The Trouton ratio

The difference between the two kinds of viscosities being considered is best illustrated by introducing the so-called Trouton ratio, defined as the ratio of the 

The fact that the Trouton ratio eventually approaches a constant value means that the shear and extensional viscosity flow curves are paraUeT in that range, and if the behaviour is power-law, then the power-law index is the same in both cases. (Notice this from the polymer melt examples shown later in the figures 13 - 20.) 

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The ratio of extensional viscosity r e to shear viscosity r s is known as the Trouton ratio, which is three for Newtonian fluids in uniaxial extension and larger than three for non-Newtonian fluids. For a viscoelastic fluid such as a polymer in solution, the uniaxial extensional viscosity characterizes the resistance of the fluid... 

This is, according Eq. (15.64), three times the zero shear viscosity, which in agreement with practice. It is a result already obtained by Trouton in 1906. Accordingly, the Trouton ratio, defined as... 

The Trouton ratio, 7r, is defined as the ratio of the extensional viscosity to the shear viscosity ... 

A solution or melt composed of such stretchable molecules can be highly springy, especially in extensional flows (Tirtaatmadja and Sridhar 1993). The kinematics of an extensional flow are described in Section 1.4.1.2. From Eqs. (1-6) and (1-9), one can show that for a Newtonian fluid (for which a = 2rjD) the Trouton ratio Tr = f)u/hQ the uniaxial extensional viscosity to the zero-shear viscosity /jo is numerically equal to 3. For polymers, Tr can be much higher than this. Figure 3-2, for example, shows Tr for a... 

Comparisons of the predictions of the FENE dumbbell model with measurements of the extensional viscosity of dilute solutions have been fairly encouraging. Figure 3-2 compares the Trouton ratio predicted by a multimode FENE dumbbell model with experimental data for a Roger fluid Good agreement is obtained if one uses a value of the... 

Such a viscoelastic or memory liquid is another example of a non-Newtonian liquid. Nearly all viscoelastic liquids are also strain rate thinning, but not all strain rate thinning liquids show significant elasticity. Deformation can, of course, be in shear or elongation, etc. However, for viscoelastic liquids, the Trouton ratios [see Eq. (5.2)] are higher, often much... 

However, for anisometric particles the Trouton ratio, q / q is a strong function of p. For example, at ( ) = 0.01 extensional viscosity of rods with aspect ratio p = 1000 is 1000 times higher than that for suspension of spheres. 

As we have seen, there exist a number of treatments of the increase in extensional viscosity of the solution corresponding to the coil-stretch transition. In particular, the Warner FENE dumbbell model (29) and the Kramers bead-rod model (30) predict increases in normalized extensional viscosity of the order of N, the number of statistical segment units in the flexible chain. The normalized extensional viscosity (T)e ) compares the increase in extensional viscosity due to the polymer to three times (the Trouton ratio) the corresponding increase in the simple shear viscosity and is given by... 

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 he was three times the shear viscosity n [Barnes et al., 1989], The ratio Mb/M is referred to as the Trouton ratio, Tr and thus ... 

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 non-Newtonian fluid ... 

The definition of the Trouton ratio given by equation (1.26) is somewhat ambiguous, since it depends on both e 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 behaviom in extension, Jones etal. [1987] proposed the following definition of the Trouton ratio ... 

It is important to note that the Trouton ratio, Tr, defined as the ratio of the extensional viscosity to the shear viscosity, involves the shear viseosity evaluated at the same magnitude of the second invariant of the rate of deformation tensor where s is the rate of extension, i.e. ... 

A commOTi behavior of polymeric solutions [9-11] and polymer melts [12] is that the Trouton ratio is greater than three at moderate and higher deformation rate. Their typical behaviors are illustrated in Fig. 2, which show that the Trouton ratio can be a function of both strain and strain rate. 

Since the Trouton ratio depends on both a and y in Eq. 10, there is some ambiguity in choosing their values. To provide a convenient estimate without this ambiguity, Jones et al. [8] proposed the following definition ... 

In qualitative agreement with the von Mises criterion, ctii y/ai2,5 was reported (Utracki 1984). The Trouton ratio... 

The dynamical behavior is striking. A steady-state stress is reached as long as 2A < 1 for all modes the Trouton ratio of 3 is attained at steady state when... 

Figure 10 The Trouton ratio (extensional viscosity divided by shear viscosity) for various kinds of liquid.

A discrepancy concerning the Trouton ratio and the measured elongational viscosity can be found for K30 solutions with 20 m% solids fraction. Here, the viscosity of the liquid is too low to yield a stable filament during the experiments [33]. Strain-hardening effects could not be detected in these measurements since the strain rate is much lower compared to strain rates expected in sprays. The existence of this effect for the polymer solutions, especially for K90 solutions, is likely in the twin-fluid atomization process under investigation. 

This important result demonstrates the value of the tensor form of Newton s viscosity law. It is directly analogous to the result in Chapter 1, that the tensile modulus is three times the shear modulus, eq. 1.5.11. The three times rule for viscosity in steady uniaxial extension is often called the Trouton ratio. We see it holds true at low rates for the polymer melt in Figure 2.1.3. The following examples give applications of the Newtonian model to more complex deformations. Further examples appear at the end of the chapter. Bird, et al. (1987, Chapter 1) or any other good fluid mechanics book contains many worked Newtonian examples. 

Note that the Trouton ratio of 3 is alteady included when ju. i is used to define uniaxial extensional viscosity, in contrast to t]u used in Chapter 2 (eq. 2.3.9) and frequently in the literature. 

This Newtonian flmd behavior was first noted by Trouton, and the quantity Tj e)l3 t]q is sometimes referred to as the Trouton ratio and used to normalize extensional viscosity data. The reader should be aware, however, that this name is also used for several other ratios, including X )/Vo Vi )l Viy = Vi )l ViY = V3 f), the last of which compares the extensional... 

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