Simple elongational flow

A simple elongational flow is developed as the filament is stretched with the following components of the rate of deformation ... 

For a Newtonian fluid in a simple elongational flow, the constitutive equation becomes... 

The same authors studied polypropylene samples PP-H-N, PP-H-R-B, PP-M-N, and PP-M-B in low-speed isothermal melt-spinning experiments at 180°C [93]. The apparent viscosities measured in that experiment are given in Figure 3.9. As seen, the viscosities are larger than those measured in simple elongational flow. Also, all the viscosities decrease with elongation rate, although those of the narrow samples do not decrease as rapidly as those of the broad-distribution samples. 

FIGURE 3.10 Time dependence of the elongational viscosity at the startup of simple elongational flow for two polystyrene samples (PS III and PS IV). (Reprinted by permission of the publisher from Miindstedt, 1980.)... 

In packed beds of particles possessing small pores, dilute aqueous solutions of hydroly2ed polyacrylamide will sometimes exhibit dilatant behavior iastead of the usual shear thinning behavior seen ia simple shear or Couette flow. In elongational flow, such as flow through porous sandstone, flow resistance can iacrease with flow rate due to iacreases ia elongational viscosity and normal stress differences. The iacrease ia normal stress differences with shear rate is typical of isotropic polymer solutions. Normal stress differences of anisotropic polymers, such as xanthan ia water, are shear rate iadependent (25,26). 

As opposed to stagnant elongational flow which necessitates highly specific flow geometries, transient elongational flow is readily obtained with some simple arrangements which will be described below. 

In simple shear flow where vorticity and extensional rate are equal in magnitude (cf. Eq. (79), Sect. 4), the molecular coil rotates in the transverse velocity gradient and interacts successively for a limited time with the elongational and the compressional flow component during each turn. Because of the finite relaxation time (xz) of the chain, it is believed that the macromolecule can no more follow these alternative deformations and remains in a steady deformed state above some critical shear rate (y ) given by [193] (Fig. 65) ... 

All these features can be rationalized with the simple model of chain scission through frictional loading previously mentioned. The series of experiments performed in transient elongational flow and reported in this review show that... 

Viscoelastic properties have been discussed in relation to molar mass, concentration, solvent quality and shear rate. Considering the molecular models presented here, it is possible to describe the flow characteristics of dilute and semi-dilute solutions, as well as in simple shear flow, independent of the molar mass, concentration and thermodynamic quality of the solvent. The derivations can be extended to finite shear, i.e. it is possible to evaluate T) as a function of the shear rate. Furthermore it is now possible to approximate the critical conditions (critical shear rate, critical rate of elongation) at which the onset of mechanical degradation occurs. With these findings it is therefore possible to tune the flow features of a polymeric solution so that it exhibits the desired behaviour under the respective deposit conditions. 

The degree of deformation and whether or not a drop breaks is completely determined by Ca, p, the flow type, and the initial drop shape and orientation. If Ca is less than a critical value, Cacri the initially spherical drop is deformed into a stable ellipsoid. If Ca is greater than Cacrit, a stable drop shape does not exist, so the drop will be continually stretched until it breaks. For linear, steady flows, the critical capillary number, Cacrit, is a function of the flow type and p. Figure 14 shows the dependence of CaCTi, on p for flows between elongational flow and simple shear flow. Bentley and Leal (1986) have shown that for flows with vorticity between simple shear flow and planar elongational flow, Caen, lies between the two curves in Fig. 14. The important points to be noted from Fig. 14 are these ... 

Elongational flow is more effective than simple shear flow for a given viscosity ratio. 

For Ca > Cacri, a drop continually stretches until it breaks. If Ca > KCacr , where k is about 2 for simple shear flow and 5 for elongational flow (Janssen, 1993), the drop undergoes affine deformation, i.e., the drop acts as a material element, and it is stretched into an extended cylindrical thread with length L and radius R according to... 

Rgure 1.10. Condition of mptur-ing in quasi static-conditions for (a) a simple shear flow and (b) a pure elongational flow. (Adapted from [139].)... 

A simple RIS model of polymers in elongational flows is developed and used to analyze the coil stretching and chain retraction as a function of polymer and flow parameters. The results are in agreement with available experimental data on dilute polymer solutions in strong elongation flows. 

If the mixing device generates a simple shear flow, as shown in Fig. 3.23, the maximum separation forces that act on the particles as they travel on their streamline occur when they are oriented in a 45° position as they continuously rotate during flow. However, if the flow field generated by the mixing device is a pure elongational flow, such as shown in Fig. 3.24, the particles will always be oriented at 0° the position of maximum force. 

Three kinds of viscometric flows are used by rheologists to obtain rheological polymer melt functions and to study the rheological phenomena that are characteristic of these materials steady simple shear flows, dynamic (sinusoidally varying) simple shear flows, and extensional, elongational, or shear-free flows. 

Figure E7.2 compares a stepwise increase in interfacial area in simple shear flow with optimal initial orientation, and simple shear flow where, at the beginning of each step, the interfacial area element is placed 45° to the direction of shear. The figure shows that, whereas in the former case the area ratio after four shear units is 4.1, in the latter case the ratio is 6.1, with a theoretical value of 7.3 when the 45° between the plane and direction of shear is maintained at all times. We note, however, that it is quite difficult to generate steady extensional flows for times sufficiently long to attain the required total elongational strain. This is why a mixing protocol of stepwise stretching and folding (bakers transformation) is so efficient. Not only does it impose elongational stretching, but it also distributes the surface area elements over the volume.

Fig. 7.23 Critical capillary number for droplet breakup as a function of viscosity ratio p in simple shear and planar elongational flow. [Reprinted by permission from H. P. Grace, Chem. Eng. Commun., 14, 2225 (1971).]...

Deformation of a Sphere in Various Types of Flows A spherical liquid particle of radius 0.5 in is placed in a liquid medium of identical physical properties. Plot the shape of the particle (a) after 1 s and 2 s in simple shear flow with y 2s1 (b) after 1 s and 2 s in steady elongational flow with e = 1 s 1. (c) In each case, the ratio of the surface area of the deformed particle to the initial one can be calculated. What does this ratio represent ... 

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