Session Details

Day 1:

Introduction to dynamical systems – linearizing nonlinear system of ODEs, Phase Plane Analysis, fixed points and their stability, Bifurcation Analysis with the damped oscillator example, Lyapunov and Asymptotic stability.

Day 2:

Motivating linear stability analysis of generic Fluid Dynamical Systems, normal mode analysis, problems with density variation having exact solutions, e.g. Rayleigh- Taylor Instability and Rayleigh–Bénard Convection, shear flow problem with exact solution – Kelvin-Helmholtz instability.

Day 3:

Stability of smooth Velocity profiles, Rayleigh’s inflexion point Theorem, Squire’s Theorem, deriving the governing equation for density stratified shear flows with Viscosity – the density stratified Orr-Sommerfeld equation (also known as the Viscous Taylor- Goldstein equation).

Day 4:

Exploring numerical solutions (using MATLAB) of the density stratified Orr- Sommerfeld equation with various examples – boundary layer instabilities in channel (or pipe) flows, Kelvin-Helmholtz instability in mixing layers, discussion of applications in industry (e.g. combustion in IC engines, multiphase flows, industrial displacement flows) and geophysics (ocean mixing, barotropic instability of atmospheric jets). Limitations of normal mode theory and pointers to recent developments in linear stability analysis, including data-driven stability analysis.