Deformations, extensional

Extensional Viscosity. In addition to the shear viscosity Tj, two other rheological constants can be defined for fluids the bulk viscosity, iC, and the extensional or elongational viscosity, Tj (34,49,100—107). The bulk viscosity relates the hydrostatic pressure to the rate of deformation of volume, whereas the extensional viscosity relates the tensile stress to the rate of extensional deformation of the fluid. Extensional viscosity is important in a number of industrial processes and problems (34,100,108—110). Shear properties alone are insufficient for the characterization of many fluids, particularly polymer melts (101,107,111,112). 

Rumscheidt and Mason [10] studied the drop fracture of Newtonian fluids according to the value of a dimensionless 8. They found that 0.1 < 6 < 1.0 is most favorable for extensional deformation and subsequently... 

Finally we will consider the two flow fields that yield extensional deformation. The first is axial elongational in which the fluid is contained between two planar surfaces in relative motion along their planar normals. With the axial extension direction taken as z, one has nzz = e = —2nxx = —2xyy and... 

However, this expression assumes that the total resistance to flow is due to the shear deformation of the fluid, as in a uniform pipe. In reality the resistance is a result of both shear and stretching (extensional) deformation as the fluid moves through the nonuniform converging-diverging flow cross section within the pores. The stretching resistance is the product of the extension (stretch) rate and the extensional viscosity. The extension rate in porous media is of the same order as the shear rate, and the extensional viscosity for a Newtonian fluid is three times the shear viscosity. Thus, in practice a value of 150-180 instead of 72 is in closer agreement with observations at low Reynolds numbers, i.e.,... 

Uniaxial extensional deformation of a material specimen of uniform cross-sectional area along its long axis by the continuous application of a sinusoidal force of constant amplitude. 

Below the yield point, however, stress/strain behaviour was found to be independent of initial cell orientation, due to the threefold symmetry of the hexagonal cellular array [54], This allows a correlation between shearing and extensional deformations to be made [55], namely that shear can be considered as elongation followed by rotation. Thus, information on one type of deformation can be obtained by solving expressions for the other. 

Pugnaloni, L.A., Ettelaie, R., Dickinson, E. (2005). Brownian dynamics simulation of adsorbed layers of interacting particles subjected to large extensional deformation. Journal of Colloid and Interface Science, 287, 401 114. 

In the following sections, I review a few, commonly-cited models of widespread compressional and extensional deformation and consider their respective elevation predictions. This review is not meant to be an exhaustive list of all possible mechanisms of elevation change due to continental deformation. Rather, the purpose is to illustrate the role of elevation as a parameter that can be used to constrain mechanisms of continental deformation. 

Fig. 12.19 Cold postextrusion micrographs as a function of the flow rate. The processing conditions were T = 177°C and no PPA. Each image is actually a composite of two micrographs in which the side and top are focused. The relative errors in throughputs are 0.05 Q = (a) 1.0, (b) 2.2, (c) 3.8, (d) 6.3, and (e) 11 g/min. The width of each image corresponds to 3 mm. [Reprinted by permission from K. B. Migler, Extensional Deformation, Cohesive Failure, and Boundary Conditions during Sharkskin Melt Fracture, J. Rheol., 46, 383 4-00 (2002).]...

Fig. 12.21 Sketch of the kinetics of the sharkskin instability, side view. [Reprinted by permission from K. B. Migler, Extensional Deformation, Cohesive Failure, and Boundary Conditions during Sharkskin Melt Fracture, J. Rheol., 46, 383 400 (2002).]...

The cohesive failure is brought about by the sudden and large axial acceleration of the melt layer next to the capillary wall, as shown by Migler et al. (49) in Fig. 12.22(a). Figure 12.22(b) shows that the addition of a flouroelastomer (PPA) additive allows the formation of a slip that is larger upstream from the exit, as shown on Fig. 12.23, which reduces the axial acceleration and the level of extensional deformation and rate. 

The mathematical formulation of the fiber-spinning process is meant to simulate and predict the hydrodynamics of the process and the relationship between spinning conditions and fiber structure. It involves rapid extensional deformation, heat transfer to the surrounding quenching environment, air drag on the filament surface, crystallization under rapid axial-orientation, and nonisothermal conditions. 

Recent advances in molecular dynamics simulations enabled Levine et al. (20) to take modeling one step further, to the molecular level. They succeeded in simulating from first principles the structure formation of 100 carbon atom polyethylene during uniaxial extension, under a variety of conditions. Figure 14.9 shows the dynamics of extensional deformation below the melting point, beautifully indicating the dynamic development of orientation and order. 

In the extruder, not only shear flow is present, but also extensional flow occurs as well. This is illustrated in Fig. 3.20 for the deformation of a fluid element. Wherever cross-sections narrow, such as at the tips or between kneading blocks and the wall, the fluid elements are compressed and extended. This effect is particularly relevant for non-homogenous polymer melts, e.g., immiscible blends, in which the disperse phase can be split by extensional deformation. For more details, see Chapter 9. 

From these results, Everage and Ballman (1974) concluded that melt fracture originates at a point where fluid elements are subjected primarily to extensional deformation. They could correlate the results of Ferrari with a critical extension rate of about 1000 s-1. 

Extensional deformation of polymer solutions is applied technically in the so-called dry spinning of polymer fibres. The literature data in this field are not as numerous as in the field of shear viscosity, so that only a qualitative picture can be given here. 

For concentrated polymer solutions, the behaviour in extensional deformation shows a great correspondence to that of polymer melts. At low rates of deformation the extensional viscosity has the theoretical value of three times the shear viscosity. At higher rates of deformation, the experimental results show different types of behaviour. In some cases, the extensional viscosity decreases with increasing rate of extension in the same way as the shear viscosity decreases with increasing shear rate. In other cases, however, a slight increase of the extensional viscosity with increasing rate of extension was observed. 

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