Fluid flows with vortices are common observations in daily life. From wash basin vortices
to oceanic eddies and cyclones, vortical structures can reside in a wide range of scales.
In engineering flows vortices usually form behind solid objects moving in a fluid medium
(e.g. wingtip vortices in aircraft wakes). Instabilities in vortices have hence been studied
extensively, owing to their fundamental and applied consequences. Small-scale linear instabilities have characteristic wavelengths smaller than the dominant vortex length scale.
These short-wavelength instabilities are a mechanism by which three-dimensional features
arise in large-scale two-dimensional flows, thus having implications in turbulent flows. Density stratification is a factor that influences vortex instability characteristics, in geophysical
and aircraft wake applications for example.
In this thesis, we investigate the effects of Schmidt number (Sc) (the ratio between
kinematic viscosity and mass diffusivity) on small-scale instabilities in vortices that occur in
a density-stratified environment. A local stability analysis based on a WKBJ approximation
is used to study these instabilities. The method is first validated by reproducing inviscid
results of centrifugal instability. An axisymmetric vortex with a stable density stratification
along its vortical axis is then considered. While a centrifugally stable axisymmetric vortex
remains stable for any Sc, it is shown that Sc can have significant effects in a centrifugally
unstable axisymmetric vortex: the range of unstable perturbations increases when Sc is taken
away from unity, with the extent of increase being larger for Sc 1 than for Sc >> 1.
Additionally, for Sc > 1, we report the possibility of oscillatory instability. Our studies will
be extended to elliptical and hyperbolic instabilities in the thesis.