Extensional flow biaxial

K. Wikstrom, L. Bohlin and A.C. Eliasson, Biaxial Extensional Flow and Wheat Quality, Anal. Trans. Nordic Rheol. Soc., 2 32-34 (1994). 

Dimensional Changes in Planar and Biaxial Extensional Flows Determine the rate of dimensional changes that have to be applied on a flat film in order to generate (a) planar extension, and (b) biaxial extension flows. 

Other forms of extensional flow are biaxial extensional flow and planar extensional flow or pure shear flow. 

Biaxial extensional flow is extensional flow in two directions with the same constant extensional strain rate qe. If the volume is constant then... 

Figure 3-28 Biaxial Extensional Flow Test Setup of Campanella et al. (1987).

A plot showing streamlines for this flow with a spherical body at the origin (calculated later in this section) is shown in Fig. 7-13. For A > 0, there is flow outward away from the sphere along the axis of symmetry and flow inward in the plane orthogonal to this axis. This flow is called uniaxial extensional flow. For E < 0, the direction of fluid motion is reversed and the undisturbed flow is known as biaxial extensional flow. In either case, with an axially... 

Figure 7-13. The streamlines for axisymmetric flow in the vicinity of a solid sphere with uniaxial extensional flow at infinity. When the direction of motion is reversed at infinity, the undisturbed flow is known as biaxial extensional flow. The stream-function values are calculated from Eq. (7-185). Contour values are plotted in equal increments equal to 0.5.

Figure 9-9. A schematic sketch showing the flow patterns for (a) uniaxial extensional flow, (b) biaxial extensional flow, and (c) hyperbolic (or 2D) extensional flow. Each part of the figure shows the flow from two perspectives one along the z axis toward the xy plane and the other along the x axis toward the yz plane. In the first two cases, the flow is axisymmetric, with the x axis being the symmetry axis. In the 2D case, the flow is invariant in the z direction.

In biaxial extensional flow, the energy per unit volume is ... 

In biaxial extensional flow, too, the dimensions of the fluid elements change drastically but tfiey change in two directions as against the one-direction in uniaxial extensional flow. Thus, biaxial extensional flow can be visualized as a deformation caused by forces acting in two directions perpendicular to the opposite faces of a plate as shown in Figure 2.4. TTie velocity field in simple biaxial extensional flow is given by. 

Figure 2.4 Schematic diagram of a fluid element in biaxial extensional flow.

Denson, C.D. and Hylton, D.C. (1980) A rheometer for measuring flie viscoelastic response of pol)mier melts in arbitrary planar and biaxial extensional flow fields, Polym. Engg Sci., 20,535-9. 

The velocity field in simple biaxial extensional flow is given by... 

Sugimoto, M., Masubuchi, Y., Takimoto, J., Koyama, K. Melt rheology of polypropylene containing small amounts of high-molecular weight chain. 2. Uniaxial and biaxial extensional flow. Macromol... 

All aspects of crystallization dynamics of polymers, the strong dependence on the temperature, cooling rate, and the application of flow, are of relevance to highspeed FS (uniaxial extensional flow) [1-3,13] and during biaxial extensional flow, that is, flhn blowing [28,93-95]. 

There are significant differences in the behavior of polymeric fluids in these two types of deformation, and each type of deformation has a different effect on the orientation of macromolecules. For example, uniaxial and planar extensional flows impart significant molecular orientation in polymers during flow compared to shear flows. On the other hand, biaxial extensional flow is a weak flow and does not lead to a strong degree of molecular orientation. Furthermore, the rheological response can be significantly different for a polymer in extensional flow versus shear flow. We demonstrate these differences later in this chapter. 

Yamane, H., K. Sasai, M. Takano, and M. Takashi, Poly(D-lactic acid) as a Rheological Modifier of Poly(L-lactic acid) Shear and Biaxial Extensional Flow Behavior , J.Rheol, 48(3), 599-609 (2004)... 

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