15 Viscoelastic fluid behaviour
In the classical theory of elasticity, the stress in a sheared body is directly proportional to the strain. For tension, Hooke's law applies and the coefficient of proportionality is known as Young's modulus, G,:
where dx is the shear displacement of two elements separated by a distance dy. When a perfect solid is deformed elastically, it regains its original form on removal of the stress. However, if the applied stress exceeds the characteristic yield stress of the material, complete recovery will not occur and 'creep' will take placethat is, the 'solid' will have flowed.
At the other extreme, in the Newtonian fluid the shearing stress is proportional to the rate of shear, equation (1.1). Many materials show both elastic and viscous effects under appropriate circumstances. In the absence of the timedependent behaviour mentioned in the preceding section, the material is said to be viscoelastic. Perfectly elastic deformation and perfectly viscous flow are, in effect, limiting cases of viscoelastic behaviour. For some materials, it is only these limiting conditions that are observed in practice. The elasticity of water and the viscosity of ice may generally pass unnoticed! The response of a material depends not only its structure but also on the conditions (kinematic) to which it has been subjected; thus the distinction between 'solid' and 'fluid' and between 'elastic' and 'viscous' is to some extent arbitrary and subjective.
Many materials of practical interest (such as polymer melts, polymer and soap solutions, synovial fluid) exhibit viscoelastic behaviour; they have some ability to store and recover shear energy, as shown schematically in Figure 1.13. Perhaps the most easily observed experiment is the 'soup bowl' effect. If a liquid in a dish is made to rotate by means of gentle stirring with a spoon, on removing the energy source (i.e. the spoon), the inertial circulation will die out as a result of the action of viscous forces. If the liquid is viscoelastic (as some of the proprietary soups are), the liquid will be seen to slow to a stop and then to unwind a little. This type of behaviour is closely linked to the tendency for a gel structure to form within the fluid; such an element of rigidity makes simple shear less likely to occurthe shearing forces tending to act as couples to produce rotation of the fluid elements as well as pure slip. Such incipient rotation produces a stress perpendicular to the direction of shear. Numerous other unusual phenomena often ascribed to fluid viscoelasticity include die swell, rod climbing (Weissenberg effect), tubeless siphon, and the development of secondary flows at low Reynolds numbers. Most of these have been illustrated photographically in a recent book [Boger and Walters, 1992]. A detailed treatment of viscoelastic fluid behaviour is beyond the scope of this book and interested readers are referred to several excellent books available on this subject, e.g. see [Schowalter, 1978; Bird et al., 1987; Carreau etal., 1997; Tanner and Walters, 1998]. Here we shall describe the 'primary' and 'secondary' normal stress differences observed in steady shearing flows which are used both to classify a material as viscoelastic or viscoinelastic as well as to quantify the importance of viscoelastic effects in an envisaged application.
Viscous liquidenergy dissipated as heat dx
Viscous liquidenergy dissipated as heat
Elastic solidenergy recoverable
Elastic solidenergy recoverable
Figure 1.13 Qualitative differences between a viscous fluid and an elastic solid
Normal stresses in steady shear flows
Let us consider the onedimensional shearing motion of a fluid; the stresses developed by the shearing of an infinitesimal element of fluid between two planes are shown in Figure 1.14. By nature of the steady shear flow, the components of velocity in the y and zdirections are zero while that in the xdirection is a function of y only. Note that in addition to the shear stress, ryx, there are three normal stresses denoted by Pxx, Pyy and Pzz within the sheared fluid which are given by equation (1.6). Weissenberg [1947] was the first to observe that the shearing motion of a viscoelastic fluid gives rise to unequal normal stresses. Since the pressure in a nonNewtonian fluid cannot be defined by equation (1.7) the differences, Pxx — Pyy = N1 and Pyy — Pzz = N2, are more readily measured than the individual stresses, and it is therefore customary to express N1 and N2 together with ryx as functions of the shear rate yyx to describe the rheological behaviour of a viscoelastic material in a simple shear flow. Sometimes, the first and second normal stress differences N1 and N2 are expressed in terms of two coefficients, and defined as follows:
Kyx 2
A typical dependence of the first normal stress difference on shear rate is shown in Figure 1.15 for a series of polystyreneintoluene solutions. Usually, the rate of decrease of ^ with shear rate is greater than that of the apparent
viscosity. At very low shear rates, the first normal stress difference, Ni, is expected to be proportional to the square of shear rate  that is, tends to a constant value this limiting behaviour is seen to be approached by some of the experimental data shown in Figure 1.15. It is common that the first normal stress difference N1 is higher than the shear stress r at the same value of shear rate. The ratio of N1 to r is often taken as a measure of how elastic a liquid is; specifically (N1/2r) is used and is called the recoverable shear. Recoverable shears greater than 0.5 are not uncommon in polymer solutions and melts. They indicate a highly elastic behaviour of the fluid. There is, however, no evidence of approaching a limiting value at high shear rates. It is fair to mention here that the first normal stress difference has been investigated much less extensively than the shear stress.
