Nonlinear viscosity

Heat Exchangers Using Non-Newtonian Fluids. Most fluids used in the chemical, pharmaceutical, food, and biomedical industries can be classified as non-Newtonian, ie, the viscosity varies with shear rate at a given temperature. In contrast, Newtonian fluids such as water, air, and glycerin have constant viscosities at a given temperature. Examples of non-Newtonian fluids include molten polymer, aqueous polymer solutions, slurries, coal—water mixture, tomato ketchup, soup, mayonnaise, purees, suspension of small particles, blood, etc. Because non-Newtonian fluids ate nonlinear in nature, these ate seldom amenable to analysis by classical mathematical techniques. 

The polymerization is carried out at temperatures of 0—80°C in 1—5 h at a soHds concentration of 6—12%. The polymerization is terminated by neutralizing agents such as calcium hydroxide, calcium oxide, calcium carbonate, or lithium hydroxide. Inherent viscosities of 2-4 dL/g are obtained at 3,4 -dianiinodiphenyl ether contents of 35—50 mol %. Because of the introduction of nonlinearity into the PPT chain by the inclusion of 3,4 -dianiinodiphenyl ether kinks, the copolymer shows improved tractabiUty and may be wet or dry jet-wet spun from the polymerization solvent. The fibers are best coagulated in an aqueous equiUbrium bath containing less than 50 vol % of polymerization solvent and from 35 to 50% of calcium chloride or magnesium chloride. 

Viscosity can also be determined from the rising rate of an air bubble through a Hquid. This simple technique is widely used for routine viscosity measurements of Newtonian fluids. A bubble tube viscometer consists of a glass tube of a certain size to which Hquid is added until a small air space remains at the top. The tube is then capped. When it is inverted, the air bubble rises through the Hquid. The rise time in seconds may be taken as a measure of viscosity, or an approximate viscosity in mm /s may be calculated from it. In an older method that is commonly used, the rate of rise is matched to that of a member of a series of standards, eg, with that of the Gardner-Holdt bubble tubes. Unfortunately, this technique employs a nonlinear scale of letter designations and may be difficult to interpret. 

PEs, as other polymers, exhibit nonlinear behavior in their viscous and elastic properties under practical processing conditions, i.e., at high-shear stresses. The MFI value is, therefore, of little importance in polymer processing as it is determined at a fixed low-shear rate and does not provide information on melt elasticity [38,39]. In order to understand the processing behavior of polymers, studies on melt viscosity are done in the high-shear rate range viz. 100-1000 s . Additionally, it is important to measure the elastic property of a polymer under similar conditions to achieve consistent product quality in terms of residual stress and/or dimensional accuracy of the processed product. 

Nonlinear viscosity growth with increase in concentration up to limiting filling of volume... 

However, dendrimeric and hyperbranched polyesters are more soluble than the linear ones (respectively 1.05, 0.70, and 0.02 g/mL in acetone). The solution behavior has been investigated, and in the case of aromatic hyperbranched polyesters,84 a very low a-value of the Mark-Houvink-Sakurada equation 0/ = KMa) and low intrinsic viscosity were observed. Frechet presented a description of the intrinsic viscosity as a function of the molar mass85 for different architectures The hyperbranched macromolecules show a nonlinear variation for low molecular weight and a bell-shaped curve is observed in the case of dendrimers (Fig. 5.18). 

The behavior of liquid flow in micro-tubes and channels depends not only on the absolute value of the viscosity but also on its dependence on temperature. The nonlinear character of this dependence is a source of an important phenomenon - hydrodynamic thermal explosion, which is a sharp change of flow parameters at small temperature disturbances due to viscous dissipation. This is accompanied by radical changes of flow characteristics. Bastanjian et al. (1965) showed that under certain conditions the steady-state flow cannot exist, and an oscillatory regime begins. 

In fluid dynamics the behavior in this system is described by the full set of hydrodynamic equations. This behavior can be characterized by the Reynolds number. Re, which is the ratio of characteristic flow scales to viscosity scales. We recall that the Reynolds number is a measure of the dominating terms in the Navier-Stokes equation and, if the Reynolds number is small, linear terms will dominate if it is large, nonlinear terms will dominate. In this system, the nonlinear term, (u V)u, serves to convert linear momentum into angular momentum. This phenomena is evidenced by the appearance of two counter-rotating vortices or eddies immediately behind the obstacle. Experiments and numerical integration of the Navier-Stokes equations predict the formation of these vortices at the length scale of the obstacle. Further, they predict that the distance between the vortex center and the obstacle is proportional to the Reynolds number. All these have been observed in our 2-dimensional flow system obstructed by a thermal plate at microscopic scales. ... 

These two examples show that regular patterns can evolve but, by definition, dissipative structures disappear once the thermodynamic equilibrium has been reached. When one wants to use dissipative structures for patterning of materials, the dissipative structure has to be fixed. Then, even though the thermodynamic instability that led to and supported the pattern has ceased, the structure would remain. Here, polymers play an important role. Since many polymers are amorphous, there is the possibility to freeze temporal patterns. Furthermore, polymer solutions are nonlinear with respect to viscosity and thus strong effects are expected to be seen in evaporating polymer solutions. Since a macromolecule is a nanoscale object, conformational entropy will also play a role in nanoscale ordered structures of polymers. 

Recently, we explored the effect of molecular weight on the pattern and employed post-dewetting processes to alter the shape of the dewetted polymer droplets. Since the viscosity of a polymer solution is nonlinear with respect to concentration and also strongly dependent on polymer weight, we expected a drastic effect. Figure 11.4... 

Viscosity Behavior. The polymeric nature of triorganotin fluorides dissolved in nonpolar solvents is outlined in the introduction. As a result of the transient polymer formation, these solutions exhibit nonlinear concentration vs. viscosity curves. 

Viscosity data are reported in Table I for a number of the polysaccharide derivatives in 5% LiCl/N,N-dimethylacetamide solutions. At low concentrations of polymers, an upward curvature in the DSp/c (reduced viscosity) vs c (concentration) plot was observed. Additionally, nonlinear increases in solvent viscosity were observed for increased lithium ion concentrations in the absence of polymer. Therefore, reduced viscosities at 0.25 dl/g are reported. 

The results showed that for all nozzle diameters the decrease in bubble volume with an increase in the continuous phase velocity is nonlinear. Further, for any particular velocity, the reduction in the bubble volume is greater as the liquid viscosity is increased. 

The disadvantage of the power law model is that it cannot predict the viscosity in the zero-shear viscosity plateau. When the zero-shear viscosity plateau is included, a nonlinear model must be specified with additional fitting parameters. A convenient model that includes the zero-shear viscosity and utilizes an additional parameter is the Cross model [30] ... 

With the least polar solvent, 9 1 MIBK/MEOH, aggregation dominates the viscosity behavior. This solvent is of intermediate quality, between pure MeOH and the 1 1 mixture. Still, the viscosity is greatest using the 9 1 mix at all temperatures, by up to a factor of four. The effect of temperature on the aggregation in the 9 1 MIBK/MeOH solution is so large that the fmj versus 1/T curve becomes significantly nonlinear. An apparent E determined when... 

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