What you'll learn

Models:

• Formulation of continuum models
• Integral and differential forms of conservation equations
• Interfacial conservation equations
• Models of diffusion
• Random walks

Mathematical Formulation:

• Formulation of continuum models and their exact and approximate solution
• Scaling, dimensional analysis, and similarity solutions
• Perturbation methods, matched asymptotic expansions
• Fourier series, eigenfunction expansions
• Fluid dynamics, waves, capillary phenomena, and instabilities
• Forced and natural convection
• Phase transformations and electrochemical transport

Scaling:

• Dimensional analysis
• Similarity solutions for linear and nonlinear diffusion
• Green functions and the method of images

Asymptotics:

• Asymptotic analysis
• Regular and singular perturbations for algebraic and differential equations
• Matched asymptotic expansions

Series Expansions:

• Fourier series and eigenfunction expansions
• Sturm–Liouville theory
• Generalized Fourier series: Bessel functions, spherical Bessel functions, and Legendre polynomials

Fluid Mechanics:

• Unidirectional Couette and Poiseuille flows
• Lubrication approximation and the Reynolds lubrication equation
• Linear and nonlinear waves and the method of characteristics
• Tensor algebra and calculus for continuum mechanics
• Navier–Stokes equations
• Creeping flow at low Reynolds number
• Inertial flow at high Reynolds number; turbulence
• Interfacial tension, wetting, and thin films
• Linear stability analysis

Convection:

• Boundary-layer and fully developed forced convection
• Taylor dispersion
• Natural convection and combined convection
• Turbulent convection

Nonequilibrium Thermodynamics:

• Nonequilirbrium phase transformations
• Calculus of variations
• Cahn–Hilliard and Allen–Cahn equations for phase separation

Electrochemical Transport:

• Electrochemical transport in neutral electrolytes
• Equilibrium double layers and linear electrokinetics in charged electrolytes