Non-Newtonian effects

There are possible different forms of analytic representation of the t versus j dependence with an allowance made for non-Newtonian effects. To expose the decisive factors of this dependence an approach suggested in [19] deserves much attent-... 

Viscosity of 12% PVB solutions. Solutions were prepared with 0.1185Kinematic viscosities were measured in a size 400 Cannon-Zhukov viscometer. In this viscometer, the apparent shear rate at the wall, y =T /iy, ranged from 2.5 to 40 sec". It is possible tliat non-Newtonian effects were present even at these low shear rates. However, such effects would only alter the data quantitatively—but not qualitatively—and the qualitative features of the data are the focus of this investigation. For further details about equipment and procedures, see Ref. 35. 

To take into consideration these non-Newtonian effects, it is common to use a viscosity which is a function of the strain rate and temperature to calculate the stress tensor in eqn. (2.42)... 

Incorporating both the effect of convection in the Him and the temperature dependence of the viscosity into the model improves the agreement between predictions and experimental measurements. It should be noted, however, that experimental conditions were such that viscous dissipation was insignificant and the temperature drop across the film was relatively small. Consequently, non-Newtonian effects, and effects due to the temperature dependence of viscosity, were less significant than were convection effects. This may not be the case in many practical situations, in particular with polymers, whose viscosity is more temperature sensitive than that of HDPE. 

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. 

Figure 9.13 Number averaged diameter of droplets d of polypropylene (M = 60,000) in polystyrene (M = 200,000) as a function of wt% polypropylene mixed in three different mixers at a nominal shear rate of around 65 sec and T — 200°C. The viscosities of the the PP and PS under these conditions are 840 and 950 Pa-s, respectively. The interfacial tension F is 5.0 dyn/cm. The error bars represent the distribution of droplet sizes, and they encompass one standard deviation in each direction from the mean. The deviation from the Taylor limit at low concentrations is attributed to non-Newtonian effects, while the increase in droplet size at higher concentrations is attributed to droplet coalescence. Note that similar droplet sizes are obtained in all three different mixers. (Reprinted with permission from Sundararaj and Macosko, Macromolecules 28 2647. Copyright 1995, American Chemical Society.)...

Now, if the Reynolds number of the flow is sufficiently small for the creeping-motion approximation to apply, it can be shown by the arguments of Subsection B.3 in Chap. 7 that no lateral motion of the drop is possible unless the drop deforms. In other words, Us = Useiin this case, though, of course, Us is not generally equal to the undisturbed velocity of the fluid evaluated at the X3 position of the drop center. The drop may either lag or lead (in principle) because of a combination of interaction with the walls and the hydrodynamic effect of the quadratic form of the undisturbed velocity profile - see Faxen s law. Because the drop deforms, however, lateral migration can occur even in the complete absence of inertia (or non-Newtonian) effects. In this problem, our goal is to formulate two... 

In steady-state shear flow the rigid dumbbells give a shear-rate dependent t] and 6, whereas these non-Newtonian effects are absent for the elastic dumbbells. 

The zero-shear viscosity rip is defined as the melt viscosity in the limit of y=0, and is a function of T and Mw. It is important to keep in mind, however, that rp) is very often not measured directly, but extrapolated from measurements at low shear rates. Such extrapolations can introduce an error in the value of r o if the range of shear rates used in the extrapolation is sufficiently high for non-Newtonian effects to begin manifesting themselves. 

Kaloni used Oldroyd model, Schtimmer a fourth order fluid model, while Wissler a nonhnear Maxwell model Employing the perturbation method, the authors observed that the inclusion of second-order perturbation terms (which bring in the non-Newtonian effects) predicted velocity profiles with superimposed secondary circulation patterns. 

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. 

In the last section we introduced the concept of two asymptotic viscosity limits for shear thinning colloidal suspensions as a function of shear rate. One is the high shear limit which corresponds to high values of the Peclet number where viscous forces dominate over Brownian and interparticle surface forces. Generally this limit is attained with non-colloidal size particles since to achieve large Peclet numbers by increase in shear rate alone requires very large values for colloidal size particles. In this limit, non-Newtonian effects are negligible for colloidal as well as non-colloidal particles. 

Figure 20. Flow resistance vs eynolds number (Re) for a PAA copolymer of ultrahigh molecular weight (M 27 X JO ) in 0.5 M NaCl a, first run b, second run after forcing the solution through the porous medium at Re = 15 c, third run after forcing the solution through at Re = 20 d, Newtonian fluid. The onset Reynolds number where the non-Newtonian effect sets in (Reo) is indicated for curve a. (After ref. 22.)...

HPAA in Pore Flow. The preceding findings lead us to reinterpret results from porous media and capillary entrance flow. Strong non-Newtonian effects have previously been attributed to the increase in extensional viscosity associated with the coil-stretch transition, occurring when the strain rate exceeds the reciprocal of the lowest order conformational relaxation time (47-49),... 

Nevertheless, a method to quantify this non-Newtonian effect was needed. The method we have adopted is as follows. The solvent contributions... 

As concentration is increased, the elongational flow behavior reveals the important role of molecular interactions, which in strong flow fields can occur at much lower concentrations than generally realized. These interactions are interpreted as the development of transient networks such networks often are responsible for extreme dilatant behavior in extensional flow. Many anomalous non-Newtonian effects reported previously in flows that contain appreciable elongational components parallel these phenomena, particularly pore flow, and are themselves due to the existence of transient networks. 

To accommodate non-Newtonian effects, a realistic constitutive equation is used for the shear-rate dependent viscosity (21-23). 

From Jenekhe s data (12) on what appears to be the same polyamic acid solution as that used in (6), the newtonian Region (p - 1.4 Pa s) persists out to a shear rate of about 300s". This indicates the flow is always in the newtonian region for the experiments reported in (6). It is, of course, difficult to estimate the boundary between newtonian and non-newtonian effects in this flow since as the solids content of the resist increases, non-newtonian behavior may be expected to set in at a lower shear rate. However, from equation (17), the shear rate experienced by the fluid decreases as well due to the increased viscosity. In order to resolve this question, more data is needed on the variation of the rheological behavior with solids content. 

Proceeding from the estimates of possible viscosity variations caused by non-Newtonian effects and rheokinetic phenomena, it is obvious that the effect of non-Newtonian factors on flow regularities always plays a minor role as a correction factor. The case, when the non-Newtonimi behavior affects qualitatively shear flow, for example, if a reactant acquires the abiUty to slip along e wall of a channel, will be discussed in detail later. 

Regardless of breakup morphology, [17] demonstrated that early drop motion obeys a constant acceleration model. Therefore, (6.6) and (6.8) can be applied directly to the calculation of the initial drop trajectory. However, (6.7) requires modification for the case of non-Newtonian liquids. Unfortunately, experimental deformation data is currently unavailable. Analytical models, such as the TAB model or its derivatives, discussed in Chap. 7, could be modified to include purely viscous or viscoelastic non-Newtonian effects. However, this has yet to be done and as a result the accuracy of such a modification is unknown. 

Non-Newtonian effects are important in numerous microfluidics applications. In some cases, microfluidic devices have been designed specifically to characterize the molecular properties and the response of a non-Newtonian liquid in a flow. Other applications involve the use of polymer molecules for specific functions, such as gene amplification and sequencing, in which the non-Newtonian behavior may also play an important role. Still others have used non-Newtonian effects to design functional microfluidic elements. We highlight a few examples here. 

Non-Newtonian Effects on Particle Size in Mixing Systems... 

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