Example---shear flow

As an example, consider the shear flow (10.86). From the symmetry of the flow, n is in the x-y plane, and can be written as 

Equations (10.94) and (10.128) give the concentration dependence of j. (Note that eqn (10.128) holds also in the isotropic solution if 5 is put to zero.) The result, shown in Fig. 10.6, indicates that the viscosity takes a maximum near the phase transition point, in agreement with the experimental results. 

This vanishes in the isotropic phase (5 = 0), but does not vanish in the nematic phase (5 0). 

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Normal Stresses in Shear Flow. The tendency of polymer molecules to curl-up while they are being stretched in shear flow results in normal stresses in the fluid. For example, shear flows exhibit a deviatoric stress defined by... 

The ratio of the fluid relaxation time to the timescale for flow If defines a dimensionless group termed the Deborah number, De = /tf. This group has been used in the literature to characterize deviations from Newtonian flow behavior in polymers [6]. Specifically, in flows such as simple steady shear flow where a single flow time tf = y can be defined, it has been observed that for De 1, a Newtonian fluid behavior is observed, whereas for De 1, a non-Newtonian fluid response is observed. However, in flows where multiple timescales can be identified, for example, shear flow between eccentric cylinders, the Deborah number is clearly not unique. In this case, it is generally more useful to discuss the effect of flow on polymer liquids in terms of the relative rates of deformation of material lines and material relaxation. In a steady flow, this effect can be captured by a second dimensionless group termed the Weissenberg number, Wi = k A, where k is a characteristic deformation rate and A is a characteristic fluid relaxation time. For polymer liquids, A is typically taken to be the longest relaxation time Ap, and for steady shear flow, k = y, which leads to Wi = y Ap. 

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... 

At least, in absolute majority of cases, where the concentration dependence of viscosity is discussed, the case at hand is a shear flow. At the same time, it is by no means obvious (to be more exact the reverse is valid) that the values of the viscosity of dispersions determined during shear, will correlate with the values of the viscosity measured at other types of stressed state, for example at extension. Then a concept on the viscosity of suspensions (except ultimately diluted) loses its unambiguousness, and correspondingly the coefficients cn cease to be characteristics of the system, because they become dependent on the type of flow. 

Flow is generally classified as shear flow and extensional flow [2]. Simple shear flow is further divided into two categories Steady and unsteady shear flow. Extensional flow also could be steady and unsteady however, it is very difficult to measure steady extensional flow. Unsteady flow conditions are quite often measured. Extensional flow differs from both steady and unsteady simple shear flows in that it is a shear free flow. In extensional flow, the volume of a fluid element must remain constant. Extensional flow can be visualized as occurring when a material is longitudinally stretched as, for example, in fibre spinning. When extension occurs in a single direction, the related flow is termed uniaxial extensional flow. Extension of polymers or fibers can occur in two directions simultaneously, and hence the flow is referred as biaxial extensional or planar extensional flow. 

In contrast to rotational shear flow, deformation and breakage occurs over the whole range of viscosity ratio in an irrotational (extension) flow produced, for example, in a 4-roll apparatus (Fig. 23) from which the data shown in Fig. 21 were obtained [76]. Comparing the critical conditions for breakage by shear and by elongation. Fig. 23 shows that for equal deformation rates, irrotational flow tends to be more damaging than rotational flow. 

This flow field is somewhat idealized, and cannot be exactly reproduced in practice. For example, near the planar surfaces, shear flow is inevitable, and, of course, the range of % and y is consequently finite, leading to boundary effects in which the extensional flow field is perturbed. Such uniaxial flow is inevitably transient because the surfaces either meet or separate to laboratory scale distances. 

Flows that produce an exponential increase in length with time are referred to as strong flows, and this behavior results if the symmetric part of the velocity gradient tensor (D) has at least one positive eigenvalue. For example, 2D flows with K > 0 and uniaxial extensional flow are strong flows simple shear flow (K = 0) and all 2D flows with K < 0 are weak flows. 

With the gel equation, we can conveniently compute the consequences of the self-similar spectrum and later compare to experimental observations. The material behaves somehow in between a liquid and a solid. It does not qualify as solid since it cannot sustain a constant stress in the absence of motion. However, it is not acceptable as a liquid either, since it cannot reach a constant stress in shear flow at constant rate. We will examine the properties of the gel equation by modeling two selected shear flow examples. In shear flow, the Finger strain tensor reduces to a simple matrix with a shear component... 

Finally, a relatively new area in the computer simulation of confined polymers is the simulation of nonequilibrium phenomena [72,79-87]. An example is the behavior of fluids undergoing shear flow, which is studied by moving the confining surfaces parallel to each other. There have been some controversies regarding the use of thermostats and other technical issues in the simulations. If only the walls are maintained at a constant temperature and the fluid is allowed to heat up under shear [79-82], the results from these simulations can be analyzed using continuum mechanics, and excellent results can be obtained for the transport properties from molecular simulations of confined liquids. This avenue of research is interesting and could prove to be important in the future. 

