Uniaxial extension Newtonian fluid

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... 

For steady uniaxial extension, eq. 2.2.19 gives the components of 2D. Using these components with eq. 2.3.2 gives the stresses in a Newtonian fluid... 

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. 

Calculate the predictions of the upper-convected Maxwell equation in (a) start Up of steady shear and (b) steady state uniaxial extension for arbitrary shear rate y and extension rate e, and compare these predictions with those for the Newtonian and second-order fluids. 

Unlike shear viscosity, extensional viscosity has no meaning unless the type of deformation is specified. The three types of extensional viscosity identified aind measured are uniaxial or simple, biaxial, and pure shear. Uniaxial viscosity is the only one used to characterize fluids. It has been employed mainly in the study of polymer melts, but also for other fluids. For a Newtonian fluid, the uniaxial extensional viscosity is three times the shear viscosity ( fe)uni = 3/ . This is confirmed at very low shear rates in Figure 13, which provides a typical example of the extensional viscosity behavior of a polymer (129). The two other extensional viscosities are used to study elastomers in the form of films or sheets. Uniaxial and biaxial extensions are important in industry (118,125-128,130,132), the former for the spinning of textile fibers and roller spattering of paints, and the latter for blow molding, vacuum forming, film blowing, and foam processes. 

A.8 Efficiencies of Mixers and Striation Thickness Reduction. Calculate the shear and extension rates required to reduce the striation thickness 1000 times in 10 s in the three mixers—pure shear, uniaxial extensional, and simple shear—and then prove Eq. 6.81 assuming Newtonian fluid behavior. 

Growth of the uniaxial extensional viscosity t) after onset of steady extension for the UCM/Lodge equation compared to a Newtonian and a second-order fluid. 

For simple Newtonian liquids (and some complex fluids at low enough extension rates, e where ip is again the longest material relaxation time or terminal time), the viscosity of a material measured in uniaxial elongation becomes independent of time upon persistent stretching and can be related to its zero-shear viscosity by the... 

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