Average shear viscosity

The combined mass flow for rotation-driven and pressure-driven flow is given by Eq. 1.28 and is the expected rate of the process. The average shear viscosity is calculated using the average shear rate in the channel for screw rotation and the bulk temperature. This method is also known as the generalized Newtonian method. 

A Newtonian shear behavior for the PVP solutions can be obtained at low shear rates, while higher concentrated PVP K30 solutions, for example, 45 m%, show shear thinning behavior at shear rate higher than 1000 s . The extensional viscosities of K30 are depicted in Fig. 19.1. The plateau value for the elongational viscosity at higher strain rates is characterized by the Trouton factor for Newtonian fluids where the extensional viscosity (plateau value) scales by a constant factor concerning the average shear viscosity. The theoretical values about three are represented by dashed lines. 

As the time scales for atomization are much smaller than the time scales for evaporation, the thermal energy transfer into the liquid might be beneficial. In rheometric measurements for PVP K30, shear viscosity decreases with increasing temperature [47]. A raise in liquid temperature by 50 K decreases the average shear viscosity at low shear rates to half of the value at room temperature. For Newtonian K30 solutions, the extensional viscosity is reduced due to temperature increase by the same factor as the shear viscosity. For K90, it has to be considered that the resistance towards elongation cannot be related to its shear viscosity. 

The shear viscosity is a tensor quantity, with components T] y, t],cz, T)yx> Vyz> Vzx> Vzy If property of the whole sample rather than of individual atoms and so cannot be calculat< with the same accuracy as the self-diffusion coefficient. For a homogeneous fluid the cor ponents of the shear viscosity should all be equal and so the statistical error can be reducf by averaging over the six components. An estimate of the precision of the calculation c then be determined by evaluating the standard deviation of these components from tl average. Unfortunately, Equation (7.89) cannot be directly used in periodic systems, evi if the positions have been unfolded, because the unfolded distance between two particl may not correspond to the distance of the minimum image that is used to calculate the fore For this reason alternative approaches are required. 

Polymer solutions are often characterized by their high viscosities compared to solutions of nonpolymeric solutes at similar mass concentrations. This is due to the mechanical entanglements formed between polymer chains. In fact, where entanglements dominate flow, the (zero-shear) viscosity of polymer melts and solutions varies with the 3.4 power of weight-average molecular weight. 

The limiting low shear or 2ero-shear viscosity T q of the molten polymer can be related to its weight-average molecular weight, by the same... 

The average Nusselt number, Nu, is presented in Fig. 4.10a,b versus the shear Reynolds number, RCsh- This dependence is qualitatively similar to water behavior for all surfactant solutions used. At a given value of Reynolds number, RCsh, the Nusselt number, Nu, increases with an increase in the shear viscosity. As discussed in Chap. 3, the use of shear viscosity for the determination of drag reduction is not a good choice. The heat transfer results also illustrate the need for a more appropriate physical parameter. In particular. Fig. 4.10a shows different behavior of the Nusselt number for water and surfactants. Figure 4.10b shows the dependence of the Nusselt number on the Peclet number. The Nusselt numbers of all solutions are in agreement with heat transfer enhancement presented in Fig. 4.8. The data in Fig. 4.10b show... 

The surface shear viscosity of a monolayer is a valuable tool in that it reflects the intermolecular associations within the film at a given thermodynamic state as defined by the surface pressure and average molecular area. These data may be Used in conjunction with II/A isotherms and thermodynamic analyses of equilibrium spreading to determine the phase of a monolayer at a given surface pressure. This has been demonstrated in the shear viscosities of long-chain fatty acids, esters, amides, and amines (Jarvis, 1965). In addition,... 

We would expect a difference between the average time and that which determines the onset of shear thinning since the low shear viscosity tends to be dominated by the longest mode of relaxation. 

The z average molecular weight has been found to correlate with the shear viscosity of polymer melts when the molecular weight distribution is very broad and where very large molecules appear to dominate the resistance to fluid flow. 

The shear viscosity shown in Fig. 3.30 is for a polymer with MJM = 1.5, and it is for the same weight average molecular weight and temperature as in Fig. 3.29. At a temperature of 543 K, the resin shown in Fig. 3.30 has the Newtonian to power law transition beginning at about 10 1/s. By decreasing the polydispersity the transition has moved almost two orders of magnitude higher. The narrow distribu-... 

Two different calculation methods were used for the simulations (1) the generalized Newtonian method as developed above, and (2) the three-dimensional numerical method presented in Section 7.5.1. The generalized Newtonian method used a shear viscosity value that was based on the average barrel rotation shear rate and temperature in the channel. The average shear rate based on barrel rotation (7ft) is provided by Eq. 7.52. Barrel rotation shear rate and the generalized Newtonian method are used by many commercial codes, and that is why it was used for this study. 

Now that the temperature is known at the /c+1 position, the pressure gradient at the end of the volume can be estimated using Eq. 7.105 as follows by evaluating the viscosity at temperature T, i and the average shear rate in the channel using Eq. 7.41 ... 

In this way, the relaxation times can be calculated when the weight average molecular weight of the polymer and its (zero shear) viscosity are known (Mt and p are the free parameters of this equation). 

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