Simple rotational flow

The study of flow and elasticity dates to antiquity. Practical rheology existed for centuries before Hooke and Newton proposed the basic laws of elastic response and simple viscous flow, respectively, in the seventeenth century. Further advances in understanding came in the mid-nineteenth century with models for viscous flow in round tubes. The introduction of the first practical rotational viscometer by Couette in 1890 (1,2) was another milestone. 

Concerning a liquid droplet deformation and drop breakup in a two-phase model flow, in particular the Newtonian drop development in Newtonian median, results of most investigations [16,21,22] may be generalized in a plot of the Weber number W,. against the vi.scos-ity ratio 8 (Fig. 9). For a simple shear flow (rotational shear flow), a U-shaped curve with a minimum corresponding to 6 = 1 is found, and for an uniaxial exten-tional flow (irrotational shear flow), a slightly decreased curve below the U-shaped curve appears. In the following text, the U-shaped curve will be called the Taylor-limit [16]. 

The Giesekus criterion for local flow character, defined as

extensional flow, 0 in 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. 

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) ... 

Mechanical rheometry requires a measurement of both stress and strain (or strain rate) and is thus usually performed in a simple rotating geometry configuration. Typical examples are the cone-and-plate and cylindrical Couette devices [1,14]. In stress-controlled rheometric measurements one applies a known stress and measures the deformational response of the material. In strain-controlled rheometry one applies a deformation flow and measures the stress. Stress-controlled rheometry requires the use of specialized torque transducers in conjunction with low friction air-bearing drive in which the control of torque and the measurement of strain is integrated. By contrast, strain-controlled rheometry is generally performed with a motor drive to rotate one surface of the cell and a separate torque transducer to measure the resultant torque on the other surface. 

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. 

Knowledge of the geometry and mathematical description of a screw Is required to understand the analysis of the functional sections of the screw and the troubleshooting of case studies. In Chapter 1 the geometry and mathematical descriptions are presented. Also In this chapter, the calculation of the rotational flow (also known as drag flow) and pressure flow rates for a metering channel Is Introduced. Simple calculation problems are presented and solved so that the reader can understand the value of the calculations. 

The ratio of the rotational flows indicated that the diameter of the extruder is not a factor for the deviation shown in Fig. 7.16. That is, for this channel with a small H/Wthe simplified analysis produces less than a 10% error when compared to the exact numerical solution. Thus, the rotational flow rate can be calculated quite reliably using the simple generalized Newtonian method at these conditions. 

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. 

In a complex, polymeric liquid, normal stresses as well as the shear stress can be present, and these contributions will influence the shape of the structure factor. The simplest rheological constitutive model that can account for normal stresses is the second-order fluid model [64], where the first and second normal stress differences are quadratic functions of the shear rate. Calculations using this model [92,93,94,90,60], indicate that the appearance of normal stresses can rotate the structure factor towards the direction of flow in the case of simple shear flow and can induce a four-fold symmetry in the case of exten-sional flow. 

Fig. E7.2 Schematic representation of interfacial area increase in simple shear flow, (a) With initial orientation of 45° to the direction of shear, and after each shear unit, the plane is rotated hack to 45° orientation, (b) With optimal initial orientation and no rotation.

Above the critical value, the viscous shear stresses overrule the interfacial stresses, no stable equilibrium exists, and the drop breaks into fragments. For p > 4, it is not possible to break up the droplet in simple shear flow, due to the rotational character of the flow. Figure 7.23 also indicates that in shear flow, the easiest breakup takes place when the... 

Melt conveying is the forward motion of the molten polymer through the extruder, due to the pumping action of the rotating screw. This simple drag flow Md is proportional to melt density, down-channel velocity, and cross-sectional area of the screw channel. In most cases, however, there is also a pressure gradient as the melt moves downstream, either... 

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