PURELY VISCOUS FLOW

We limit ourselves here to two-dimensional deformations. A detailed three-dimensional treatment of rheology is beyond the scope of this book. Several excellent treatises are available.  

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Mack (58, 59) points out that asphaltenes from different sources in the same petro-lenes give mixtures of approximately the same rheological type, but sols of the same asphaltenes in different petrolenes differ in flow behavior. Those in aromatic petrolenes show viscous behavior and presumably approach true solution. Those in paraffinic media show complex flow and are considered to be true colloidal systems. Pfeiffer and associates (91) consider that degree of peptization of asphaltene micelles determines the flow behavior. Thus, a low concentration of asphaltenes well peptized by aromatic petrolenes leads to purely viscous flow. High concentrations of asphaltenes and petrolenes of low aromatic content result in gel-type asphalts. All shades of flow behavior between these extremes are observed. 

For purely viscous flow, the equations in Table I can be used directly. If necessary an iteration using p can be carried out. 

The effect of slip flow can be treated either as an extension of a pure viscous flow or as an extension of a Knudsen flow. The simplest method is by adding an additional term (R/2p) (P/RT) dP/dz to Eq. (9.2), with P being the slip coefficient which is proportional to P. 

On the other hand, in a non-Newtonian fluid, the viscosity depends on the shear rate. Besides showing very high non-Newtonian viscosities, polymers exhibit a complex viscoelastic flow behavior, that is, their flow exhibits memory , as it includes an elastic component in addition to the purely viscous flow. Rheological properties are those that define the flow behavior, such as the viscosity and the melt elasticity, and they determine how easy or difficult is to process these materials, as well as the performance of the polymer in some applications. The rheology of the polymers and its effect on the processing of these materials are studied in Chapters 22 and 23. 

However, no real material shows either ideal elastic behavior or pure viscous flow. Some materials, for example, steel, obey Hooke s law over a wide range of stress and strain, but no material responds without inertial effects. Similarly, the behavior of some fluids, like water, approximate Newtonian response. Typical deviations from linear elastic response are shown by rubber elasticity and viscoelasticity. 

Viscoelastic material such as polymers combine the characteristics of both elastic and viscous materials. They often exhibit elements of both Hookean elastic solid and pure viscous flow depending on the experimental time scale. Application of stresses of relatively long duration may cause some flow and irrecoverable (permanent) deformation, while a rapid shearing will induce elastic response in some polymeric fluids. Other examples of viscoelastic response include creep and stress relaxation, as described previously. 

As Equation 14.7 shows, the Maxwell element is merely a linear combination of the behavior of an ideally elastic material and pure viscous flow. Now let us examine the response of the Maxwell element to two typical experiments used to monitor the viscoelastic behavior of polymer. 

A creep test was performed to examine the viscoelastic properties of the CNT-grease. A shear stress of 5 Pa was used for 5.0 and 7.0 wt% nanotube loading, while 40 and 100 Pa was used for 10.0 wt% and 12.0 wt% nanotube loading. These stresses were considered to be below that of instantaneous pure viscous flow for each sample. Each creep test was sequenced for 1000 s, followed by a 100 s creep-recovery analysis. 

The yield stress of each CNT grease was determined via an increasing shear stress ramp. The yield stress in these experiments shall be defined as the stress at which the sample s instantaneous viscosity rapidly decreases from an increasing or constant value. This will determine the shear stress required for the onset of pure viscous flow. Again, the range of shear stress was chosen to compliment each individual sample. 

The viscoelastic deviations from ideal elasticity or purely viscous flow depend on both the experimental conditions (particiflarly melt temperature with its four temperature regions and magnitudes and rates of application of stress or strain). They also depend on the basic polymer structure particularly molecular weight (MW), molecular weight distribution (MWD), crystallinity, crosslinking, and branching. 

It has been found that the type of flow depends chiefly on the properties and quantity of the maltenes. If these are aromatic, pure viscous flow is observed, indicating the peptisation of the asphaltenes to be to such an extent (Fig. 4a) that these do not hinder each others motions. If however the maltenes are aliphatic or if they... 

For the present section, the gel-state of Fig. 21c is of special interest. The sol exhibits a purely viscous flow, but when the gel-state is introduced by subsequent polymerisation, the flow curve gradually approaches the form shown by Kruyt... 

Darcy s empirical law for laminar flow. Equation (3.I-I7) for laminar flow in packed beds shows that the flow rate is proportional to Ap and inversely proportional to the viscosity p and length AL. This is the basis for Darcy s law as follows for purely viscous flow in consolidated porous media. 

Svabik, Samsonkova, and Perdikoulias [45] proposed another explanation for the interface distortion in coextrusion of fluids with equal viscosity. They performed three-dimensional flow analysis of coextrusion flow and found that even in coextrusion with Newtonian fluids with equal viscosity layer distortion takes place. Obviously, this type of distortion cannot be caused by normal stress differences since these do not occur in Newtonian fluids. Also, with the viscosities being equal the distortion cannot be caused by viscosity differences. The predicted layer distortion is schematically illustrated in Fig. 9.43. The authors call the distortion resulting from purely viscous flow geometrical encapsulation. 

TablelV. The numerical values of activation energy shown in Table IV are larger than that 4 Kcal/mol (at 25 C) for viscous flow of water (22). Thus the water flow in the membrane is considered to be different from a purely viscous flow. The nature of the microheterophase structure in the membranes should play an important role for the water permeability through the membrane. The most remarkable characteristic of the block copolymer membranes compared to homopolymer membranes is that the values of K for the former are highly changable by selecting casting solvents between quite broad regions, and the K value for MBM-14 membrane cast from CF-TFA 8 1 mixture is almost 100 folds of K value for MBM-14 membrane cast from CF-TFE 2 1 mixture is almost the same as that for PMLG-12 cast from the same solvent mixture. Such drastic increase in the K values for the block copolymer membranes should be attributed to the specific structure of the interfacial zone between A and B...

We begin our treatment of rheology with a discussion of purely viscous flow. For our purposes, this will be defined as a deformation process in which all the... 

Consider, for example, the creep response of the four-parameter model (Fig. 18.8). For this model, a logical choice for A would be the time constant for its Voigt-Kelvin component, Jja/Gz- For De> 1 (t - Ac), the Voigt-Kelvin element and dashpot 1 will be essentially immobile, and the response will be due almost entirely to spring 1, that is, almost purely elastic. For De 0 (t, > A ), the instantaneous and retarded elastic response mechanisms have long since reached equilibrium, so the only remaining response will be the purely viscous flow of dashpot 1, and the deformation due to viscous flow will completely overshadow that due to the elastic response mechanisms (imagine the creep... 

As discussed above, polymer chains tend to be disentangled and oriented under shear. One important result of the disentanglement and molecular orientation under shear is the decrease of viscosity with increasing shear rate. However, the flow of polymers is not pure viscous flow, and it has elastic component since the change of chain conformations is not completely irreversible. Upon the release of the shear, the polymer chains tend to recoil and be pulled back by the restraining force. Such elastic response has significant effect on the fiber formation processes. 

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