Extensional viscosity defined

The Apparent Extensional Viscosity The evolution of filament diameter is driven by the capillary pressure and resisted by the extensional stress in the fluid. Thus, measurements can also be represented in terms of an apparent extensional viscosity, defined as... 

In an analoguous manner the time dependent extensional viscosity , defined as... 

Extensional Viscosity. In addition to the shear viscosity Tj, two other rheological constants can be defined for fluids the bulk viscosity, iC, and the extensional or elongational viscosity, Tj (34,49,100—107). The bulk viscosity relates the hydrostatic pressure to the rate of deformation of volume, whereas the extensional viscosity relates the tensile stress to the rate of extensional deformation of the fluid. Extensional viscosity is important in a number of industrial processes and problems (34,100,108—110). Shear properties alone are insufficient for the characterization of many fluids, particularly polymer melts (101,107,111,112). 

Since the flow is axially symmetric, the stress is also symmetric and analogous to shear viscosity, the extensional viscosity can be defined as... 

We define a material function rj, commonly called the elongational or extensional viscosity, through the primary normal stress difference % — %iT, thus, for the case of F e), it is given by... 

An elongational or extensional viscosity (%) develops as a result of a conformational transition when disperse systems are forced through constrictions, or compressed or stretched (Kulicke and Haas, 1984 Rinaudo, 1988 Barnes et al., 1989 Odell et al., 1989 Clark, 1992). The intuitive logic is that the random coils resist the initial distortion. % is believed to elicit the human sensation of stringiness (Clark, 1995). If shear viscosity is denoted iq, rheologists define a Trouton ratio as %/ti, wherein % > T) by a factor approximating 3 for uniaxial extension and 6 for biaxial extension. Alternatively stated, the Newtonian ly calculates to one-third to one-sixth % (Steffe, 1992). 

Eq. (15.98) strongly resembles Eq. (15.61), that we obtained for shear stress build up after starting a steady shear flow with shear rate q at time t = 0. Accordingly, the elongational or extensional viscosity, which is defined to be... 

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

Often, it is not possible to reach a steady state in extension and it is convenient to define a transient extensional viscosity, tje, that is a function of time, t, and the extensional strain rate, e, (Barnes et al., 1989). 

The biaxial extensional viscosity ( b) was defined as the ratio of the normal stress difference and the radial extension rate assumed to be half the strain rate ... 

When describing dilatant behavior, the maximum stretch rate, e, in the converging flow at the contraction is a better parameter, but more difficult to be calculated. Instead of the term stretch rate, other authors also used deformation rate (e.g., Chauveteau, 1981) or elongational rate (e.g.. Sorbic, 1991). The shear-thickening viscosity is also called elongational viscosity (often referred to as the Trouton viscosity Sorbie, 1991) or extensional viscosity in the literature. James and McLaren (1975) reported that for a solution of polyethylene oxide (a flexible coil, water-soluble polymer physically similar to HPAM), the onset of elastic behavior at maximum stretch rates was of the order of 100 s and shear rates of the order of 1000 s. In this instance, the stretch rate is about 10 times lower than the shear rate. However, some authors use shear rate instead of stretch rate in defining the Deborah number—for example, Delshad et al. (2008). 

Furthermore, when this flow is looked upon as biaxial stretching, it is thought of as being generated by a radial, tensile stress rather than an axial, compressive stress. Thus, the biaxial extensional viscosity is defined as... 

In marked contrast to measurements of shear rheological properties, such as apparent viscosity in steady shear, or of complex viscosity in small amplitude oscillatory shear, extensional viscosity measurements are far from straightforward. This is particularly so in the case of mobile elastic liquids whose rheology can mitigate against the generation of well-defined extensional flow fields. 

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. ... 

In sim le elongational flow we define only the extensional viscosity (q) as follows... 

For the uniaxial extensional flow the extensional viscosity is defined as (Dealy 1994)... 

Here, and are, respectively, strain of the dispersed phase (defined as a ratio of the long axis to the original drop diameter) and that of the matrix, and is the extensional viscosity ratio. The relation was experimentally verified for < 1. 

In extensional flow, the diagonal components of are non-zero (i.e. T,y = 0 for i j). In the case of uniaxial extension, Th is the primary stress that can be measured, while T22 and T33 are generally equal to the pressiure of the environment. Thus, the uniaxial extensional viscosity rj is defined by. 

By the same token, the biaxial extensional viscosity ri can be defined... 

Following Newton (1640), the viscosity is defined as the ratio of the stress over the deformation rate. Whether a shear or a simple extensional flow is considered, one has then the shear or the extensional viscosity, and if such quantities are rate dependent, one deals with shear or extensional viscosity functions. Experimentally,... 

As indicated in Figure 7.1.2, there are several types of extensional deformation all can be described using the convention given in Chapter 4, which defines two transient extensional viscosities (Meissner, 1985)... 

From the equations for stress and rate we can readily define the extensional viscosity following eq.7.1.1 with m = —0.5. 

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. 

From this relation and eq. 7.5.5 we can define an apparent uniaxial extensional viscosity for the fiber spinning experiment... 

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