The spectrum of the elastic rayleigh equation

Abstract

Fluids have fascinated many generations of scientists and engineers. Although a considerable amount of research has been devoted to the study of uids of low molecular weight (well described by the Navier-Stokes equations), many challenging problems in both theory and applications still remain. But, even more challenging are non-Newtonian uids, whose motions cannot be described by the Navier-Stokes equations. The present work would be of fundamental importance to the dynamics of fast ows of a class of such uids (dilute polymeric solutions). Here, we consider a well-known theoretical model (Oldroyd-B uid) in order to represent, in the simplest possible manner, the polymer-solvent coupling in a dilute polymer solution, and study how the resulting non-Newtonian rheology a ects the known structure of the continuous spectrum in the inviscid limit. In general, the equation governing the evolution of small-amplitude perturbations to inviscid shearing ows, the Rayleigh equation, is singular, and the continuous spectra associated with the Rayleigh equation owe their origin to such singular points. Additional continuous spectra exist with the introduction of elasticity, and it has already been shown, in the inertialess limit, that the nature of the continuous spectrum is sensitive to the base-state velocity pro le and the particular constitutive model used (UCM v/s Oldroyd-B; Wilson et al. (1999)). The viscoelastic continuous spectra owe their origin to the `simple uid' assumption underlying almost all constitutive equations used in polymer rheology. The fact that the polymeric stress only depends on the evolution of the polymer conformation along a particular streamline, and is not in uenced by the polymer molecules convected by streamlines in the immediate vicinity, supports the existence of continuous spectra in elastic liquids, and this is independent of the Reynolds number. In this thesis, we study the nature of the elastic continuous spectrum at large Reynolds number which serves as a complement to the aforementioned study of Wilson et al. (1999) in the absence of inertia.