Uniaxial extensional flow

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. 

Flows that produce an exponential increase in length with time are referred to as strong flows, and this behavior results if the symmetric part of the velocity gradient tensor (D) has at least one positive eigenvalue. For example, 2D flows with K > 0 and uniaxial extensional flow are strong flows simple shear flow (K = 0) and all 2D flows with K < 0 are weak flows. 

M. J. Menosveta and D. A. Hoagland, Light scattering from dilute poly(styrene) solutions in uniaxial extensional flow, Macromolecules, 24,3427 (1991). 

Figure 3.2 (Case 1) shows a simple uniaxial extensional flow created by the uniform stretching of a rectangular or a thin filament in the 1 direction. For this flow, 22 = - n /2, and because of the incompressibility assumption, 22 = 33. Thus, in Eq. 3.1-1, m = —0.5, giving the following rate of deformation matrix...

For this simple uniaxial extensional flow to be steady, the instantaneous rate of change of the 1 direction length (/) must be constant... 

Thus, in order to create a steady simple uniaxial extensional flow, the rheometer must cause the thin filament length to increase exponentially in time. 

Agglomerate breakup will occur at Z > 2 in shear flow in biaxial extensional and uniaxial extensional flow, it will occur at Z > 1 and Z > 0.5, respectively. Breakup does not depend on agglomerate size, but on the size of the primary particle. Clearly, the smaller the primary particle is, the higher the shear stresses needed to reach breakup. It is worth noting... 

It has already been shown before that in uniaxial extensional flow or stretching flow the length in the direction of the flow changes according to... 

The maximum strain rate (e < Is1) for either extensional rheometer is often very slow compared with those of fabrication. Fortunately, time-temperature superposition approaches work well for SAN copolymers, and permit the elevation of the reduced strain rates kaj to those comparable to fabrication. Typical extensional rheology data for a SAN copolymer (h>an = 0.264, Mw = 7 kg/mol,Mw/Mn = 2.8) are illustrated in Figure 13.5 after time-temperature superposition to a reference temperature of 170°C [63]. The tensile stress growth coefficient rj (k, t) was measured at discrete times t during the startup of uniaxial extensional flow. Data points are marked with individual symbols (o) and terminate at the tensile break point at longest time t. Isothermal data points are connected by solid curves. Data were collected at selected k between 0.0167 and 0.0840 s-1 and at temperatures between 130 and 180 °C. Also illustrated in Figure 13.5 (dashed line) is a shear flow curve from a dynamic experiment displayed in a special format (3 versus or1) as suggested by Trouton [64]. The superposition of the low-strain rate data from two types (shear and extensional flow) of rheometers is an important validation of the reliability of both data sets. 

A few rheometers are available for measurement of equi-biaxial and planar extensional properties polymer melts [62,65,66]. The additional experimental challenges associated with these more complicated flows often preclude their use. In practice, these melt rheological properties are often first estimated from decomposing a shear flow curve into a relaxation spectrum and predicting the properties with a constitutive model appropriate for the extensional flow [54-57]. Predictions may be improved at higher strains with damping factors estimated from either a simple shear or uniaxial extensional flow. The limiting tensile strain or stress at the melt break point are not well predicted by this simple approach. 

Birefringence setups can be designed to characterize molten materials undergoing isothermal homogeneous flow. The ranges of strains and strain rates also often coincide with those of rheometers, and consequently may be limited relative to those used in fabrication. Similarly, time-temperature superposition approaches may be used to expand the rate window. State-of-the-art setups suitable for rapid screening of new materials with research-scale quantities (5-20 g) are available for shear flow [72] and startup of uniaxial extensional flow [73,74]. 

M.Takahashi, T.Isaki, T.Takigawa, T.Masuda, Measurement of biaxial and uniaxial extensional flow of polymer melts at constant strain rates, J. Rheol. 31 (1993), 827-846. 

While these functions have been adjusted to describe shear and uniaxial extensional flows, they seem to work poorly for planar extension of LDPE (Samurkas et al. 1989). Planar extensional flow represents a particularly difficult test for K-BKZ-type constitutive equations, since fits to shear data fix all the model parameters required for planar extension, and there is therefore no wiggle room left to obtain a fit to the latter. (This is because I = I2 in both shear and planar extension.) A recent non-K-BKZ molecular constitutive equation derived from reptation-related ideas shows improved qualitative agreement with planar extensional data (McLeish and Larson 1998). 

For a steady uniaxial extensional flow, the velocity gradient tensor is given by Eq. (1-9) ... 

For a steady uniaxial extensional flow, we can assume by symmetry that the stress tensor contains only diagonal components. We can then evaluate the terms in Eq. (A3-13) containing the velocity gradient by using Eq. (A3-1) ... 

Since Vv is a diagonal tensor, and hence symmetric, the term 0 Vv equals Vv Thus, for a steady uniaxial extensional flow, Eq. (A3-13) can be written as... 

While the fiber contribution to the steady-state stress tensor at steady-state is modest for shearing flow, its contribution to the stress in extensional flow is large at steady state. In a uniaxial extensional flow, the fibers orient in such a way that the viscous dissipation is maximized. Large values of the extensional viscosity are the result from Batchelor s (1971) theory the uniaxial extensional viscosity is... 

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.

Problem 9-9. Solid Sphere in a Uniaxial Extensional Flow at Re solid sphere whose center is located at the origin so that... 

However, rheological measurements are also performed with other types of flow or stress fields. If a uniaxial extensional flow field is applied to a material, the stress distribution can be described by... 

During shear or uniaxial extensional flow, the initially spherical drop deforms into a prolate ellipsoid with the long axis, a and two orthogonal short axes... 

It is convenient to express the capillarity number in its reduced form K = K / K, where the critical capillary number, K., is defined as the minimum capillarity number sufficient to cause breakup of the deformed drop. Many experimental studies have been carried out to establish dependency of K on X. For simple shear and uniaxial extensional flow, De Bruijn [1989] found that droplets break most easily when 0.1 4 ... 

The uniaxial extensional flow, with regard to describing both the deformation and the resulting stresses, is uniform shear free flow, in which the strain rate is the same for every material element, and there is no relative... 

Figure 5.1 5 Tensile stress vs. tensile strain rate (log scales) for the uniaxial extensional flow of LDPE, ethylene propylene copolymer and PMMA,...

Fihpe, S., M. Gidade, and J. Maia. 2006. Uniaxial extensional flow behavior of immiscible and compatibUized polypropylene/liquid crystalline polymer blends. Rheologica Acta 45 281-289. 

For simplicity, consider the behavior of an incompressible fluid element which is being elongated at a constant strain rate e in the x-direction (uniaxial extensional flow). For an incompressible fluid, the volume of the element must remain constant and therefore it must contract in both the y- and z-directions at the rate of (e/2) for a system symmetrical in these directions. The three components of the velocity vector Vi are given by... 

L.A. Utracki, P. Saimnut, On the uniaxial extensional flow of polystyrene/polyethylene blends. Polym. Eng. Sci. 30(17), 1019-1026 (1990a)... 

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