Outline
Algebra and Calculus of Tensors (3 lectures)
- Dyadic Product of Vectors
- Gradient of a Scalar Field: Implicit Definition
- Partial and Material Derivatives
- Surface-Integral Defintions for Divergence, Curl and Gradient of a Vector Field
- Physical Meaning of Divergence and Curl of Fluid Velocity Field
Vector Field Theory (1 lectures)
- Divergence Theorem and the Continuity Equation
- Reynolds Transport Theorem
- Stokes Theorem and Kelvin’s Velocity Circulation
- Irrotational, Conservative, and Derivable from a Potential
- Divergence of a Second-Rank Tensor
Introduction to Continuum Mechanics (4 lectures)
- Continuum Hypothesis and Constitutive Equations
- Hydrostatic Equilibrim
- Flow of Ideal Fluids: Euler’s Equation of Motion
- Kelvin’s Circulation Theorem
- Irrotational Flow of an Incompressible Fluid (How Airplanes Fly)
- Potential Flow Around a Sphere
- d’Alembert’s Paradox: Lift but No Drag is Predicted
- Velocity Potential and Stream Function
Viscous Fluids (3 lectures)
- Nonideal Behavior Caused by Viscous Friction
- With Nonideality, Stress is a Tensor of Rank Two
- Analogy Between Viscous Stress and the Diffusive flux of Momentum
- Response of Elastic Solids to Stress: Hooke’s Law
- Response of Viscous Fluids to Stress: Newton’s Law of Viscosity
- Navier Stokes Equation and Boundary Conditions
Exact Solutions of NSE (1 lecture)
- Fully Developed Flow in a Cylindrical Duct
- Flow of Thin Film Down an Inclined Plane
- Unsteady Flow Near a Plate Suddenly Brought into Lateral Motion
- Flow Near a Rotating Disk
Creeping-Flow Approximation (3 lectures)
- The Cone-and-Plate Viscometer
- Stokes Flow Around a Sphere
- Whitehead’s Paradox and Oseen’s Approximation for Leading Term for Inertia
- Proudman & Pearson’s Matched Asymptotic Expansions
- Slow Flow Around a Cylinder: Stokes Paradox and Lamb’s Solution
Boundary-Layer Approximation (4 lectures)
- Origin and Mathematical nature of Boundary Layers
- Method of matched Asymptotic Expansions
- Boundary-Layer Equations for 2-D Flows
- Boundary Layer for Flow Tangent to Flat Plate
- Boundary Layer for Flow Normal to Cylinder: Boundary-Layer Separation
Lubrication Approximation (2 lectures)
- Sliding Motion Between Slightly Nonparallel Plates
- 2-D Sliding Motion Between a Cylinder and a Plate
- Cavitaion
- Sliding Motion Between a Sphere and a Plate
- Squeezing Flow
- Reynolds Equation
Turbulence (2 lectures)
- Origin and nature of Turbulence
- Description of Turbulent Flow in Pipes
- Time-Smoothed Equations of Motion: Reynolds Stress
- Prandtl’s Mixing Length Theory
Electrokinetic Phenomena (as time permits)