Maxwell limiting viscosity

This result, that the low frequency limit of the in phase component of the viscosity equates to the viscosity of the dashpot, means that for a single Maxwell model it is possible to replace rj by rj(0). Thus far we have concentrated on the description of experimental responses to the application of a strain. Similar constructions can be developed for the application of a stress. For example the application of an oscillating stress to a sample gives rise to an oscillating strain. We can define a complex compliance J which is the ratio of the strain to the stress. We will explore the relationship between different experiments and the resulting models in Section 4.6. 

From the foregoing it becomes evident that only a measure of the magnitude of the form birefringence, but not of its influence on the extinction angle, can be given. For this purpose the reader may be reminded that the stress-optical coefficient of an infinitely dilute solution can be expressed by one half of the ratio of Maxwell constant to intrinsic viscosity [eq. (2.33)]. In the absence of the form birefringence the limiting... 

Fig. 5.4. Ratio of Maxwell-constant to intrinsic viscosity as a function of molecular weight for linear oligomers of poly oxy-propylene glycol in cyclo-hexanol at 20° C (full circles). Ratio of birefringence to shear stress in the limit of zero shear stress for the same oligomers in bulk (temperatures 20—60° C) (open squares). The experimental point at the righthand side is obtained on a high molecular weight sample, Tsvetkov, Garmonova and Stankevich (166)...

The Poisson-MaxweU theory that the viscous flow of a hquid is analogous to the yieldihg of a solid under forces exceeding the elastic limit ( 3.IX F) has received much attention recently. According to Maxwell, if P is the shearing force per unit area, i the time, 6 the deformation, n the shear modulus ( 4.IX F), then for a solid free from viscosity ... 

The values of the dimensionless parameters 2 and CO for the most classic collision models are given in Table 1. The Maxwell molecules (MM) model assumes a linear relationship between viscosity and temperature, although for the hard sphere (HS) model, the viscosity is proportional to the square root of the temperature. These models could be roughly considered as limits for the real behavior of gases, and the variable hard sphere (VHS) model proposed by Bird [2] is much more accurate. Another sophistication has been proposed by Koura and Matsumoto who developed the variable soft... 

The only parameter in (11) having dimensions of time is C. Although this is not a relaxation time per se, it can be associated with a "Maxwell-type relaxation time, as follows. Although the linear Maxwell model predicts a constant (Newtonian) viscosity, it may be generalized by utilizing a co-rotational reference frame which follows the local rotation and translation of each fluid element [9]. When a term is added to account for the high shear limiting behavior, the result is the co-rotational form of the Jeffreys model ... 

Here, we neglect the elastic part of the deformation and still suppose 8 < tu- This limit is quite imrealistic for commonly produced shapes and the nonlinear solution is to be considered. Adding the elastic and viscous solutions (Eqs. (6.39) and (6.40)) results in a Maxwell modelling of the plate behaviom as detailed in Section 6.4. Equation (6.40) cannot be integrated simply unless the temperature (the viscosity) is supposed to be constant, which is rarely the case when an industrial process is considered. 

It is seen that the material functions obtained from the covariant convected derivative of a are different from those obtained from the contravariant convected derivative of a. Experimental results reported to date indicate that the magnitude of N2 is much smaller than that of (say -A 2/ i 0.2-0.3). Therefore, the rheology community uses only the contravariant convected derivative of a when using Eq. (3.4), which is referred to as the upper convected Maxwell model. However, the limitations of the upper convected Maxwell model lie in that, as shown in Eq. (3.6), (1) it predicts shear-rate independent viscosity (i.e., Newtonian viscosity, t]q), (2) is proportional to over the entire range of shear rate, and (3) N2 = 0. There is experimental evidence (Baek et al. 1993 Christiansen and Miller 1971 Ginn and Metzner 1969 Olabisi and Williams 1972) that suggests Nj is negative. Also, as will be shown later in this chapter, and also in Chapter 5, in steady-state shear flow for many polymeric liquids, (1) l (k) follows Newtonian behavior at low y and then decreases as y increases above a certain critical value, and (2) increases with at low y and then increases with y (l < n < 2) as y increases further above a certain critical value. 

This modification of the convected Maxwell model contains one constant G (an elastic modulus) and the non-Newtonian viscosity function rj y). It describes the shear-rate dependence of the viscosity perfectly and the first normal stress coefficient rather well. In steady elongational flow it gives an infinite elongational viscosity, and does not simplify properly in the linear viscoelastic limit. Nonetheless it has been found to be useful in exploratory flow calculations aimed at assessing the interaction of shear thinning and memory. 

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