FLUID MECHANICS
Undergraduate Physics Course - UERJ
Prof. Marcelo Santos Guimaraes
Based on David Tong’s lecture notes (University of Cambridge)
1st semester 2026 - Mondays and Wednesdays - 1h40 per class
Course Overview
This course covers the foundations and main applications of fluid mechanics in 25 classes of 1h40 each, running from 09/03 to 10/06/2026. The material follows the style of David Tong’s notes, with emphasis on symmetries, conservation laws, and the underlying physical structure.
| Module | Theme | Classes | Dates | Count |
|---|---|---|---|---|
| I | Fundamentals | Classes 1-4 | 09/03-18/03 | 4 |
| II | Ideal Fluids | Classes 5-10 | 23/03-13/04 | 6 |
| III | Viscous Fluids | Classes 11-15 | 15/04-29/04 | 5 |
| IV | Waves and Instabilities | Classes 16-20 | 06/05-20/05 | 5 |
| V | Rotating and Geophysical Fluids | Classes 21-23 | 27/05-03/06 | 3 |
| VI | Turbulence | Classes 24-25 | 08/06-10/06 | 2 |
| Total | 25 classes | 09/03-10/06 | 25 |
After the last class (10/06), the calendar includes two general review sessions (17/06 and 22/06), five assessments distributed throughout the semester, and four reserve dates in July for make-up classes or extra activities.
Assessment System
Assessment is done entirely through 5 problem sets, with no traditional exams. On each assessment date, students solve selected problems from the corresponding set on the board. Each set has three levels: (A) direct application, (B) concept combination, and (C) challenging problem.
| Eval. | Date | Set | Covered content | Notes |
|---|---|---|---|---|
| A1 | 08/04 (Wed) | Set 1 | Modules I and II (partial) - classes 1-9 | Fundamentals + Ideal Fluids up to potential flow |
| A2 | 04/05 (Mon) | Set 2 | Module II (complete) + Module III - classes 5-15 | Ideal Fluids (aerodynamics) + complete Viscous Fluids |
| A3 | 25/05 (Mon) | Set 3 | Module IV - classes 16-20 | Surface waves, surface tension, instabilities |
| A4 | 15/06 (Mon) | Set 4 | Modules V and VI - classes 21-25 | Rotation, geophysics, and turbulence |
| A5 | 06/07 (Mon) | Set 5 | Supplementary Notes: Sound Waves | Sound waves |
Full Calendar
| # | Date | Mod. | Type | Activity | Tong (ref.) |
|---|---|---|---|---|---|
| 1 | 09/03 Mon | I | Class | Introduction to Fluid Mechanics | Ch. 1, introduction and Sec. 1.1 |
| 2 | 11/03 Wed | I | Class | Fluid Kinematics | Ch. 1, Sec. 1.1.1-1.1.4 |
| 3 | 16/03 Mon | I | Class | Forces and Stresses in Fluids | Ch. 2, Sec. 2.1.1-2.1.3; Ch. 3, Sec. 3.1-3.1.1 |
| 4 | 18/03 Wed | I | Class | Euler Equation | Ch. 2, Sec. 2.1-2.1.2 |
| 5 | 23/03 Mon | II | Class | Bernoulli Theorem and Applications | Ch. 2, Sec. 2.1.4, 2.3.3, 2.4.2 |
| 6 | 25/03 Wed | II | Class | Vorticity I: Definition and Fundamental Theorems | Ch. 2, Sec. 2.2-2.2.2 |
| 7 | 30/03 Mon | II | Class | Vorticity II: Vortex Dynamics | Ch. 2, Sec. 2.2.1-2.2.2 (+ supplementary) |
| 8 | 01/04 Wed | II | Class | Potential Flow I: Foundations ($\phi$, $\psi$, superposition) | Ch. 1, Sec. 1.1.4; Ch. 2, Sec. 2.3-2.4.1 |
| 9 | 06/04 Mon | II | Class | Potential Flow II: Complex Functions and Transformations | Ch. 2, Sec. 2.4.1-2.4.2 (+ supplementary) |
