Steady Shear Viscosity and Normal Stresses

The simplest case to consider is steady flow of a dilute suspension of Newtonian drops or bubbles in a Newtonian medium. If the capillary number y a / F is small, so that the drops or bubbles do not deform under flow, then at steady state the viscosity of the suspension is given by Taylor s (1932) extension of the Einstein formula for solid spheres  

For higher concentrations, outside of the dilute regime. Pal (1992) has proposed an empirical equation for the zero-shear viscosity of an emulsion  

Here rjr.o = Va/Vs is the zero-shear-rate relative viscosity, and 

If the droplet or bubble is deformable (Ca is not negligible), then the emulsion will be viscoelastic, even if the two fluids composing it are both Newtonian. The characteristic 

For a viscosity ratio M of order unity or less, r is the relaxation time of the droplet shape and of the resulting viscoelastic stress in the dispersion. Thus, for rjs I cP, F 10 dyn/cm, a (xm, we obtain t 10 sec, and the stress in the emulsion relaxes almost instantly after cessation of flow. However, when the fluid medium is more viscous and the droplets are bigger, say, rjg 100 P, a 10 jxm, with the same F, we obtain T 0.01 sec. Although this latter value of r corresponds to rather fast relaxation, it can produce appreciable normal stress differences at high shear rates thus (Choi and Schowalter 1975) 

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Phan-Thien and Tanner used data of Meissner for the low-density polyethylene in Figure 9.9 to test the model, with = 0.2 and s = 0.01. The fit to steady shear viscosity and normal stresses is good, and the transients are fit reasonably well the shear data are insensitive to the value of s as expected. The fit to the extensional data at three stretch rates is shown in Figure 9.13. The agreement is quite good, especially when it is recalled that there is probably an experimental artifact at short times that causes the data to be high. The model predicts an approach to a steady extensional viscosity that scales as 1/e, but the data do not extend sufficiently far to test the prediction. 

Steady Shear Viscosity and Normal Stress Difference... 

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