles, and hence the low shear limiting viscosity by modifying Equation (3.56). The liquid/solid transition of these charged particles will occur at... 

The intrinsic viscosity is the Einstein value [rj] = 2.5 and the packing fraction cpm(0) is that in the low shear limit. As the volume fraction approaches the maximum packing fraction, the viscosity rapidly... 

For this to happen we know that f(y) = y in the low shear limit. As the shear stress is increased we also know that we want our viscosity to fall so we need to multiply our strain by a damping function that reduces from unity at low strains to a lesser value at high strains. A good candidate for... 

Intuitively we might associate the low shear limit with the order-disorder transition at literature data for the packing fraction in this limit is more widely scattered. We must remember that the approach to equilibrium in these systems can take a while to progress. So it is feasible that some systems have been measured away from the equilibrium state when the samples have been transferred and placed in the measuring geometry on an instrument. We could... 

A plot of rj(o) versus the log of reduced stress shows a linear slope between the high and low shear limits. We can use this feature with our master curve to define another value of b. This slope is given by... 

If we extrapolate this slope toward low rates, where the tangent equates with the low shear viscosity and assume the Peclet number here is unity we eventually obtain a value of b = 2.55. This defines the Peclet number as unity, at a stress somewhere just after the curvature of the viscosity curve deviates from the low shear limit. This seems quite an appropriate reference system. By setting the Peclet number to the appropriate value of b we can determine the variation in packing fraction with stress between the high and low shear limits to the viscosity ... 

At the percolation limit, the rheological behavior of the suspension changes from Newtonian to either the Cross equation with a low shear limit viscosity or the Bingham plastic equation with an apparent yield... 

This then gives a direct link between the thermodynamic stress and the osmotic pressure (or the compressibility) of the suspension. As a result of this stress, the viscosity will depend directly upon the structure, and the interpartide potential, V(ry). Using this interrelationship Batchelor has been able to evaluate the ensemble averages of both the mechanical and thermodynamic stresses by renormalizing the integrals. As a result, he has developed truncated series expressions for the low shear limit viscosity, and the high shear limit viscosity, t) , corresponding to... 

Figure 11. Relative low shear limit viscosity variation with volume fraction.

In deriving equation 32 or equation 33, it is assumed that 0max is the solid volume fraction at which the suspended particles cease moving. Thus, the forces, such as shearing, that can disturb the suspension structure and hence improve the mobility of particles will have an effect on the value of max. This is confirmed by the fact that a value of kH = 6.0 is observed at low shear limit, that is, y 0 and at high shear limit, y - oo, kH = 7.1 is found. Typical values of max have been found with the use of Quemada s equation as 0max = 0.63 0.02 in the low shear limit and high shear limit for submicrometersized sterically stabilized silica spheres in cyclohexane (72, 85, 88). 

Figure 13 shows a typical plot of the steady shear relative viscosity versus the Peclet number for polystyrene spheres of various sizes suspended in various fluids. The success of the Peclet number scaling is well observed. One can also observe that the viscosity is higher when the shear rate is small, and at both high and low shear limits, the viscosity curve shows a plateau, corresponding to the high and low shear limit Newtonian behavior. The explanation for this behavior has been, in part, discussed earlier for the random packing limit of the particles. 

Let the relative viscosity (normalized by the suspending fluid viscosity) as measured in the direction of the cylinder axis (longitudinal direction) as prL and the relative transverse viscosity be / rT. At the low shear limit and in a dilute system, the viscosity is expected to be isotropic. Eshelby (100) obtained... 

The Einstein constant for a fiber suspension is also a function of shear rate. The ratio of low to high shear limit Einstein constant is rj 1.17 for rods and re/31 for discs (114). The low shear limit Einstein constant is given by... 

Figure 17. Low shear limit viscosity variation with solid volume fraction and aspect ratio for suspensions of spheroids and cylinders (118).