| 10 | 08/04 Wed | — | Assessment | ASSESSMENT 1 - Set 1 (Modules I and II, classes 1-9) | — |
| 11 | 13/04 Mon | II | Class | Aerodynamics: Airfoils, Kutta-Joukowski Theorem, and Lift | Ch. 2, Sec. 2.1.4, 2.4.2 (+ supplementary) |
| 12 | 15/04 Wed | III | Class | Navier-Stokes Equation | Ch. 3, intro and Sec. 3.1-3.3.1 |
| 13 | 20/04 Mon | III | Class | Exact Solutions I: Poiseuille and Couette | Ch. 3, Sec. 3.2.1-3.2.3 |
| 14 | 22/04 Wed | III | Class | Exact Solutions II: Stokes, Lubrication, and Drag | Ch. 3, Sec. 3.4-3.4.4 (+ supplementary) |
| 15 | 27/04 Mon | III | Class | Boundary Layer I: Prandtl Equations and Blasius Solution | Ch. 3, Sec. 3.5-3.5.2 |
| 16 | 29/04 Wed | III | Class | Boundary Layer II: Separation, Transition, and Drag | Ch. 3, Sec. 3.5.3-3.5.4 |
| 17 | 04/05 Mon | — | Assessment | ASSESSMENT 2 - Set 2 (full Module II + Module III) | — |
| 18 | 06/05 Wed | IV | Class | Surface Waves I: Linear Perturbations and Dispersion Relation | Ch. 4, Sec. 4.1-4.1.2 |
| 19 | 11/05 Mon | IV | Class | Surface Waves II: Group Velocity, Dispersion, and Shallow-Water Waves | Ch. 4, Sec. 4.1.2 and 4.3.1 |
| 20 | 13/05 Wed | IV | Class | Surface Tension, Capillarity, and Capillary Waves | Ch. 4, Sec. 4.1.3 |
| 21 | 18/05 Mon | IV | Class | Instabilities I: Rayleigh-Taylor and Rayleigh-Plateau | Ch. 5, Sec. 5.1.4 and 5.2-5.2.1 |
| 22 | 20/05 Wed | IV | Class | Instabilities II: Kelvin-Helmholtz and Richardson Criterion | Ch. 5, Sec. 5.1-5.1.3 (+ supplementary) |
| 23 | 25/05 Mon | — | Assessment | ASSESSMENT 3 - Set 3 (Module IV) | — |
| 24 | 27/05 Wed | V | Class | Fluids in a Rotating Frame: Coriolis, Rossby, and Geostrophic Balance | Ch. 4, Sec. 4.3-4.3.2 |
| 25 | 01/06 Mon | V | Class | Taylor-Proudman Theorem and Taylor Columns | Ch. 4, Sec. 4.3 (+ supplementary) |
| 26 | 03/06 Wed | V | Class | Rossby Waves and Geophysical Applications | Ch. 4, Sec. 4.3.1-4.3.4 |
| 27 | 08/06 Mon | VI | Class | Turbulence I: Phenomenology, Reynolds Averages, and Reynolds Stresses | Ch. 6, Sec. 6.1-6.1.1 |
| 28 | 10/06 Wed | VI | Class | Turbulence II: Kolmogorov Cascade and the 5/3 Law | Ch. 6, Sec. 6.2-6.3.4 |
| 29 | 15/06 Mon | — | Assessment | ASSESSMENT 4 - Set 4 (Modules V and VI) | — |
| 30 | 17/06 Wed | — | Review | General Review I - Ideal and Viscous Fluids (classes 1-15) | — |
| 31 | 22/06 Mon | — | Review | General Review II - Waves, Rotation, and Turbulence (classes 16-25) | — |
| 32 | 24/06 Wed | — | Reserve | Reserve / Make-up | — |
| 33 | 29/06 Mon | — | Reserve | Reserve / Make-up | — |
| 34 | 01/07 Wed | — | Reserve | Reserve / Make-up | — |
| 35 | 06/07 Mon | — | Assessment | ASSESSMENT 5 - Set 5 (Sound waves) | — |
| 36 | 08/07 Wed | — | Reserve | Reserve / Make-up | — |
Detailed Plan by Module
Module I - Fundamentals
| Class | Title | Main topics | Tong | Secondary ref. |
|---|---|---|---|---|
| 1 | Introduction to Fluid Mechanics | What is a fluid. Continuum hypothesis. Density, pressure, velocity. Eulerian and Lagrangian descriptions. | Ch. 1, introduction and Sec. 1.1 | L&L Sec. 1; Acheson Sec. 1.1 |
