Phenomenological Theory of Non-Newtonian Flow

Abstract

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 scale¹⁾. There have been made many experiments that show the relation quantitatively
η(κ=γω)=η'(ω), 1<γ<2 (1)
where η (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 report³⁾. Denoting the observable displacement tensor and the internal strain tensor by a and λ, respectively, we have the fundamental equation of our Maxwell model:
dλ/dt=da/dt·a^-1·λ+λ·a^+-1·da⁺/dt+(dλ/dt)^* (2)
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
(dλ/dt)^*=-β/1+θ(1+θλ)·(λ-1)=-β[1+φ(λ-1)]·(λ-1) (3)
with two constant parameters β>0 and θ≥0 or φ=θ/(1+θ). On the other hand, the stress tensor σ may be written in the following form attending to the condition σ_e=σ-P1→0 for λ→1:
σ_e=G[1+ν(λ-1)]·(λ-1) (4)
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
(dλ/dt)^*=-β/G[σ_e+Gφ(1-ε)(λ-1)·(λ-1)] (5)
where ε=ν/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 gradient dependence of the viscosity coefficient for a series of values of the parameters φ and ε. In these figures the curves denoting as“Dynamic”show the dynamic viscosity vs. angular velocity relation for which the abscissa should be read as the value of the reduced 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
Δ₁σ=σ₁₁+σ₂₂-2σ₃₃ (6)
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 investigators⁴⁾.