This course develops the principles of hydrodynamics from fundamental statistical and continuum viewpoints, progressing toward complex systems in engineering, biology, geophysics, and astrophysics. Emphasis is placed on scaling laws, similarity solutions, singular behaviors, along with modern applications across the physical and biological sciences, from electron flows to geophysical fluid dynamics, from active matter to combustion and turbulence.
Statistical mechanics and hydrodynamic emergence; Boltzmann to Navier-Stokes; Knudsen number, local equilibrium, fluctuation–dissipation; continuum mechanics and constitutive laws; Eulerian and Lagrangian descriptions; kinematics of flow; Euler’s equation, Bernoulli integral; vorticity and Kelvin’s theorem; Navier-Stokes equations and boundary conditions; Stokes flows and lubrication theory; boundary layers, lift, drag, separation; similarity solutions and asymptotics; finite-time singularities, jet pinch-off, drop break-up; self-similarity, cusp formation, shocks; Kolmogorov theory, intermittency, law of the wall; transitions to turbulence, chaos, excitable models, directed percolation; Navier-Stokes existence; non-Newtonian fluids, viscoelasticity, shear-thinning; suspensions, colloids, granular flows; emulsions, foams, multiphase flows; surface tension, thin films, Marangoni flows; Instabilities; linear and nonlinear waves; GFD and shallow water theory, tsunamis; rotating flows, Coriolis effects, Taylor columns; stratification, internal waves, convection; hurricanes and atmospheric vortices; viscous electron fluids, Gurzhi effect; MHD, Alfven waves, reconnection; plasma hydrodynamics, Langmuir waves; combustion, flame theory, deflagration, detonation; Stefan problems and phase change; low Reynolds number swimming; cilia, flagella, slender-body theory; active matter, nematodynamics, towards ML and data-driven hydrodynamics.