| 2 | Fluid Kinematics | Material derivative D/Dt. Streamlines, trajectories, streaklines. Continuity equation. Incompressible flow. | Ch. 1, Sec. 1.1.1-1.1.4 | L&L Sec. 1; Kundu Sec. 3.1-3.3 |
| 3 | Forces and Stresses in Fluids | Body and surface forces. Stress tensor. Hydrostatic pressure. Newtonian fluids: viscous stress tensor. | Ch. 2, Sec. 2.1.1-2.1.3; Ch. 3, Sec. 3.1-3.1.1 | L&L Sec. 2; Kundu Sec. 2.3-2.4 |
| 4 | Euler Equation | Derivation from momentum conservation. Ideal fluids. Boundary conditions. Interpretation as a field equation for u. | Ch. 2, Sec. 2.1-2.1.2 | L&L Sec. 2; Acheson Sec. 1.2-1.3 |
Module II - Ideal Fluids
| Class | Title | Main topics | Tong | Secondary ref. |
|---|---|---|---|---|
| 5 | Bernoulli Theorem and Applications | Bernoulli along streamlines. Generalized form. Venturi, efflux, simplified lift. d’Alembert paradox. | Ch. 2, Sec. 2.1.4, 2.3.3, 2.4.2 | L&L Sec. 5; Acheson Sec. 1.4 |
| 6 | Vorticity I: Definition and Fundamental Theorems | $\omega = \nabla \times \mathbf{u}$. Vorticity equation. Kelvin’s theorem (circulation). Helmholtz theorem (vortex tubes). Topological conservation. | Ch. 2, Sec. 2.2-2.2.2 | L&L Sec. 8; Acheson Sec. 2.1-2.4 |
| 7 | Vorticity II: Vortex Dynamics | 2D point vortices. Vortex pair. Rankine vortex. Vortex rings. Vorticity stretching. Helicity as a topological invariant. | Ch. 2, Sec. 2.2.1-2.2.2 (+ supplementary) | L&L Sec. 9; Acheson Sec. 2.5-2.7 |
| 8 | Potential Flow I | Irrotational flow: potential $\phi$ and stream function $\psi$. Superposition: source, sink, dipole, vortex. Flow around a cylinder. | Ch. 1, Sec. 1.1.4; Ch. 2, Sec. 2.3-2.4.1 | Acheson Sec. 4.1-4.4; Batchelor Sec. 2.9 |
| 9 | Potential Flow II: Complex Functions | Complex potential $w(z)=\phi+i\psi$. Complex velocity. Joukowski transformation. Conformal mapping. Circle to airfoil. | Ch. 2, Sec. 2.4.1-2.4.2 (+ supplementary) | Acheson Sec. 4.5-4.9; Batchelor Sec. 2.10 |
| 10 | Aerodynamics: Airfoils and Lift | Kutta condition. Kutta-Joukowski theorem: $L=\rho U\Gamma$. Circulation and lift. Induced drag. Reynolds number in aerodynamics. | Ch. 2, Sec. 2.1.4, 2.4.2 (+ supplementary) | Acheson Sec. 4.6-4.9; Kundu Sec. 14 |
Module III - Viscous Fluids
| Class | Title | Main topics | Tong | Secondary ref. |
|---|---|---|---|---|
| 11 | Navier-Stokes Equation | Dynamic viscosity $\mu$ and kinematic viscosity $\nu=\mu/\rho$. N-S derivation. Reynolds number $\mathrm{Re}$. Limits $\mathrm{Re}\to 0$ (Stokes) and $\mathrm{Re}\to\infty$ (Euler). Dissipation. | Ch. 3, introduction and Sec. 3.1-3.3.1 | L&L Sec. 15-17; Kundu Sec. 9.1-9.2 |
| 12 | Exact Solutions I: Poiseuille and Couette | Poiseuille flow in tube and channel (parabolic profile, Hagen-Poiseuille law). Couette flow. Pressure-flow relation. | Ch. 3, Sec. 3.2.1-3.2.3 | L&L Sec. 17; Acheson Sec. 7.1-7.2 |
| 13 | Exact Solutions II: Stokes, Lubrication, and Drag | Stokes flow around a sphere: Stokes drag formula. Lubrication theory (Reynolds). Validity for $\mathrm{Re}\ll 1$. | Ch. 3, Sec. 3.4-3.4.4 (+ supplementary) | L&L Sec. 20; Acheson Sec. 7.3-7.4 |
