Second normal stress negative coefficient

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]. 

Rg. 9.20 First and second normal stress differences plotted against the shear rate on a logarithmic scale, nsing a = 1 as a typical example. The second coefficient is negative and its ratio is virtually constant over a wide range of shear rates, (a) Pq = 0, (b) = 1. (Reprinted with permission from... 

Second normal stress coefficients have not yet been reliably determined, and thus eq. 11.3.17 is difficult to evaluate at this time. They do seem to be finite, small ( 0.1 N ), and negative. 

Figure 3.37 Negative ratio of the second to the first normal stress coefficients versus stress ratio. Vi /a for various dilute solutions (concentration < 0.6 wt% closed symbols) and entangled solutions (concentration > 1 wt% open symbols). (Reprinted with permission from Magda et al.. Macro-molecules 26 1696. Copyright 1993, American Chemical Society.)...

When each component of one vector is linearly related to each component of another vector the coefficients of proportionality are the components of a second-rank tensor. Stress, defined as force per unit area, is the quotient of two vectors, and is an example of a second-rank tensor. Note that the condition defining positive in the direction of the outward normal means that a hydrostatic pressure must be negative. Strain is also a tensor, and in the most general case both stress and strain can be expressed in terms of nine tensor components. Stress and strain are both examples of second-rank tensors, which have nine components vectors, which have three components, are sometimes referred to as first-rank tensors single-valued scalars are zero rank tensors. 

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