Viscoelasticity secondary flows

Many materials of practical interest (such as polymer solutions and melts, foodstuffs, and biological fluids) exhibit viscoelastic characteristics they have some ability to store and recover shear energy and therefore show some of the properties of both a solid and a liquid. Thus a solid may be subject to creep and a fluid may exhibit elastic properties. Several phenomena ascribed to fluid elasticity including die swell, rod climbing (Weissenberg effect), the tubeless siphon, bouncing of a sphere, and the development of secondary flow patterns at low Reynolds numbers, have recently been illustrated in an excellent photographic study(18). Two common and easily observable examples of viscoelastic behaviour in a liquid are ... 

Ide and White W studied the viscoelastic effects in agitating polystyrene solutions with a turbine. At concentrations below 50% PS, flow was normal. Abovfe 35%, the viscoelastic forces caused the flow to reverse, moving away from the impeller along the axis. At 30 to 35% PS, both occurred, causing a segregated secondary flow around the turbine. 

Flow irregularities at gap angles of 30° were observed in viscoelastic liquids [94]. It has been indicated in theoretical treatments that the possibility of secondary flows [96,97] in rotational devices is to be expected if the gap angle is much greater than 5°. For viscoelastic fluids deviations from laminar flow have only been reported in cone-and-plate geometries with gap angles above 10°. 

Almost every biological solution of low viscosity [but also viscous biopolymers like xanthane and dilute solutions of long-chain polymers, e.g., carbox-ymethyl-cellulose (CMC), polyacrylamide (PAA), polyacrylnitrile (PAN), etc.] displays not only viscous but also viscoelastic flow behavior. These liquids are capable of storing a part of the deformation energy elastically and reversibly. They evade mechanical stress by contracting like rubber bands. This behavior causes a secondary flow that often runs contrary to the flow produced by mass forces (e.g., the liquid climbs the shaft of a stirrer, the so-called Weissenberg effect ). 

The same statement can be made about inelastic non-Newtonian fluids, such as the Power Law fluid, from a mathematical solution point of view. In reality, most non-Newtonian fluids are viscoelastic and exhibit normal stresses. For fluids such as those (i.e., fluids described by constitutive equations that predict normal stresses for viscometric flows), theoretical analyses have shown that secondary flows are created inside channels of nonuniform cross section (78,79). Specifically it can be shown that a zero second normal stress difference is a necessary (but not sufficient) condition to ensure the absence of secondary flow (79). Of course, the analyses of flows in noncircular channels in terms of constitutive equations—which, strictly speaking, hold only for viscometric flows—are expected to yield qualitative results only. Experimentally low Reynolds number flows in noncircular channels have not been investigated extensively. In particular, only a few studies have been conducted with fluids exhibiting normal stresses (80,81). Secondary flows, such as vortices in rectangular channels, have been observed using dyes in dilute aqueous solutions of polyacrylamide. Interestingly, these secondary flow vortices (if they exist) seem to have very little effect on the flow rate. 

It is well known in polymer rheology that a torsional parallel-plate flow cell develops certain secondary flow and meniscus distortion beyond some stress level [ 14]. For viscoelastic melts, this can happen at an embarrassingly low stress. The critical condition for these instabilities has not been clearly identified in terms of the shear stress, normal stress, and surface tension. It is very plausible that the boundary discontinuity and stress intensification discussed in Sect. 4 is the primary source for the meniscus instability. On the other hand, it is well documented that the first indication of an unstable flow in parallel plates is not a visually observable meniscus distortion or edge fracture, but a measurable decay of stress at a given shear rate [40]. The decay of the average stress can occur in both steady shear and frequency-dependent dynamic shear. 

The validity of Eq. 10.62 has been confirmed by the experiments of Wheeler and Wissler [50], Hartnett et al. [54], and Hartnett and Kostic [55] for fully developed laminar flow of aqueous polymer solutions in rectangular channels (Fig. 10.7). Given the fact that these solutions are viscoelastic, a number of analytical studies that take elasticity into account predict that the presence of normal forces produces secondary flows [56-60]. However, these analytical studies, along with the previously cited pressure drop measurements, indicate that if such secondary flows exist, they have little effect on the laminar friction factor. 

Mashelkar, R. A. and G. V. Devarajan, Secondary Flows of non-Newtonian Fluids Pt Il-Frictional Losses in Laminar Flow of Purely Viscous and Viscoelastic Fluid through Coiled Tube, Trans, Instn. Chem. Engrs., 54, 108-114 (1976). 

Figure 10.15. Flow of a high-impact polystyrene resin (Dow Styron 484) through a 61-cm channel with a square cross section that is 0.9525 cm on a side. A special feedblock brings in concentric layers, which are colored for contrast, (a) At the channel entrance, (b) At the channel exit, (c) Computed secondary flow. Reprinted with permission from J. Dooley, Viscoelastic Flow Effects in Multilayer Polymer Coextrusion, Ph.D. dissertation. Technical University of Eindhoven, the Netherlands, 2002.

