The invariant forms of rheological equations of state for a homogeneous continuum, suitable for application to all conditions of motion and stress, are discussed. The right invariance properties can most readily be recognized if the frame of reference is a co-ordinate system convected with the material, but it is necessary to transform to a fixed frame of reference in order to solve the equations of state simultaneously with the equations of continuity and of motion. An illustration is given of the process of formulating equations of state suitable for universal application, based on non-invariant equations obtained from a simple experiment or structural theory. Anisotropic materials, and materials whose properties depend on previous rheological history, are included within the scope of the paper.

In this paper we are concerned with a material which can support a finite stress elastically without flow and which flows with constant mobility(1) (or plastic fluidity) when the stresses are sufficiently great. Following Bingham(1) and Houwink(2), such a material is called a Bingham solid and the type of flow (purely) plastic. The transition from elastic to plastic behaviour takes place at the yield point.

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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.

A calculation is made of the elastic properties of a dilute emulsion of one incompressible viscous liquid in another, arising from the interfacial tension between the two phases. A linear relation between the stress tensor, the rate-of-strain tensor and their first time derivatives defines the behaviour at small rates of strain; the three constants involved are expressed as functions of the viscosities of the two components, the drop size (assumed small and uniform), the interfacial tension and the concentration. The relaxation time and retardation time for the system vary directly as the drop diameter and inversely as the interfacial tension. The effect of slip at the interfaces, which might be associated with the presence of a film of a third component introduced as a stabilizer, is also calculated. The values of the rheological constants are appreciably altered if the frictional coefficient specifying the degree of slipping is sufficiently small, but the type of elastic behaviour is unchanged. In the case of a suspension of elastic solid particles in a viscous liquid, slip at the solid-liquid interfaces can cause a change in the type of elastic behaviour.