Divergence shear viscosity

As a result, we find for sols that the divergence of the above zero shear viscosity rj0 and of two other linear viscoelastic material functions, first normal stress coefficient and equilibrium compliance 7°, depends on the divergence... 

Fig. 5. Schematic of the divergence of zero-shear viscosity, rj0, and equilibrium modulus, Ge. The LST is marked by pc...

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. 

The zero-shear viscosity and the dynamic viscosity (at low frequencies) diverge at high concentration, while they are constant at low concentration [99,100,102-105],... 

The divergence of the longest relaxation time does not perturb the measurement. In comparison, steady state properties (the steady shear viscosity, for instance) would probe an integral over all relaxation modes and, hence, fail near the gel point. 

Measurement of the equilibrium properties near the LST is difficult because long relaxation times make it impossible to reach equilibrium flow conditions without disruption of the network structure. The fact that some of those properties diverge (e.g. zero-shear viscosity or equilibrium compliance) or equal zero (equilibrium modulus) complicates their determination even more. More promising are time-cure superposition techniques [15] which determine the exponents from the entire relaxation spectrum and not only from the diverging longest mode. 

Measurement of the diverging steady shear viscosity is an appealing experiment because of its simplicity. Even the torque on a processing machine might serve as an estimate of the diverging viscosity. It has, however, severe disadvantages that need to be considered [149] ... 

However, this expression assumes that the total resistance to flow is due to the shear deformation of the fluid, as in a uniform pipe. In reality the resistance is a result of both shear and stretching (extensional) deformation as the fluid moves through the nonuniform converging-diverging flow cross section within the pores. The stretching resistance is the product of the extension (stretch) rate and the extensional viscosity. The extension rate in porous media is of the same order as the shear rate, and the extensional viscosity for a Newtonian fluid is three times the shear viscosity. Thus, in practice a value of 150-180 instead of 72 is in closer agreement with observations at low Reynolds numbers, i.e.,... 

The last terms in each of Eqn. (5.2-9) and Eqn. (5.2-10) represent the divergence of the deviatoric stress including viscosity and pseudo-turbulence. The quantities /n L and uG are effective viscosities of each phase including bulk and shear viscosities. Fl and Fg represent the volume-averaged forces exerted on the liquid-and gas-phase (respectively) by the other phases across the common interfaces. 

FIGURE 12.13 Low and high shear viscosity as a function of volume fraction, showing the divergence at different volume fractions. Data fi m Kreiger [42],... 

Biroad Size Distributioiis Suspensions of broad size distributions of ceramic particles show in general that the low shear viscosity, 7jo> and the high shear viscosity, 7j , diverge according to... 

A similar calculation for discxjtics yields Aq ca) co"i The original work of Mazenko et argued that one of the shear viscosities should also diverge, but Milner and Martin showed that this was not the case. 

Scaling laws hold not only for geometrical properties but also for viscoelastic properties. The steady-state (zero shear) viscosity (2.54) diverges as... 

Equilibrium Rheological Measurements. The appearance of an equilibrium modulus or the divergence of the steady shear viscosity might be used to estimate the position of the gel point by extrapolation. Extrapolation is necessary because these measurements fail in the close vicinity of GP. Measurement of the equilibrium modulus (51,52) is extremely difficult because its value remains below the detection limit for a considerable time and it, theoretically, requires an infinite time to perform the measurement. The diverging steady shear viscosity (53-55) indicates the location of GP. The simplicity of this experiment may lead to neglect of the severe shortcomings that can be summed up as follows ... 

In eq 7.102, the difinsion coefficient, which controls the life time of the fluctuations, is assumed to follow eq 7.101 for a Brownian particle, where the radius R is replaced by the correlation length The shear viscosity also diverges at the critical point as but very weakly, with z 0.07. ... 

Cheng Z.D., Zhu J.X., Chaikin P.M., Phan S.E., Russel W.B. Nature of the divergence in low shear viscosity of coUoidal hard-sphere dispersions. Phys. Rev. E 2002 65(4) A041405, 1-8 Chow M., Zukoski C.E. Gold sol formation mechanisms Role of coUoidal stability. J. CoUoid Interface Sci. 1994 165(1) 97-109... 

In the previous section, detail of the frequency power law was described for PVC solution as an example of experimental gelation system. Moreover, rheological studies are seen in this section by focusing on how the zero shear viscosity, Tjo, diverges when the gelling system goes close to the gel point from the solution of low concentration. 

If there is a phase transition from a nematic to a smectic phase, pretransitional effects are observed in the neighbourhood of the transition [9, 44, 49-51]. Figure 10 [9] shows this behaviour for 4-/7-octyloxy-4 -cyanobiphenyl (80CBP) with a nematic-smectic phase transition at 340.3 K. Pretransitional effects cause a divergence of the shear viscosity coefficient 772 and the rotational viscosity /j. 

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