PU-307 · Fluid Dynamics
Starting from the continuum hypothesis and conservation laws, the unit builds the Navier-Stokes equations as the governing dynamics of a deforming material and then explores their structure through the two great limits that make fluids tractable: inviscid, high-Reynolds-number flow (Euler, Bernoulli, vorticity, potential flow) and viscous-dominated, low-Reynolds-number flow (Stokes drag). It closes by confronting where clean theory breaks down — boundary layers, waves, instability, and turbulence — showing how dimensional reasoning and scaling recover predictive power when exact solutions fail.
Lectures
| L01 | The Continuum Hypothesis and the Fluid Description — |
| L02 | Kinematics: Material Derivative, Streamlines, and Deformation — |
| L03 | The Reynolds Transport Theorem |
| L04 | Conservation of Mass: The Continuity Equation |
| L05 | Stress in a Continuum: The Cauchy Stress Tensor |
| L06 | Cauchy's Momentum Equation |
| L07 | Constitutive Relations and the Navier-Stokes Equations |
| L08 | Boundary Conditions and the No-Slip Condition — |
| L09 | Exact Solutions: Couette and Poiseuille Flow |
| L10 | Energy, Dissipation, and the First Law for Fluids |
| L11 | The Inviscid Limit: Euler's Equations |
| L12 | Bernoulli's Theorem and Its Applications |
| L13 | Vorticity and Its Transport |
| L14 | Kelvin's Circulation Theorem |
| L15 | Helmholtz Vortex Theorems and Vortex Dynamics |
| L16 | Potential Flow and Laplace's Equation |
| L17 | Complex Potentials and Flow Past a Cylinder |
| L18 | D'Alembert's Paradox |
| L19 | Lift, Circulation, and the Kutta-Joukowski Theorem |
| L20 | Dimensional Analysis and the Reynolds Number |
| L21 | Creeping Flow and Stokes Drag |
| L22 | High Reynolds Number: Prandtl's Boundary Layer |
| L23 | Boundary-Layer Separation and Form Drag |
| L24 | Compressible Flow and Sound Waves |
| L25 | Surface Gravity Waves |
| L26 | Hydrodynamic Instability: Kelvin-Helmholtz and Rayleigh-Taylor |
| L27 | Introduction to Turbulence: The Energy Cascade |
| L28 | Kolmogorov's Theory and the Four-Fifths Law |
Derivations homed in this unit
Reynolds Transport Theorem
Derives the rate of change of an extensive quantity over a moving material volume by applying the Leibniz rule to a control volume with a moving boundary.
The Continuity Equation
Derives the local mass-conservation law ∂ρ/∂t + ∇·(ρu) = 0 by applying the Reynolds transport theorem to total mass.
Existence of the Cauchy Stress Tensor
Proves via the tetrahedron argument that the surface traction depends linearly on the outward normal, establishing the second-rank stress tensor.
Cauchy's Momentum Equation
Derives the differential momentum balance ρ Du/Dt = ∇·σ + ρf from the transport theorem plus the divergence of the stress tensor.
Navier-Stokes from the Newtonian Constitutive Law
Closes Cauchy's equation by positing a linear, isotropic stress–strain-rate relation, yielding the incompressible Navier-Stokes equations.
Euler's Equations as the Inviscid Limit
Recovers the inviscid Euler equations by taking viscosity to zero and discusses the singular nature of that limit near boundaries.
Bernoulli's Theorem, Steady and Unsteady
Integrates Euler's equation along streamlines for steady flow and over the whole field for irrotational flow to obtain the two Bernoulli relations.
The Vorticity Transport Equation
Takes the curl of Navier-Stokes to obtain the evolution of vorticity with its stretching, tilting, and diffusion terms.
Kelvin's Circulation Theorem
Proves that circulation around a material loop is conserved in barotropic, inviscid flow with conservative body forces.
Potential Flow and Laplace's Equation
Shows that incompressible irrotational flow admits a velocity potential satisfying Laplace's equation, linearising the problem.
D'Alembert's Paradox
Demonstrates that steady potential flow past a body produces zero net drag, exposing the need for viscosity and boundary layers.
Stokes Drag on a Sphere
Solves the creeping-flow (Stokes) equations around a sphere to obtain the drag force F = 6πμaU.
The Reynolds Number from Nondimensionalisation
Nondimensionalises Navier-Stokes to reveal the Reynolds number as the sole governing parameter and grounds dynamic similarity via the Buckingham Pi theorem.
Energy Balance and Viscous Dissipation
Derives the mechanical-energy equation and identifies the positive-definite viscous dissipation function as the rate of irreversible heating.
The Acoustic Wave Equation and Speed of Sound
Linearises the compressible Euler and continuity equations about a rest state to obtain the wave equation with c² = (∂p/∂ρ)_s.
Prandtl's Boundary-Layer Equations
Uses matched scaling of Navier-Stokes at high Reynolds number to reduce the near-wall dynamics to the boundary-layer equations.
Dispersion of Surface Gravity Waves
Solves potential flow with linearised free-surface conditions to obtain ω² = gk tanh(kh) and its deep- and shallow-water limits.
Kolmogorov's Four-Fifths Law
Derives the exact -4/5 scaling of the third-order longitudinal velocity structure function from the Kármán-Howarth equation under homogeneous isotropic turbulence.