Viscosity Trouton

Combining Eqs. 3.1-31 and3.1-26, we obtain the so-called Trouton relation, which defines the Trouton viscosity (13). 

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). 

For steady elongational flow (ic = constant) it is conventional to define a quantity tf, called the elongational viscosity (or Trouton viscosity ), as... 

Elongational viscosity (extensional viscosity, Trouton viscosity) n. The viscosity that characterizes an element undergoing elongational flow (above). It is equal to the tensile stress divided by the rate of elongation and for polymers it depends on the rate, but may increase with rate, unlike the usual reduction of shear viscosity... 

The models developed here are visualized in tension, with tensile stress cr, tensile strain s, and Young s modulus E. However, the same theory holds true for pure shear (viscometric) deformation, where a shear stress t results in a shear strain y with proportionality constant G (Hooke s modulus), rj represents the Newtonian (shear) viscosity, while the elongational (Trouton) viscosity is given by rjg. 

To describe this nonviscometric deformation, a new material function the elongational or Trouton viscosity, is needed to relate tensile stress to the rate of tensile strain, ctu = T7e(su) iis or equivalently the function (t — 0 11 (an). For incompressible Newtonian fluids, it may be shown that = Sjj,... 

In addition to the observations in shear that have just been discussed, there is considerable interest in the elongational response in polymeric fluids. The elon-gational or Trouton viscosity ije is three times the zero shear rate viscosity in the linear viscoelastic range. However, its behavior is more complex and, as shear rates get higher, the viscosity can even go through a maximum with increasing shear rate as illustrated in Figure 22 (77). 

The two functions and 2 depend on the parameter b as well as on the elongation rate e in steady shear-free flows. When b = 0, the function 2 is zero, and is replaced by the symbol tj, which is called the elongational viscosity . The elongational viscosity describes the resistance to elongational flow if 8 is positive and the resistance to biaxial stretching if e is negative (the terms extensional viscosity and Trouton viscosity have also been used for ). 

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