Zero-shear second normal stress coefficient

A review by Bird and Wiest [6] gives a more complete list of existing viscoelastic models. The upper convective model and the White-Metzner model are very similar with the exception that the White-Metzner model incorporates the strain rate effects of the relaxation time and the viscosity. Both models provide a first order approximation to flows, in which shear rate dependence and memory effects are important. However, both models predict zero second normal stress coefficients. The Giesekus model is molecular-based, non-linear in nature and describes thepower law region for viscosity andboth normal stress coefficients. The Phan-Thien Tanner models are based on network theory and give non-linear stresses. Both the Giesekus and Phan-Thien Tanner models have been successfully used to model complex flows. 

Here we have three parameters r/o the zero-shear-rate viscosity, Ai the relaxation time and A2 the retardation time. In the case of A2 = 0 the model reduces to the convected Maxwell model, for Ai = 0 the model simplifies to a second-order fluid with a vanishing second normal stress coefficient [6], and for Ai = A2 the model reduces to a Newtonian fluid with viscosity r/o. If we impose a shear flow,... 

Here t, 4, and 4 2 are three important material functions of a nonnewtonian fluid in steady shear flow. Experimentally, the apparent viscosity is the best known material function. There are numerous viscometers that can be used to measure the viscosity for almost all nonnewtonian fluids. Manipulating the measuring conditions allows the viscosity to be measured over the entire shear rate range. Instruments to measure the first normal stress coefficients are commercially available and provide accurate results for polymer melts and concentrated polymer solutions. The available experimental results on polymer melts show that , is positive and that it approaches zero as y approaches zero. Studies related to the second normal stress coefficient 4 reveal that it is much smaller than 4V and, furthermore, 4 2 is negative. For 2.5 percent polyacrylamide in a 50/50 mixture of water and glycerin, -4 2/4 i is reported to be in the range of 0.0001 to 0.1 [7]. 

To obtain the steady state results, we set the time derivative to zero. We find immediately that T33 = T22 = 0. With this result for T22. we find that T12 = mY> f >m which we can obtain Th = 2riokY. This result implies that the shear viscosity is a constant tjo. the fitst normal stress coefficient is also a constant equal to 2X> o> end the second normal stress coefficient is zero. This is the same result that we obtained in the second-order fluid limit of the UCM equation ... 

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 ... 

Here h(/i, 12), the damping function , is a function of the invariants of the Finger strain tensor given in equations (31) and (32) the damping function is determined by requiring the constitutive equation to describe shear and elongational flow data. Extensive comparisons with experimental data show that this rather simple empiricism is extremely useful. Equation (47) gives a value of zero for the second normal stress coefficient. 

Vrc y,y r 11 o - volume fraction of dispersed and matrix phase, respectively - volume fraction of the crosslinked monomer units - volume fraction of phase i at phase inversion - maximum packing volume fraction - percolation threshold - shear strain and rate of shearing, respectively - viscosity - zero-shear viscosity - hrst and second normal stress difference coefficient, respectively... 

The Maxwell model results in a second normal stress, Ni = Oyy - azz, equal to zero. In determining the contribution of the stress field to the effective diffusion coefficient, we also need to consider how the shear rate is perturbed by the presence of concentration fluctuations. If the relative velocities are eliminated from Eqs. 57 and 58, then an important constraint on the stress tensor can be found. 

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