Deiber and Schowalter (1981) found an increased resistance to flow over that predicted for a purely viscous fluid and concluded that this was because of elasticity and not because of secondary flow. Lagrangean unsteadiness was caused by the variations in the amplitude of the diameter, and hence they concluded that the Deborah number alone cannot predict the onset of viscoelastic effects. Gupta and Sridhar (1985) considered theoretically the... 

Vortices are induced by the viscoelastic characteristics of the converging fluid [71,72]. When extensional viscosity values are large, the balance of forces makes radial flow impossible, thereby giving rise to secondary flows in the form of vortices in order to provide stress relief. 

Xue S C, Phan-Thien N and Tanner RI (1995), Numerical study of secondary flows of viscoelastic fluid in straight pipes by an implicit finite volume method . Journal of Non-Newtonian Fluids Mechanics, 59,191-213. 

This model has been found to be useful in studying the slow motion of viscoelastic fluids around particles, droplets and bubbles, and in predicting the directions of secondary flows in rotating systems. Furthermore it is very helpful in providing a framework for the presentation of kinetic theory results obtained by perturbation theories. The retarded motion expansion is not, however, useful for most industrial flow problems, in which large velocity gradients or rapid time responses are generally encountered. 

Material parameters defined by Equations (1.11) and (1.12) arise from anisotropy (i.e. direction dependency) of the microstructure of long-chain polymers subjected to liigh shear deformations. Generalized Newtonian constitutive equations cannot predict any normal stress acting along the direction perpendicular to the shearing surface in a viscometric flow. Thus the primary and secondary normal stress coefficients are only used in conjunction with viscoelastic constitutive models. 

From the results obtained in [344] it follows that the composites with PMF are more likely to develop a secondary network and a considerable deformation is needed to break it. As the authors of [344] note, at low frequencies the Gr(to) relationship for Specimens Nos. 4 and 5 (Table 16) has the form typical of a viscoelastic body. This kind of behavior has been attributed to the formation of the spatial skeleton of filler owing to the overlap of the thin boundary layers of polymer. The authors also note that only plastic deformations occurred in shear flow. 

TTie presence of crystallites is very important for product performance because crystallites serve as physical crosslinking points which reinforce material and improve its viscoelastic properties. It is therefore very important to notice that the mechanism peculiar to PVC allows for melting of crystallites and their formation on cooling. Melting of crystallites improves PVC processability (lowers melt viscosity and improves its flow), and recovery of crystallites on cooling improves performance characteristics of PVC. The gelation depends on the melting enthalpies of primary and secondary crystallites. 

Polypropylene melts are viscoelastic fluids. As such, the melts exhibit non-Newtonian viscosity, normal stresses in shear flow, excessive entrance-and-exit pressure drop, die swell, secondary entrance flows, melt fracture, and draw resonance. (Newtonian fluids also exhibit draw resonance.) Polypropylene melts are more viscoelastic than melts of nylon and polyester. 

Creep krep [ME crepen, fr. OE creopan akin to ON krjupa to creep] (before 12c) vt. Due to its viscoelastic nature, a plastic subjected to a load for a period of time tends to deform more than it would from the same load released immediately after application. The degree of this deformation increases with the duration of the load and with rising temperature. Creep is the permanent deformation resulting from the prolonged apphcation of a stress below the elastic limit. This deformation, after any time under stress is partly recoverable (primary creep) upon the release of the load and partly unrecoverable (secondary creep). Creep at room temperature is sometimes called cold flow. Elias HG (2003) An introduction to plastics. John Wiley and Sons, New York. Rosato DV (ed) (1992) Rosato s plastics encyclopedia and dictionary. Hanser-Gardner Publications, New York. 

To help understand and quantitatively evaluate the secondary movement shown above, Debbaut et al. [75, 77] augmented this experimental work with a three-dimensional flow simulation that incorporated viscoelastic effects. The finite element method, using a 4-mode Giesekus model as the viscoelastic constitutive equation, was used for the simulation. The polymer used for the experiment and simulation was a low-density polyethylene. Figures 12.20 and 12.21 show the experimental observations and the numerical predictions of the deformations of the interface for the rectangular straight channel [78], and for the teardrop channel [75], respectively. 

The viscoelastic forces that produce the remarkable manifestations illustrated above are properly characterised in shear flow by the so-called first and second normal-stress differences, Ni and N2, which occur in addition to the shear stress CT (with which we are already familiar) —note that occasionally Ni and N2 are called the primary and secondary normal stress differences. The complete stress distribution in a flowing viscoelastic Hquid may be written down formally as follows,... 

Extruder, Elastic-Melt n (elastodynamic extmder) A type of extmder in which the material is fed into a fixed gap between stationary and rotating, vertical disks, is melted by frictional heat, and flows in a spiral path toward the center of rotation, from which it is discharged into a secondary device that can develop the high pressure required for extmdate shaping. Only mbbery polymers with certain viscoelastic properties are suitable for the process. 

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