非ニュートン流動の現象論 : 高分子液体の流動性
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概要
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It has been pointed out by several authors that in some polymeric substances there exists an interesting relation between steady viscosity and dynamic viscosity, namely that the velocity gradient dependence of steady viscosity is similar to the angular velocity dependence of dynamic viscosity, at least in the range of a long time scale. There have been made many experiments that show the relation quantitatively [numerical formula] whereη (κ) denotes the steady shear viscosity as a function of velocity gradient κ and η' (ω) denotes the dynamic viscosity as a function of angular velocity ω. In this paper we shall consider such a relation phenomenologically by making use of the three dimensional Maxwell model described in the previous report. Denoting the observable displacement tensor and the internal strain tensor by α and λ, respectively, we have the fundamental equation of our Maxwell model : [numerical formula] where (dλ/dt) is the term due to the dissipation mechanism. Attending to the non-negativity of the dissipation energy - (1/2)Sp[(dλ/dt)・λ^<-1>・σ] (σ is the stress tensor), we assume the dissipation term in the form [numerical formula] with two constant parameters β>0 and θ>__-0 or φ=0/(1+θ). On the other hand, the stress tensor σ may be written in the following form attending to the condition [numerical formula] [numerical formula] where ν is a constant parameter which may be non-negative in high-polymeric system consisting of the so-called Langevin chains. From Eqs. (3) and (4), we have [numerical formula] where ε=υ/φ gives non-linearity between the dissipation term and the stress. The dissipation term may increase more rapidly than the stress with increasing strain, so that we assume 0<__-ε<__-1. In Figs. 1〜3 we can find the velocity grandient dependence of the viscosity coefficient for a series of values of the parameters φ and εε. In these figures the curves denoting as "Dynamic" show the dynamic viscosity vs. angular velocity ω/β instead of the reduced velocity gradient κ/β. These figures show that the rather strange relation, Eq(1) is qualitatively reproduced by our non-linear model. In the case of φ=1, we have η'(ω=κ)=η(κ), irrespective of the value of ε. In Figs. 4〜6 is shown the normal stress difference [numerical formula] which is measurable directly by the so-called Weissenberg rheogoniometer of cone and plate type. This behavior is qualitatively in good agreement with the experimental results obtained by several investigators.
- 社団法人日本材料学会の論文
- 1964-05-15
著者
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