A material is isotropic if its properties are the same in all directions. Gases and simple liquids are isotropic but liquids having complex, chain-like molecules, such as polymers, may exhibit different properties in different directions. For example, polymer molecules tend to become partially aligned in a shearing flow. 

Most examples of flow in nature and many in industry are turbulent. Turbulence is an instability phenomenon caused, in most cases, by the shearing of the fluid. Turbulent flow is characterized by rapid, chaotic fluctuations of all properties including the velocity and pressure. This chaotic motion is often described as being made up of eddies but it is important to appreciate that eddies do not have a purely circular motion. 

It is important to be aware of the fact that most turbulence models have been developed for and validated against turbulent shear flows parallel to walls. Models for separated and recirculating flows have received much less attention. The turbulent impinging jet shown in Fig. 4.6 is a good example of a flow used to test the shortcomings of widely employed turbulence models. The unique characteristics of this flow include the following.16... 

Flows with shear do not exhibit simple gradient transport. See, for example, the homogeneous shear flow results reported by Rogers et al. (1986). 

Other examples of interpolymer complexation due to hydrogen bonding include complexes of poly(ethylene-co-methacrylic acid) with polyethers, poly(2-vinylpyridine), and polyethyloxazoline [107-109], and PAA with poly(vinyl alcohol-co-vinyl acetate) [110]. PAA has even been shown to form stable interchain hydrogen bonded association under high shear flow [111]. 

Various investigations have considered the effects of titanate treatments on melt rheology of filled thermoplastics [17,41]. Figure 10, for example, shows that with polypropylene filled with 50% by weight of calcium carbonate, the inclusion of isopropyl triisostearoyl titanate dispersion aid decreases melt viscosity but increases first normal stress difference. This suggests that the shear flow of the polymer is promoted by the presence of titanate treatment, and is consistent with the view that these additives provide ineffective coupling between filler particles and polymer matrix [42]. 

This characterizes the time taken for the restoration of the equilibrium microstructure after a disturbance caused, for example, by convective motion, i.e., this is the relaxation time of the microstructure. The time scale of shear flow is given by the reciprocal of the shear rate, 7. The dimensionless group formed by the ratio (tD ff/tShear) is the Peclet number... 

The response of simple fluids to certain classes of deformation history can be analyzed. That is, a limited number of material functions can be identified which contain all the information necessary to describe the behavior of a substance in any member of that class of deformations. Examples are the viscometric or steady shear flows which require, at most, three independent functions of the shear rate (79), and linear viscoelastic behavior (80,81) which requires only a single function, in this case a relaxation function. The functions themselves must be determined experimentally for each substance. 

Viscoelastic behavior is classified as linear or non-linear according to the manner by which the stress depends upon the imposed deformation history (SO). Insteady shear flows, for example, the shear rate dependence of viscosity and the normal stress functions are non-linear properties. Linear viscoelastic behavior is obtained for simple fluids if the deformation is sufficiently small for all past times (infinitesimal deformations) or if it is imposed sufficiently slowly (infinitesimal rate of deformation) (80,83). In shear flow under these circumstances, the normal stress differences are small compared to the shear stress, and the expression for the shear stress reduces to a statement of the Boltzmann superposition principle (15,81) ... 

This brought a bout a keen interest in other methods of intensification in processing. Lately, the directed effect of physical (mechanical) fields on molten polymers has become one such area. These effects, as demonstrated in many works published in the 1970s and in the 1980s, (see for examples [6-9]) result in altered parameters of micro- and macrostress of the system. Molding under conditions of directed physical fields, in particular, in the case of mechanical and acoustic vibration effects upon melts, is performed so that an additional stress superimposed on the polymer s main shear flow and the state of material is characterized by combined stress. 

In example 2.2 we obtained that for steady shearing flows the viscometric functions for this constitutive equation are defined by... 

Comment how the viscometric functions for the shear flow of a Lodge rubber-liquid develop in Example 2.4 compare with experimental observations. 

Hence, when solving a non-isothermal problem the question arises -is this a problem where the equations of motion and energy are coupled To address this question we can go back to Example 6.1, a simple shear flow system was analyzed to decide whether it can be addressed as an isothermal problem or not. In a simple shear flow, the maximum temperature will occur at the center of the melt. By substituting y = h/2 into eqn. (6.5), we get an equation that will help us estimate the temperature rise... 

A common flow problem in polymer processing is a shear flow with a temperature gradient as depicted in Fig. 6.58. For example, this type of flow occurs within the melt film that develops during melting with drag flow removal, as will be discussed later in this chapter. 

Example 52 On the basis of the kinetic theory, which is used to model collision-dominated gas-solid flows, derive a general expression of solid stresses of elastic spheres in a simple shear flow. 

In the presence of flow, the correlation function will be affected by gradients in the velocity. For example, in the case of simple shear flow,... 

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