First normal stress difference negative values

Figure 11.12 Shear viscosity and first normal stress difference versus shear rate for 17% PBLG (molecular weight 350,000) in /n-cresol. The circled triangles are negative N values. (From Kiss and Porter, reprinted with permission fromMol. Cryst. Liq. Cryst. 60 267, Copyright 1980, Gordon and Breach Publishers.)...

Accordingly, the value of Ni at very low shear rates is twice that of the storage relaxation modulus at very low frequencies. The second normal stress difference, 2 = 7111—5133, is negative and smaller in magnitude than the first normal stress difference. The ratio —Ni/Nx lies in the interval 0.1A).3. 

Normal stress measurements for some MLC nematics was reported to be consistent with that of a second-order fluid, that the low frequency limit of G /co equaled the low shear limit of N /(dy/dty [36]. Coleman and Markowitz demonstrated that for a second-order fluid in slow Couette flow, the viscoelastic contribution to the normal thrust must have a sign opposite to the inertial contribution on thermodynamic grounds [37]. A textbook by Walters stated that the measurements of first normal stress difference have invariably led to a positive quantity except for one case which was later found to be in error [38]. Adams and Lodge reported the possible observation of a negative value for Nj for solutions of poly isobutylene + decalin [39]. This result was obtained by a combination of obtained from radial... 

One of the distinctive phenomena observed in the flow of liquid crystal polymers in the nematic state is that of a negative steady-state first normal stress difference, Ni, in shear flow over a range of shear rates. Ni is zero or positive for isotropic fluids at rest over all shear rates, which means that the force developed due to the normal stresses, tends to push apart the two surfaces between which the material is sheared. In liquid crystalline solutions, positive normal stress differences are found at low and high shear rates, with negative values occurring at intermediate shear rates (Kiss G. and Porter R. S. 1978). 

The negative first normal stress difference (Ni) could not be predicted by Leonov s nematodynamics model. The values of Ni for filled LCPs are shown as a trend beyond 10 s . ... 

In addition, the second normal stress difference, which for conventional polymers is in general negative with an absolute value of not more than 20% of the first normal stress... 

Realizable k-e Model. The realizable k-e model (Shih et al., 1995) is a fairly recent addition to the family of two-equation models. It differs from the standard k-e model in two ways. First, the turbulent viscosity is computed in a different manner, making use of eq. (5-16) but using a variable for the quantity C, . This is motivated by the fact that in the limit of highly strained flow, some of the normal Reynolds stresses, u , can become negative in the k-e formulation, which is unphysical, or unrealizable. The variable form of the constant C, is a function of the local strain rate and rotation of the fluid and is designed to prevent unphysical values of the normal stresses from developing. 

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