Equivalent Newtonian viscosity

E. Broyer, C. Gutfinger, and Z. Tadmor, Evaluating Flows of a Non-Newtonian Fluid by the Method of Equivalent Newtonian Viscosity, AIChE J., 21, 198-200 (1975). 

The constant 49 is applicable to the selected standard fluid for scaling, but the exponent is independent of the standard chosen. Further, its low value means that to double the perceived thickness would require a 20-fold increase in the objective value of N- From Equation 7.3, one can derive an expression for the perceived thickness of Newtonian foods that can be used to compare equivalent Newtonian viscosity with... 

The effective oral shear rates at which the objective viscosity was equal to the equivalent Newtonian viscosity, calculated from the above equation were in good agreement with the results of Shama and Sherman (1973) (Figure 7-7) (Cutler et al., 1983) the only exceptions to this observation were the highly shear-thinning xanthan gum dispersions that deviated slightly from the Shama and Sherman s results. 

One of the simplest and cheapest viscometers on the market today is the Brookfield viscometer based on a thin disc rotating in large volume of liquid. While the viscosity is normally quoted as the equivalent Newtonian viscosity as a function of rotation speed in rpm for a given spindle, it is possible to evaluate the power-law parameters k and n for suitable liquids using the following equations [4]... 

Many on-line instruments are calibrated with Newtonian liquids, so that the results are presented as equivalent Newtonian viscosity—this can be far from the truth Very few viscometers give the actual imambiguous (non-Newtonian) viscosity as a function of the... 

The Cambridge piston viscosity sensor (Cambridge Applied Systems, Cambridge, MA, USA) is ingenious in that the piston is lifted to the top of a slot let into the side waU of a pipe, and allowed to drop, and the cycle then being repeated. The speed of faU is then related to the viscosity. The novelty here is that the action of the cylinder moving up and down which pumps fresh liquid in and out of the tube containing the piston. However, the one number so obtained can only )tield the equivalent Newtonian viscosity. 

B.7 Equivalent Newtonian Viscosity. It has been suggested by Broyer and co-workers (1975) that the solutions to non-Newtonian flow problems can be obtained by using the Newtonian solution with fx replaced by an equivalent Newtonian viscosity, fc. For isothermal flow between parallel plates carry out the following ... 

As mentioned earlier, ideally the best answer should come from the determination of the 5-function. Unfortunately at present the rotary diffusion coefficient is usually the least reliable quantity in all hydrodynamic measurements because of errors inherent in the physical methods of flow birefringence and perhaps also non-Newtonian viscosity (see Section IV). (Electric birefringence also may not give the same rotary diffusion coefficient as the other two methods, since the equivalent ellipsoids can be different under shearing stress and under electrical field.) Edsall (1954) has also illustrated the impossibility of evaluating the axial ratio from the 5-function. The latter was about 0.80 for fibrinogen which corresponded to a prolate ellipsoid with an axial ratio of more than 300. If the rotary diffusion coefficient were only about 15% greater than that listed in Table V the calculated axial ratio would decrease to between ten and twenty. 

The theory of non-Newtonian viscosity for ellipsoidal particles was first explicitly stated by Kuhn and Kuhn (1945), using Peterlin s distribution function (Peterlin, 1938) and Jeffery s hydrodynamic treatment (Jeffery, 1922-1923) [Eq. (10)]. More elegant treatments have recently been developed by Saito (1951), using the same ellipsoidal model, and also by Kirkwood and his co-workers (Kirkwood, 1949 Kirkwood and Auer, 1951 Kirkwood and Plock, 1956 Riseman and Kirkwood, 1956) for rodlike particles. The equivalence of the three theories has also been demonstrated by Saito and Sugita (1952). The general solution of Eq. (10) for the viscosity increment, v, can be expressed in the form... 

In this review we have briefly discussed the theoretical and experimental aspects of both Newtonian and non-Newtonian viscosities of polymer solutions. To protein chemists one of the interesting developments is no doubt the re-examination of the (Newtonian) viscosity treatments of protein solutions. There are many assumptions involved in the effective use of intrinsic viscosity measurements for evaluating the asymmetry of the protein molecules, however attractive the conventional treatment may have appeared for the past two decades. Carefully interpreted, the intrinsic viscosity (at zero gradient) can still provide a reasonable estimate of the axial ratios of the protein molecules. The concept of equivalent hydrodynamic volume, sound in principle, has put the viscometry of protein solutions in a proper perspective, although the quantitative aspects of this new approach still... 

Several attempts have been made to derive predictive relationships for the Newtonian viscosity of semidilute and concentrated polymer solutions. Simha and co-workers [Simha, 1952 Utracki and Simha, 1963 Simha and Somcynsky, 1965 Simha and Chan, 1971 Utracki and Simha, 1981] explored the possibility of developing a principle of corresponding states based on the c,M) scaling equivalent to the packing of hard spheres ... 

Figure 5, pertadning to the reduced temperature = 2, shows the behavior typical for the isotropic phase, viz. a first Newtonian viscosity, r/ = 1.1 in reduced units for smadl shear rates, a strong shear thinning for intermediate ones and the approach to the second Newtonian viscosity r/iao = 0.1 for high shear rates. Calculations with imposed sheaur rate and imposed shear stress give equivalent results. 

Thus, the rate of increase of the shear strain (shear rate, for short) is equivalent to a velocity gradient. If the magnitude of the shear rate imder steady-flow conditions is directly proportional to the magnitude of the shear stress, the material is a linear viscous (Newtonian) fluid and the proportionality constant is the (Newtonian) viscosity ... 

Effective viscosity is a term used to relate non-Newtonian behavior to that of some hypothetical equivalent Newtonian fluid, just as is apparent viscosity. However, the term effective viscosity really relates more to the behavior of the bulk of the fluid and is, as a concept, equal to the Newtonian viscosity that would give the same relationship. ... 

Distributed Parameter Models Both non-Newtonian and shear-thinning properties of polymeric melts in particular, as well as the nonisothermal nature of the flow, significantly affect the melt extmsion process. Moreover, the non-Newtonian and nonisothermal effects interact and reinforce each other. We analyzed the non-Newtonian effect in the simple case of unidirectional parallel plate flow in Example 3.6 where Fig.E 3.6c plots flow rate versus the pressure gradient, illustrating the effect of the shear-dependent viscosity on flow rate using a Power Law model fluid. These curves are equivalent to screw characteristic curves with the cross-channel flow neglected. The Newtonian straight lines are replaced with S-shaped curves. 

Procedure Use a suitable cone and plate rheometer [Con-traves Rheomat 115A (cone CP-6), Physica Rheolab MC 100 (cone MK23), or equivalent] maintained at 37.8° and capable of measuring the non-Newtonian flow curve hysteresis for ascending and descending shear rates programmed from 0 to 800 s-1. Hold the rheometer at 0 s-1 for 120 s, raise it to 800 s-1 in 7.5 min, hold for 1 s, then decrease to 0 s-1 in 7.5 min to measure the thixotropic area. Check the accuracy of the rheometer with viscosity standards (Cannon ASTM Certified Viscosity Standards, S-2000 and N-350, or equivalent). The measured viscosity must be within 0.20% of the stated viscosity at 37.8°, or the rheometer s cone factor must be recalculated. 

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