Hinch-Leal closure

A closer look at this concentration dependence of the first positive maximum N, as well as the shear rate at which this occurs, was conducted by Baek et al [46]. Both were seen to be monotonically increasing functions up to concentrations of 40%. They demonstrated qualitative agreement with the predictions of Doi theory with the Hinch-Leal closure and the Maier-Saupe potential. They also note that the ratio of shear rate at which becomes negative to the shear rate of the first positive maximum remains constant at about 3.5. (Our data from 1978 and 1980 yielded an average ratio of 3.2 with a standard deviation of 0.9 for nine PBG solutions and an average ratio of 2.2 with a standard deviation of 0.3 for three PCBZL solutions.) They determined that the rapid increase in (dy/dt) with concentration cannot be attributed to a decrease in molecular relaxation time with... 

Figure 11.25 (a) Shear rate ymax at which the first normal stress difference reaches its positive maximum Nimax. versus reduced concentration U/U predicted by the Smoluchowski equation using an exact spherical harmonic solution and using the Hinch-Leal approximate closure (see Baek et al. 1993b). These predictions use... 

The following composite closure was derived by Hinch and Leal (1976) ... 

To obtain the expressions from Eq. (12.23) for bulk shear stress and bulk first normal stress difference in steady-state shear flow, for instance, one must evaluate the quantity (uuuu) defined by Eq. (12.20). For this, information on f(u, t) must be provided. Once /(u, t) is known, in principle one can calculate (uuuu) fromEq. (12.20). Several different ways of approximating (uuuu), referred to as closure (or decoupling) approximation, have been suggested in the literature (Advani and Tucker 1980, 1987 Doi 1981 Hand 1962 Hinch and Leal 1976 Marrucci and Grizzuti 1984) (see Chapter 9). The simplest form of approximation is... 

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See also in sourсe #XX -- [ ]

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