The viscosity reduction by water addition is not due to the presence of the surfactant (HAB). For the sand-in-bitumen suspension, the viscosity variation is shown in Figure 31. It can be observed from Figure 31 that the sand-in-bitumen suspensions are slightly shear thinning type. A low shear limit viscosity is observed, although a yield stress may be assumed if Casson s model is used (194). The shear thinning behavior is more severe when the solid volume fraction is increased. 

One unique scaling behavior has been uncovered in this chapter for the low shear limit viscosity, equation 64, Bingham yield stress, equation 65, and the compressive yield stress-osmotic pressure, equation 74, for... 

For colloidal particles, the dimensionless parameters are generally small and non-Newtonian effects dominate. Considering the same example as above, but with particles of radius a = 1 /xm, the parameters take on the values Pe = y, N y = 10 y, and N = 10 y so that for shear rates of 0.1 s or less they are all small compared to unity. The limit where the values of the dimensionless forces groups are very small compared to unity is termed the low shear limit. Here the applied shear forces are unimportant and the structure of the suspension results from a competition between viscous forces. Brownian forces, and interparticle surface forces (Russel et al. 1989). If only equilibrium viscous forces and Brownian forces are important, then there is well defined stationary asymptotic limit. In this case, there is an analogue between suspensions and polymers which is similar to that for the high shear limit, wherein the low shear limit for suspensions is analogous to the zero-shear-rate viscosity limit for polymers. 

With surface forces absent, in the limit of Pe l, the distribution of particles is only slightly altered from the Einstein limit. To order cj> which takes into account two-particle interactions, Batchelor (1977) calculated the effect of Brownian motion on the stress field in a suspension of hard spheres and determined the low shear limit relative viscosity to be given by the Einstein relation with an added term equal to 6.14>. This result is found to agree satisfactorily with experiment for surface forces, including questions as to the existence of a uniquely defined asymptotic limit, we choose not to discuss this case further, instead referring the reader to Russel et al. (1989) and van de Ven (1989). 

Shear Rate Dependence. Experiments were performed on the In-stron rheometer and covered the shear-rate range 10 to 10 s Data for a 0.75-g dL" solution of polymer D-47 in 5 X 10 mol dm NaCl are shown in Figure 5. A low-shear limiting Newtonian viscosity, 1 (0), was observed. At high shear rates, the usual power-law behavior of polymers was observed with a power-law index of -0.86. This value is close to that predicted theoretically and observed for concentrated polymer solutions and undiluted polymers (13) and indicates an extensive network. 

This model includes four parameters the low-shear limiting viscosity, tie the high-shear limiting viscosity. a shear index. p and a time constant, An interesting aspect of this model is that, after convenient simplifications, different typical flow models can be obtained. As an example, let us consider the case for intermediate shear rates, in which rj. can be neglected) and the... 

Figure 11.2a shows that the low frequency limit of dynamic viscosity is very similar to the low shear limit of steady shear viscosity throughout the concentration sequence of isotropic to biphasic to anisotropic. At higher frequencies the dynamic viscosity peak is reduced in magnitude [24] for lyotropic cellulose esters + DMAC, just as viscosity is reduced at higher shear rates (Figure 11.2b). However the concentration of the peak was reported not to decrease with frequency. Thus it appears that dynamic shear, unlike steady shear, does not drive the thermodynamic transition for formation of an anisotropic phase to a lower concentration.

Normal stress measurements for some MLC nematics was reported to be consistent with that of a second-order fluid, that the low frequency limit of G /co equaled the low shear limit of N /(dy/dty [36]. Coleman and Markowitz demonstrated that for a second-order fluid in slow Couette flow, the viscoelastic contribution to the normal thrust must have a sign opposite to the inertial contribution on thermodynamic grounds [37]. A textbook by Walters stated that the measurements of first normal stress difference have invariably led to a positive quantity except for one case which was later found to be in error [38]. Adams and Lodge reported the possible observation of a negative value for Nj for solutions of poly isobutylene + decalin [39]. This result was obtained by a combination of obtained from radial... 

---