Extensional flow

Another class of flows, in which the fluid is not sheared, is known as shear-free flows (Bird et a/., 1987a). In this case all of the off-diagonal elements in the rate of strain tensor are zero and it has form  

In practice, the significance of both shear and elongational flow is appreciated when considering the flow of a polymer through a capillary with 

For a Newtonian liquid, the viscosity measured in a shearing flow can be used to predict the stress in other types of deformation. This, in general, is not so for complex fluids. In a 

The instantaneous extensional stress W(e, t) is the force F(e, t) along the cylinder axis required to pull the cylinder ends apart, divided by the instantaneous cross-sectional area A s, t) of the cylinder thus a(e, r) = F e, t)/A(e, t). The time-dependent extensional viscosity, rj(e, t), is then ct( , t)/e. If this viscosity reaches a time-independent value within the duration of the experiment, that value is called the steady-state extensional viscosity, r ( ). 

Many devices other than those mentioned above have been designed for measuring extensional viscosities. These include crossed-slot devices, opposed jets, and others described in Keller and Odell (1985) and Macosko (1994). 

It is well recognized that extensional flow is advantageous to deform and break up droplets with large viscosity difference. As shown in Fig. 7.9 [22], shear flow can lead to the droplet break-up only under the condition of a viscosity ratio of up to 3.8. This suggests that one should utilize sufficient extensional flow as well as high shear flow [22]. For this reason, the simulation of extensional flow to obtain the optimum screw design and the use of an additional, extensional mixer are considered as significant tools [23]. 

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. 

---

Extensional flows occur when fluid deformation is the result of a stretching motion. Extensional viscosity is related to the stress required for the stretching. This stress is necessary to increase the normalized distance between two material entities in the same plane when the separation is s and the relative velocity is ds/dt. The deformation rate is the extensional strain rate, which is given by equation 13 (108) ... 

Section B This section is tapered so there will be pressure losses due to both shear and extensional flows. 

D. H. King, D. F. James. Analysis of the Rouse model in extensional flow. J Chem Phys 72 4749 754, 1983. 

R. G. Larson. The unraveling of a polymer chain in a strong extensional flow. Rheol Acta 29 371-384, 1990. 

The Giesekus criterion for local flow character, defined as

simple shear flow and — 1 in solid body rotation [126]. The mapping of J> across the flow domain provides probably the best description of flow field homogeneity current calculations in that direction are being performed in the authors laboratory. 

Muller AJ (1989) Extensional flow of macromolecules in solution, Ph D Thesis, University of Bristol, Physics Department... 

Muller AJ, Odell JA, Carrington S (April 1991) In Degradation of polymer solutions in extensional flow, Proceedings of the polymer physics a Conference to mark the retirement of A Keller, Bristol UK 3-5... 

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. 

As previously discussed in earlier topics, extensional flow occurs when the material is not in contact with solid boundaries, as is the case during drawing of filaments, film, sheets, or inflating... 

It is well known that LCB has a pronounced effect on the flow behavior of polymers under shear and extensional flow. Increasing LCB will increase elasticity and the shear rate sensitivity of the melt viscosity ( ). Environmental stress cracking and low-temperature brittleness can be strongly influenced by the LCB. Thus, the ability to measure long chain branching and its molecular weight distribution is critical in order to tailor product performance. 

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. 

The same flow-aligning side-chain liquid crystalline polymer has been studied [43] in extensional flow using a rheo-NMR method in which selective excitation of... 

Extensional flow describes the situation where the large molecules in the fluid are being stretched without rotation or shearing [5]. Figure 4.3.3(b) illustrates a hypothetical situation where a polymer material is being stretched uniaxially with a velocity of v at both ends. Given the extensional strain rate e (= 2v/L0) for this configuration, the instantaneous extensional viscosity r e is related to the extensional stress difference (oxx-OyY), as... 

Fig. 13. Streamlines and velocity profiles for two-dimensional linear flows with varying vorticity. (a) K = -1 pure rotation, (b) K = 0 simple shear flow, (c) K = 1 hyperbolic 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. 

In the context of the preceding model, a drop is said to break when it undergoes infinite extension and surface tension forces are unable to balance the viscous stresses. Consider breakup in flows with D mm constant in time (for example, an axisymmetric extensional flow with the drop axis initially coincident with the maximum direction of stretching). Rearranging Eq. (26) and defining a characteristic length Rip113, we obtain the condition, for a drop in equilibrium,... 

Fig. 18. Graphical interpretation of the criterion for breakup of pointed drops in an extensional flow (Khakhar and Ottino, 1986c).

For the case of a thread breaking during flow, the analysis is complicated because the wavelength of each disturbance is stretched along with the thread. This causes the dominant disturbance to change over time, which results in a delay of actual breakup. Tomotika (1936) and Mikami et al. (1975) analyzed breakup of threads during flow for 3D extensional flow, and Khakhar and Ottino (1987) extended the analysis to general linear flows. Each of these works uses a perturbation analysis to describe an equation for the evolution of a disturbance. 

Illustration Breakup time for threads during flow. Consider a thread being deformed in a 2D extensional flow, with the following material and process parameters ... 

Consider drops of different sizes in a mixture exposed to a 2D extensional flow. The mode of breakup depends on the drop sizes. Large drops (R > Caa,tal/xcy) are stretched into long threads by the flow and undergo capillary breakup, while smaller drops (R Cacri,oV/vy) experience breakup by necking. As a limit case, we consider necking to result in binary breakup, i.e., two daughter droplets and no satellite droplets are produced on breakup. The drop size of the daughter droplets is then... 

Fig. 20. Radius of drops produced on capillary breakup in hyperbolic extensional flow (Rdrops), radius of the thread at which the disturbance that causes breakup begins to grow (Rent), and the time for growth of the disturbance (fgrow) for different values of the dimensionless parameters p and /xc feao/tr. The time for capillary breakup of the extending thread ((break) can be obtained from these graphs (see Illustration for sample calculations) (Janssen and Meijer, 1993).

Fig. 22. Radius of drops produced by capillary breakup (solid lines) and binary breakup (dotted lines) in a hyperbolic extensional flow for different viscosity ratios (p) and scaled shear rate (p,cylo) (Janssen and Meijer, 1993). The initial amplitude of the surface disturbances is ao = 10 9 m. Note that significantly smaller drops are produced by capillary breakup for high viscosity ratios.

Oscillatory shear experiments are the preferred method to study the rheological behavior due to particle interactions because they directly probe these interactions without the influence of the external flow field as encountered in steady shear experiments. However, phenomena that arise due to the external flow, such as shear thickening, can only be investigated in steady shear experiments. Additionally, the analysis is complicated by the different response of the material to shear and extensional flow. For example, very strong deviations from Trouton s ratio (extensional viscosity is three times the shear viscosity) were found for suspensions [113]. 

---