Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids

Normal-stress effects and the variation of apparent viscosity with rate of shear in simple types of steady flow of certain idealized elastico-viscous liquids are discussed. The liquids are those whose behaviour at sufficiently small variable shear stresses can be characterized by three constants (a coefficient of viscosity, a relaxation time and a retardation time) and whose invariant differential equations of state for general motion (involving eight independent physical constants) are linear in the stresses and include terms of no higher degree than the second in the stresses and velocity gradients together. The normal stresses which, in addition to shear stresses, are present in such a liquid in a state of simple shearing flow, or in flow in a circular pipe, or between rotating cylinders, are investigated; and the conditions under which the Weissenberg climbing effect will occur, in a positive or negative sense, are examined. In many liquids of this class, steady rectilinear flow under a uniform pressure gradient is not always possible in a straight pipe of arbitrary section, nor is steady flow in horizontal circles in a region bounded by arbitrary surfaces of revolution in relative rotation about common vertical axis. The behaviour of these idealized liquids when sheared in a narrow gap between a rotating wide-angled cone and a flat plate is compared with the observations of Roberts (1952, 1953) on some real elastico-viscous liquids. Certain liquids of this class, characterized by six independent constants satisfying certain inequalities, exhibit rheological behaviour which is, at least qualitatively, similar to the behaviour of many real elastico-viscous liquids in the following respects: the behaviour at small variable shear stresses, the variation of apparent viscosity with rate of steady shearing, the climbing effect up a vertical rod rotated in the liquid, and a distribution of normal stresses equivalent to an extra tension along the streamlines (with an isotropic state of stress in the plane normal to the streamlines) which is present in all the simple types of steady shearing flow investigated. These liquids can flow steadily in straight lines through a straight pipe of any section.