Free surface problems in rheological fluid mechanics

Abstract

In this paper we review our work on the theory of domain perturbations of the rest state of a viscoelastic fluid and its applications to the science of rheometry. To explain the principle behind the domain perturbation analysis we begin the paper with a discussion of a model problem, free of rheological complications, in which the principles involved may be clearly demonstrated. In Chapter two we formulate the analysis for rheological problems and we present some new, previously unpublished, simplifications of the theory. The canonical forms for stress tensors which perturb the state of rest are given in Chapter three. Chapter four is devoted to the problem of steady rod climbing. There we discuss the main physical features of the motion, the phenomenon of the critical radius, the effects of surface tension and temperature, secondary motions, the applicability of theory and experiment for rheological measurements, and an interesting new “normal stress anplifier”. Unsteady problems of rod climbing are considered in Chapter five. We review recent results on the free surface induced by torsional oscillations of a rod, and we show how to use these results to find the form of the material functions in the canonical forms of the stress. The breathing instability of steady axisymmetric rod climbing (aHopf bifurcation) and the flower instability of time-periodic climb induced by the oscillating rod (a symmetry-breakingPoincare bifurcation) are also described in Chapter five. In Chapter six we consider other free surface problems: the free surface on a fluid between oscillating planes, die swell and edge effects in rheometers.