Some steady flows of the general elastico-viscous liquid

The equations of state of an elastico-viscous liquid are taken to be any set of integro-differential equations (with the appropriate invariance properties) relating deformation history, stress history and temperature history, subject only to the absence of compressibility, of any inherent anisotropy, and of any special reference configuration with a permanent significance as a ground state. A rheological history (i.e. deformation, stress and temperature histories taken together) that satisfies the equations of state can be found, in terms of stress functions capable of experimental determination, by considering the general liquid in some very simple state of flow; of which steady isothermal simple shearing and steady isothermal pure shearing are discussed in detail and are shown to involve essentially different rheological histories. Then such particular solutions of the equations of state can be used to solve other flow problems. A rheological history to time t is most easily represented as a convected metric tensor, stress tensor and temperature, all functions of past time t' ($\leqslant$ t), at a particle $\xi^i$ in a convected system of reference. Steady rectilinear flow, as in a pipe of arbitrary section, and steady rotational flow in which each particle moves in a circle about an axis of symmetry, caused by the rotation of coaxial surfaces of revolution of arbitrary section, are discussed, and in each case the stress distribution is determined and a general condition is found for that type of flow to be possible. If the condition is not satisfied, some secondary flow transverse to the assumed primary streamlines is inevitable. Helical flow under a constant pressure gradient, along an annular gap between circular cylinders in relative rotation, is possible in all liquids; but some other flows that seem at first sight no more complicated cannot be analyzed from a knowledge only of the rheological histories of elements in steady simple and pure shearing. Liquids of the Roberts (1953) type, with a normal stress difference equivalent to an extra simple tension along the streamlines in simple shearing, are capable of steady rectilinear flow in all circumstances. But these liquids, in common with other types, can in general be expected to develop a radial-axial secondary flow (as is often observed) in the mound of liquid 'climbing' up near the inner cylinder, when the Weissenberg climbing effect is being demonstrated in a liquid contained between coaxial vertical cylinders in relative rotation.