Abstract:
Hydrodynamic stability, an important branch of fluid mechanics, owes its popularity to the
successful prediction of the transition of (unstable) laminar states in a wide class of flows. While
the traditional approach has been reliant on an analysis within a modal framework, the merits of
a nonmodal approach have been recognized in the last two and a half decades particularly in the
context of shearing flows. Although the non-modal approach typically involves the solution of
an initial value problem, the resulting temporal response, for short times and at large Reynolds
numbers (Re), may also be understood in terms of the dynamics of the underlying inviscid
continuous spectra (CS). The equations governing the evolution of small-amplitude perturbations
in shearing flows in this limit are usually singular, and the continuous spectra owe their origin
to such singular points. The thesis is mainly concerned with the structure of the singular
eigenfunctions comprising such inviscid continuous spectra in rotating flows, and to a lesser
extent, with the singular eigenfunctions in homogeneous and stratified parallel shearing flows,
and rotational flows in the presence of elasticity. The manner in which such eigenfunctions may
be superposed to obtain a solution of the initial value problem is also considered. The detailed
analysis is devoted to the singular modes of a Rankine vortex both in two and three dimensions;
an analytically soluble problem that nevertheless offers insight into the singular eigenfunctions
associated with more general vorticity profiles. The Rankine analysis is then extended to smooth
vortices. In three dimensions, such an extension is made by drawing an analogy with (stably)
stratified shear flows and the associated continuous spectra. The final part of the thesis discusses
the continuous spectrum of an elastic vortex column in the limit of high Reynolds and Deborah
numbers. Further, a novel two-dimensional instability of an elastic vortex column, that arises
from the resonant interaction of elastic shear waves, is analyzed in detail both numerically and
via an asymptotic analysis valid in the limit of weak elasticity.