Even less attention has been given to the study and measurement of the second normal stress difference. The most important points to note about N2 are that it is an order of magnitude smaller than N1, and that it is negative. Until recently, it was thought that N2 = 0; this socalled Weissenberg hypothesis is no longer believed to be correct. Some data in the literature even seem to suggest that N2 may change sign. Typical forms of the dependence of N2 on shear rate are shown in Figure 1.16 for the same solutions as used in Figure 1.15.
103 F
103 F
Figure 1.16 Representative second normal stress difference data for polystyreneintoluene solutions at 298K [Kulicke and Wallabaum, 1985]
Figure 1.16 Representative second normal stress difference data for polystyreneintoluene solutions at 298K [Kulicke and Wallabaum, 1985]
The two normal stress differences defined in this way are characteristic of a material, and as such are used to categorise a fluid either as purely viscous (N1 ~ 0) or as viscoelastic, and the magnitude of N\ in comparison with ryx, is often used as a measure of viscoelasticity.
Aside from the simple shearing motion, the response of viscoelastic materials in a variety of other welldefined flow configurations including the cessation/initiation of flow, creep, small amplitude sinusoidal shearing, etc. also lies in between that of a perfectly viscous fluid and a perfectly elastic solid. Conversely, these tests may be used to infer a variety of rheological information about a material. Detailed discussions of the subject are available in a number of books, e.g. see Walters [1975] and Makowsko [1994].
Elongational flow
Flows which result in fluids being subjected to stretching in one or more dimensions occur in many processes, fibre spinning and polymer film blowing being only two of the most common examples. Again, when two bubbles coalesce, a very similar stretching of the liquid film between them takes place until rupture occurs. Another important example of the occurrence of extensional effects is the flow of polymer solutions in porous media, as encountered in the enhanced oil recovery process, in which the fluid is stretched as the extent and shape of the flow passages change. There are three main forms of elongational flow: uniaxial, biaxial and planar, as shown schematically in Figure 1.17.
Figure 1.17 Schematic representation of uniaxial (a), biaxial (b) and planar (c) extension
Fibre spinning is an example of uniaxial elongation (but the stretch rate varies from point to point along the length of the fibre). Tubular film blowing which involves extruding of polymer through a slit die and pulling the emerging sheet forward and sideways is an example of biaxial extension; here, the stretch rates in the two directions can normally be specified and controlled. Another example is the manufacture of plastic tubes which may be made either by extrusion or by injection moulding, followed by heating and subjection to high pressure air for blowing to the desired size. Due to symmetry, the blowing step in an example of biaxial extension with equal rates of stretching in two directions. Irrespective of the type of extension, the sum of the volumetric rates of extension in the three directions must always be zero for an incompressible fluid.
Naturally, the mode of extension affects the way in which the fluid resists deformation and, although this resistance can be referred to loosely as being quantified in terms of an elongational or extensional viscosity (which further depends upon the type of elongational flow, i.e. uniaxial, biaxial or planar), this parameter is, in general, not necessarily constant. For the sake of simplicity, consideration may be given to the behaviour of an incompressible fluid element which is being elongated at a constant rate e in the xdirection, as shown in Figure 1.18. For an incompressible fluid, the volume of the element must remain constant and therefore it must contract in both the y and zdirections at the rate of (e/2), if the system is symetrical in those directions. The normal stress Pyy and Pzz will then be equal. Under these conditions, the three components of the velocity vector V are given by:
and clearly, the rate of elongation in the xdirection is given by:
In uniaxial extension, the elongational viscosity /xE is then defined as:
e e or Pyy and ryy can be replaced by Pzz and rzz respectively.
