Biaxial stretching flow

If each of the three epsilons are zero we recover the boundary conditions of planar Couette (shear) flow. For y = 0, there are three basic elongation flow fields defined as follows if one of the epsilons is also zero, the flow field is referred to as planar elongation flow if one of the epsilons is negative while the other two are positive and equal, the flow field is referred to as biaxial stretching flow and if one of the epsilons is positive while the other two are negative and equal, the flow field is referred to as uniaxial stretching flow. 

B.3 Squeezing Flow Between Lubricated Disks. Biaxial stretching flow can be generated in lubricated squeezing flow which is shown in Figure 3.36. A thin layer of lubricant applied to the upper and lower disks prevents the fluid from sticking to the plates. 

Compression of a weakly structured food between parallel plates may achieve squeezing flow (Steffe, 1996). When lubricated parallel plates are used, the result is a form of biaxial extension. Biaxial extension may be used to measure biaxial viscosity, which is a reflection of resistance to radial stretching flow in a plane. Lubricated squeezing flow of a semi-solid... 

Figure 9.2. Energy consumption as a function of change of the interface area in extensional (uniaxial, biaxial and plane strain) and simple shear flows. The most efficient is the biaxial stretch, the worst (by a factor of 500,000 ) is shear (after Erwin, 1991).

Furthermore, when this flow is looked upon as biaxial stretching, it is thought of as being generated by a radial, tensile stress rather than an axial, compressive stress. Thus, the biaxial extensional viscosity is defined as... 

The melt bubble is stretched vertically and circumferentially by a factor of 2 or more, so that an initial melt thickness of about 1 mm is reduced to between 250 and 100 pm. In the biaxial tensile flow, the melt stress in the hoop (H) direction can be calculated from the pressure p inside the bubble, the current bubble radius r and thickness t, using Eq. (C.22) of Section C.3. 

The only way to generate data for this type of unsteady biaxial tensile flow is to instrument a blown film machine. The tensile viscosity, defined by Eq. (5.16), hardly changes with the tensile strain rate. Figure 5.15 shows data for the uniaxial stretching of an LDPE and an HDPE. The apparent tensile viscosity increases with strain rate for the more elastic LDPE, in contrast with the non-Newtonian reduction in viscosity in shear flows. 

Well known examples of orientation induced crystallization are the stretching of rubber, fiber orientation in polyamides and polyesters, and the biaxial stretching of polycarbonate filuK. An example of thermal effects during induct crystallization in rubber is shown in Fig. 45, where the heat flow (Q) for successive steps of elongation and retraction is plotted as a function of time. 

After the tube has been biaxially stretched, the film needs to be annealed. This process can be done in several ways (Fig. 10). The most common is to insert into the tube a set of parallel bars. These bars have air flowing out on their outer surfaces that allow the film to slide over them. These bars are in an oven or ir-heating chamber. As the tube reaches the end of the annealing zone, it is split open and sent to the winder. 

The orientation of polydomain polymers by mechanical or viscous flow fields can be achieved easiest if a macroscopic chain anisotropy that coincides with the local symmetry of the LC phase structure is induced and fixed by chemical crosslinking. Eor nematic or Sa main chain polymers which locally show a prolate (see Sect. 3) chain anisotropy, a uniaxial deformation leads to a globally prolate chain conformation. If the chain conformation of the LC polymer is locally oblate, a globally oblate chain conformation can be induced by either uniaxial compression or -equivalently - biaxial stretching of the sample (Lig. 9). 

Uniaxial extension is not the only kind of stretching flow that one can visualize. If Cj = 2 = Cfi, 3 = 2 b, and 0, one has equal biaxial extension physically this can be realized in a sheet-stretching experiment. If one dimension of the sheet is constrained not to change so that 2, say, is zero, then 3 = i. This is called planar extension or pure shear. 

Another stretching flow that has been used to characterize the nonlinear behavior of melts is equibiaxial extension, usually called simply biaxial extension. This flow can be generated by clamping a circular sample around its rim and stretching it radially, as demonstrated by Hachmann and Meissner [160]. The biaxial strain gg is given by ... 

The two functions and 2 depend on the parameter b as well as on the elongation rate e in steady shear-free flows. When b = 0, the function 2 is zero, and is replaced by the symbol tj, which is called the elongational viscosity . The elongational viscosity describes the resistance to elongational flow if 8 is positive and the resistance to biaxial stretching if e is negative (the terms extensional viscosity and Trouton viscosity have also been used for ). 

Flow is generally classified as shear flow and extensional flow [2]. Simple shear flow is further divided into two categories Steady and unsteady shear flow. Extensional flow also could be steady and unsteady however, it is very difficult to measure steady extensional flow. Unsteady flow conditions are quite often measured. Extensional flow differs from both steady and unsteady simple shear flows in that it is a shear free flow. In extensional flow, the volume of a fluid element must remain constant. Extensional flow can be visualized as occurring when a material is longitudinally stretched as, for example, in fibre spinning. When extension occurs in a single direction, the related flow is termed uniaxial extensional flow. Extension of polymers or fibers can occur in two directions simultaneously, and hence the flow is referred as biaxial extensional or planar extensional flow. 

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