Planar 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. 

This flow field is somewhat idealized, and cannot be exactly reproduced in practice. For example, near the planar surfaces, shear flow is inevitable, and, of course, the range of % and y is consequently finite, leading to boundary effects in which the extensional flow field is perturbed. Such uniaxial flow is inevitably transient because the surfaces either meet or separate to laboratory scale distances. 

Figure 3.2 shows planar extensional flow generated by the uniform stretching of a thin wide sheet or film in one direction only, while allowing the thickness in the perpendicular direction to decrease. Thus, / 11 = —e33 and 22 = 0. Therefore, m = 0 in Eq. 3.1-1, giving... 

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. 

Example 14.2 Inflation of a Cylindrical Uniform Parison Assuming Simple Planar Extensional Flow Following Denson (83), an approximate description of the inflation of a cylindrical parison of uniform radius R, and thickness hl to that R0 and h0 can be obtained by assuming that (a) the flow is a planar extension (b) the flow is isothermal and (c) /i/rhoop stress t — PR(i)/h t). Experimentally, planar extensional visc-... 

Winter et al. [119, 120] studied phase changes in the system PS/PVME under planar extensional as well as shear flow. They developed a lubrieated stagnation flow by the impingement of two rectangular jets in a specially built die having hyperbolic walls. Change of the turbidity of the blend was monitored at constant temperature. It has been found that flow-induced miscibility occurred after a duration of the order of seconds or minutes [119]. Miscibility was observed not only in planar extensional flow, but also near the die walls where the blend was subjected to shear flow. Moreover, the period of time required to induce miscibility was found to decrease with increasing flow rate. The LCST of PS/PVME was elevated in extensional flow as much as 12 K [120]. The shift depends on the extension rate, the strain and the blend composition. Flow-induced miscibility has been also found under shear flow between parallel plates when the samples were sheared near the equilibrium coexistence temperature. However, the effect of shear on polymer miscibility turned out to be less dramatic than the effect of extensional flow. The cloud point increased by 6 K at a shear rate of 2.9 s. ... 

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

Planar extensional flow or pure shear flow is extensional flow with the same but opposite rates of strain in two directions in the third direction, there is no flow ... 

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. 

MIXED FLOW. Other flows with extensional components also have coil-stretch transitions. The smaller the extensional component is relative to the overall strain rate, the higher the overall strain rate at which the transition takes place (Giesekus 1962, 1966) A steady planar flow, for example, can be considered to be a mixture of a shearing and an extensional flow in such a mixed flow, the velocity gradient tensor, Vv, can be expressed as (Fuller and Leal 1980, 1981)... 

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). 

This formula applies to planar extensional flow as well as to shear, if the shear rate y in Eq. (9-11) is replaced by 2e, where is the extension rate. Taylor predicted that droplet breakup should occur when the viscous stresses that deform the droplet overwhelm the surface tension forces that resist deformation this occurs when D reaches a value Db given approximately by... 

Figure 9.7 Photographs of droplet shapes in planar extensional flow for various viscosity ratios M of the dispersed to the continuous phase. The droplets are viewed in the plane normal to the velocity gradient direction. The critical capillary numbers Cac and droplet deformation parameters Dc at breakup are also given. The droplet fluids are silicon oils with viscosities ranging from 5 to 60,000 centistokes, while the continuous fluids are oxidized castor oils both phases are Newtonian. (From Bentley and Leal 1986, with permission from Cambridge University Press.)...

The first observation of shear-induced increase of the LCST was reported for PS/PVME by Mazich and Carr [1983]. The authors concluded that shear stress can enhance miscibility by 2-7°C. Larger effects, AT < 12°C, were reported for the same system in hyperbolic flow [Katsaros et al., 1986]. In a planar extensional flow at 8 = 0.012 - 26 s the phase separated PS/PVME was homogenized at temperatures 3 to 6°C above... 

The extensional flow is a deformation that involves stretching along streamlines. According to the resulting deformation, it can be classified as uniaxial, biaxial, or planar extension. 

It is now possible to compare the changes in interfacial area produced in unidirectional shear, equation (11.7), uniaxial elongation, equation (11.19) and planar extension, equation (11.20). However, as pointed out be Cheng, this comparision needs to be done on a rational basis. For example, it is possible to examine the area ratios at equal strain defined by y, = Te = 7pe and it is clear from this viewpoint that effectiveness in generating new surface area increases in the order simple shear, uniaxial extoision to planar extension. It is clear that at large strains the advantage is heavily in favour of the extensional flows. If a value of strain equal to 10 is considered (Mohr states that most shear mixers exceed this value) then the area ratios SISn are given in Table 11.1. However,... 

Flow-Based Particle Trapping and Manipulation, Fig. 7 Time-lapse image of the relaxation dynamics of an individual single-stranded DNA molecule (ssDNA, L = 18 jm) studied by the hydrodynamic trap based on a planar extensional flow. A single fluorescently labeled... 

Planar extensional flow is the kind of flow where there is no deformation in one direction and the velocity field is represented as follows ... 

Extensive reviews [6-10] and a monograph [11] summarize the literature covering significant aspects of extensional flows in various commercial processes, theoretical treatment for ttie hydrod)mamics of such flows and different methods of determining material functions such as uniaxial, biaxial and planar extensional viscosities. 

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. 

If we want to find out how a fluid behaves under extension, we have to somehow grip and stretch it. Experimentally, this is much more difficult than the shear arrangement, especially if the fluid has a low viscosity. Earlier (see Section 5) we saw that it is possible to classify steady extensional flows under the categories of uniaxial, biaxial and planar flows. We will now examine uniaxial testing, since this mode is more commonly employed as a routine characterization tool. Here we encounter two approaches the first seeks to impart a uniform extensional field and back out a true material function, while the second employs a mixed flow field that is rich in its extensional component (e.g. converging flows) and use it to back out a measured property of the fluid which is somehow related to its extensional viscosity. 

---