Figure 1.18 Uniaxial extensional flow
The earliest determinations of elongational viscosity were made for the simplest case of uniaxial extension, the stretching of a fibre or filament of liquid. Trouton [1906] and many later investigators found that, at low strain (or elongation) rates, the elongational viscosity mE was three times the shear viscosity m [Barnes et al., 1989]. The ratio mE/m is referred to as the Trouton ratio, Tr and thus:
me M
The value of 3 for Trouton ratio for an incompressible Newtonian fluid applies to values of shear and elongation rates. By analogy, one may define the Trouton ratio for a nonNewtonian fluid:
The definition of the Trouton ratio given by equation (1.26) is somewhat ambiguous, since it depends on both S and y, and some convention must therefore be adopted to relate the strain rates in extension and shear. To remove this ambiguity and at the sametime to provide a convenient estimate of behaviour in extension, Jones et al. [1987] proposed the following definition of the Trouton ratio:
i.e., in the denominator, the shear viscosity is evaluated at y = /3s. They also suggested that for inelastic isotropic fluids, the Trouton ratio is equal y x z
to 3 for all values of S and p, and any departure from the value of 3 can be ascribed unambiguously to viscoelasticity. In other words, equation (1.27) implies that for an inelastic shearthinning fluid, the extensional viscosity must also decrease with increasing rate of extension (socalled "tensionthinning"). Obviously, a shearthinning viscoelastic fluid (for which the Trouton ratio will be greater than 3) will thus have an extensional viscosity which increases with the rate of extension; this property is also called "strainhardening". Many materials including polymer melts and solutions thus exhibit shearthinning in simple shear and strainhardening in uniaxial extension. Except in the limit of vanishingly small rates of deformation, there does not appear to be any simple relationship between the elongational viscosity and the other rheological properties of the fluid and, to date, its determination rests entirely on experiments which themselves are aften constrained by the difficulty of establishing and maintaining an elongational flow field for long enough for the steady state to be reached [Gupta and Sridhar, 1988; James and Walters, 1994]. Measurements made on the same fluid using different methods seldom show quantitative agreement, especially for low to medium viscosity fluids [Tirtaatmadja and Sridhar, 1993]. The Trouton ratios for biaxial and planar extensions at low strain rates have values of 6 and 9 respectively for all inelastic fluids and for Newtonian fluids under all conditions.
Mathematical models for viscoelastic behaviour
Though the results of experiments in steady and transient shear or in an elongational flow field may be used to calculate viscous and elastic properties for a fluid, in general the mathematical equations need to be quite complex in order to describe a real fluid adequately. Certainly, the most striking feature connected with the deformation of a viscoelastic substance is its simultaneous display of 'fluidlike' and 'solidlike' characteristics. It is thus not at all surprising that early attempts at the quantitative description of viscoelastic behaviour hinged on the notion of a linear combination of elastic and viscous properties by using mechanical analogues involving springs (elastic component) and dash pots (viscous action). The Maxwell model represents the cornerstone of the socalled linear viscoelastic models; though it is crude, nevertheless it does capture the salient features of viscoelastic behaviour.
A mechanical analogue of this model is obtained by series combinations of a spring and a dashpot (a vessel whose outlet contains a flow constriction over which the pressure drop is proportional to flow rate), as shown schematically in Figure 1.19. If the individual strain rates of the spring and the dashpot respectively are ■y1 and y2, then the total strain rate p is given by the sum of these two components:
df df
Figure 1.19 Schematic representation of the Maxwell model
Combining equation (1.28) with the Hooke's law of elasticity and Newton's law of viscosity, one can obtain:
where r is the time derivative of r; ^ is the viscosity of the dashpot fluid; and A(= ^/G), the relaxation time, which is a characteristic of the fluid. It can easily be seen from equation (1.29) that if a Maxwell model fluid is strained to a fixed point and held there, the stress will decay as exp(—t/X). An important feature of the Maxwell model is its predominantly fluidlike response. A more solidlike behaviour is obtained by considering the socalled Voigt model which is represented by the parallel arrangement of a spring and a dashpot, as shown in Figure 1.20.
Figure 1.20 Schematic representation of the Voigt model
In this case, the strain in the two components is equal and the equation describing the stressstrain behaviour of this system is:
If the stress is constant at r0 and the initial strain is zero, upon the removal of the stress, the strain decays exponentially with a time constant, A(= ^/G).
The more solidlike response of this model is clear from the fact that it does not exhibit unlimited nonrecoverable viscous flow and it will come to rest when the spring has taken up the load.
One of the main virtues of such linear models is that they can be conveniently superimposed by introducing a spectrum of relaxation times, as exhibited in practice, by polymeric systems or by including higher derivatives. Alternatively, using the idea of superposition, one can assume the stress to be due to the summation of a number of small partial stresses, each pertaining to a partial strain, and each stress relaxing according to some relaxation law. This approach yields the socalled 'integral' models. In addition, many other ideas have been employed to develop elementary models for viscoelastic behaviour including the dumbbell, beadspring representations, network and kinetic theories. Invariably, all such attempts entail varying degrees of idealisation and empiricism; their most notable limitation is the restriction to small strain and strain rates.
The next generation of viscoelastic fluid models has attempted to relax the restriction of small deformation and deformation rates, thereby leading to the socalled nonlinear models. Excellent critical appraisals of the developments in the field, together with the merits and demerits of a selection of models, as well as some guidelines for selecting an appropriate equation for an envisaged process application, are available in the literature, e.g. see references [Bird etal., 1987; Tanner, 1988; Larson, 1988; Barnes etal., 1989; Macosko, 1994; Bird and Wiest, 1995].
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