Second-order fluid extensional

Example 4.3.2 Uniaxial Extensional Viscosity for a Second-Order Fluid... 

Thus for steady state uniaxial extensional flow, in contrast to steady state shearing flow, the second-order fluid result agrees with the UCM prediction only at small strain rates. The UCM, second-order fluid, and Newtonian fluid equations all differ in their predictions of the strain rate dependences of the extensional viscosity, though the strain rate dependences of the shear viscosity are the same for all three equations. This result typifies the usual finding that constitutive equations differ among themselves more strongly in their predictions of extensional viscosities than in their predictions of shear viscosities. 

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. 

We noted in Section 10.7.2 that the second-order fluid approximation for flows only marginally removed from the rest state indicates that the first and second normal stress differences are second order in the shear rate, so that the first and second normal stress coefficients Pj q and T z 0 approach non-zero limiting values at vanishing shear rate. The second-order approximation also predicts that the net stretching stress in uniaxial extension is second order in the Hencky strain rate, and this implies that the extensional viscosity approaches its limiting zero-strain-rate value 3t7o with a non-zero slope ... 

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