Newtonian flow Polymer melts

Converging flow occurs in a wedge or tapering tube (restrained converging flow) and in the drawing of a molten filament (unrestrained converging flow). Polymer melts often behave very differently from Newtonian fluids under these circumstances. 

Polymers owe much of their attractiveness to their ease of processing. In many important teclmiques, such as injection moulding, fibre spinning and film fonnation, polymers are processed in the melt, so that their flow behaviour is of paramount importance. Because of the viscoelastic properties of polymers, their flow behaviour is much more complex than that of Newtonian liquids for which the viscosity is the only essential parameter. In polymer melts, the recoverable shear compliance, which relates to the elastic forces, is used in addition to the viscosity in the description of flow [48]. 

Flow behaviour of polymer melts is still difficult to predict in detail. Here, we only mention two aspects. The viscosity of a polymer melt decreases with increasing shear rate. This phenomenon is called shear thinning [48]. Another particularity of the flow of non-Newtonian liquids is the appearance of stress nonnal to the shear direction [48]. This type of stress is responsible for the expansion of a polymer melt at the exit of a tube that it was forced tlirough. Shear thinning and nonnal stress are both due to the change of the chain confonnation under large shear. On the one hand, the compressed coil cross section leads to a smaller viscosity. On the other hand, when the stress is released, as for example at the exit of a tube, the coils fold back to their isotropic confonnation and, thus, give rise to the lateral expansion of the melt. 

Pseudoplastic fluids are the most commonly encountered non-Newtonian fluids. Examples are polymeric solutions, some polymer melts, and suspensions of paper pulps. In simple shear flow, the constitutive relation for such fluids is... 

Capillary viscometers are useful for measuring precise viscosities of a large number of fluids, ranging from dilute polymer solutions to polymer melts. Shear rates vary widely and depend on the instmments and the Hquid being studied. The shear rate at the capillary wall for a Newtonian fluid may be calculated from equation 18, where Q is the volumetric flow rate and r the radius of the capillary the shear stress at the wall is = r Ap/2L. 

At the same time it is not surprising that polymer melts are non-Newtonian and do not obey such simple rules. Fortunately, if we make certain assumptions, it is possible to analyse flow in certain viscometer geometries to provide measurements of both shear stress (t) and shear rate (7) so that curves relating the two (flow curves) may be drawn. 

In the specific case of polymer melts these almost invariably are of the pseudoplastic type. In such cases the flow behaviour index n is less than 1 the greater the divergence from Newtonian behaviour the lower its value. 

As a complication some sources define a flow index as the reciprocal of that defined above so that some care has to be taken in interpretation. In such cases the values are greater than unity for polymer melts and the greater the value the greater the divergence from Newtonian behaviour.)... 

A polymer melt is injected into a circular section channel under constant pressure. What is the ratio of the maximum non-isothermal flow length to the isothermal flow length in the same time for (a) a Newtonian melt and (b) a power law melt with index, n = 0.3. 

Chapter 5 deals with the aspects of the flow behaviour of polymer melts which are relevant to the processing methods. The models are developed for both Newtonian and Non-Newtonian (Power Law) fluids so that the results can be directly compared. 

Chapter 4 describes in general terms the processing methods which can be used for plastics and wherever possible the quantitative aspects are stressed. In most cases a simple Newtonian model of each of the processes is developed so that the approach taken to the analysis of plastics processing is not concealed by mathematical complexity. Chapter 5 deals with the aspects of the flow behaviour of polymer melts which are relevant to the processing methods. The models are developed for both Newtonian and Non-Newtonian (Power Law) fluids so that the results can be directly compared. 

Since non-Newtonian flow is typical for polymer melts, the discussion of a filler s role must explicitly take into account this fundamental fact. Here, spoken above, the total flow curve includes the field of yield stress (the field of creeping flow at x < Y may not be taken into account in the majority of applications). Therefore the total equation for the dependence of efficient viscosity on concentration must take into account the indicated effects. 

Though the accuracy of description of flow curves of real polymer melts, attained by means of Eq. (10), is not always sufficient, but doubtless the equation of such a structure based on the idea of relaxation mechanism of non-Newtonian polymer flow, correctly reflects the main peculiarities of viscous properties. Therefore while discussing the effect a filler has on the viscosity properties of polymer melts, besides the dependences Y(filler modifies the characteristic time of relaxation. According to [19], a possible form of the X versus

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Though experimental data on suspensions of fibers in Newtonian dispersion media give more or less regular picture, a transition to non-Newtonian viscoelastic liquids, as Metzner noted [21], makes the whole picture far or less clear. Probably, the possibility to make somewhat general conclusions on a longitudinal flow of suspensions in polymer melts requires first of all establishing clear rules of behavior of pure melts at uniaxial extension this problem by itself has no solution as yet. 

Rheology deals with the deformation and flow of any material under the influence of an applied stress. In practical apphcations, it is related with flow, transport, and handling any simple and complex fluids [1], It deals with a variety of materials from elastic Hookean solids to viscous Newtonian liquid. In general, rheology is concerned with the deformation of solid materials including metals, plastics, and mbbers, and hquids such as polymer melts, slurries, and polymer solutions. 

On a molecular level, partially crystalline to amorphous polymers are normally used. As the material is heated, Brownian motion occurs resulting in a more random chain arrangement. When a unidirectional force is applied to a resting polymer melt, the chains tend to move away from the applied force. If the applied force is slow enough to allow the Brownian movement to continue to keep the polymers in a somewhat random conformation, the movement of the polymer melt is proportional to the applied stress, i.e., the flow is Newtonian. 

It is possible to derive an expression for the pressnre profile in the x direction using a simple model. We assnme that the flow is steady, laminar, and isothermal the flnid is incompressible and Newtonian there is no slip at the walls gravity forces are neglected, and the polymer melt is uniformly distribnted on the rolls. With these assnmptions, there is only one component to the velocity, v dy), so the equations of continuity and motion, respectively, reduce to... 

Analyzing the flow in a single screw extruder using analytical solutions can only be done if we assume a Newtonian polymer melt. As can be seen in these sections, the flow inside the screw channel is a three dimensional flow made up of a combination of pressure and drag flows. Non-Newtonian flow can be solved for using numerical techniques and will be covered in Chapters 8 to 11 of this book. 

To simplify the problem, we can assume that the polymer bar moves at a constant speed Usy, and that a film of constant thickness, 5, exists between the bar and the heated plate. In addition, we assume that the polymer melt is Newtonian and that the viscosity is independent of temperature. The Newtonian assumption is justified by low rates of deformation that develop in this relatively slow flow problem. Furthermore, due to these low rates of deformation we can assume that the convective and viscous dissipation effects are negligible. 

The lubrication approximation as previously derived is valid for purely viscous Newtonian fluids. But polymer melts are viscoelastic and also exhibit normal stresses in shearing flows, as is discussed in Chapter 3 nevertheless, for many engineering calculations in processing machines, the approximation does provide useful models. 

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