| 14 | Boundary Layer I: Prandtl Equations and Blasius Solution | Prandtl’s scale-separation idea. Derivation of boundary-layer equations. Similarity. Blasius solution: profile and thickness $\delta\sim(\nu x/U)^{1/2}$. | Ch. 3, Sec. 3.5-3.5.2 | Acheson Sec. 8.1-8.3; Kundu Sec. 10.2 |
| 15 | Boundary Layer II: Separation, Transition, and Drag | Boundary-layer separation: adverse pressure gradient. Pressure drag vs viscous drag. Laminar-turbulent transition. $\mathrm{Re}_{\mathrm{crit}}$ criterion. | Ch. 3, Sec. 3.5.3-3.5.4 | Acheson Sec. 8.4; Kundu Sec. 10.3-10.4 |
Module IV - Waves and Instabilities
| Class | Title | Main topics | Tong | Secondary ref. |
|---|---|---|---|---|
| 16 | Surface Waves I: Linear Perturbations | Linearization around rest state. Free-surface boundary conditions. Normal-mode solution. Dispersion relation $\omega^2=gk\tanh(kh)$. | Ch. 4, Sec. 4.1-4.1.2 | L&L Sec. 12; Acheson Sec. 3.1-3.3 |
| 17 | Surface Waves II: Dispersion and Wave Groups | Deep- and shallow-water limits. Phase velocity $c$ and group velocity $c_g$. Dispersion: spreading of wave packets. Energy in gravity waves. | Ch. 4, Sec. 4.1.2 and 4.3.1 | L&L Sec. 12; Acheson Sec. 3.4-3.5 |
| 18 | Surface Tension, Capillarity, and Capillary Waves | Molecular origin of surface tension $\gamma$. Young-Laplace equation. Contact angle. Capillary waves: dispersion relation with gravity and surface tension. Weber number. | Ch. 4, Sec. 4.1.3 | Kundu Sec. 1.9; Acheson Sec. 3.9 |
| 19 | Instabilities I: Rayleigh-Taylor and Rayleigh-Plateau | Linear stability analysis: perturbation $\to$ growth equation. Rayleigh-Taylor instability: heavy fluid over light fluid. Rayleigh-Plateau instability: liquid column. | Ch. 5, Sec. 5.1.4 and 5.2-5.2.1 | Acheson Sec. 9.1-9.2; Kundu Sec. 11.1 |
| 20 | Instabilities II: Kelvin-Helmholtz and Richardson Criterion | Kelvin-Helmholtz instability: shear layer. Richardson criterion (stabilizing stratification). Relevance in atmosphere, ocean, and astrophysics. | Ch. 5, Sec. 5.1-5.1.3 (+ supplementary) | Acheson Sec. 9.3-9.4; Kundu Sec. 11.3 |
Module V - Rotating and Geophysical Fluids
| Class | Title | Main topics | Tong | Secondary ref. |
|---|---|---|---|---|
| 21 | Fluids in a Rotating Frame | Fictitious forces: Coriolis $-2\Omega\times\mathbf{u}$ and centrifugal. Rossby number $\mathrm{Ro}=U/(\Omega L)$. Equations in rotating frame. Geostrophic balance. Atmospheric applications. | Ch. 4, Sec. 4.3-4.3.2 | Kundu Sec. 13.1-13.4 |
| 22 | Taylor-Proudman Theorem and Taylor Columns | Demonstration of the Taylor-Proudman theorem: $\partial u/\partial z=0$. Taylor columns: experimental visualization. Extension to stratified fluids. | Ch. 4, Sec. 4.3 (+ supplementary) | Kundu Sec. 13.5-13.6; Acheson Sec. 8.9 |
| 23 | Rossby Waves and Geophysical Applications | Physical mechanism of Rossby waves: potential vorticity conservation. Dispersion relation. Oceanic and atmospheric currents. Jet stream (introduction). | Ch. 4, Sec. 4.3.1-4.3.4 | Kundu Sec. 13.7-13.8 |
Module VI - Turbulence
| Class | Title | Main topics | Tong | Secondary ref. |
|---|---|---|---|---|
| 24 | Turbulence I: Phenomenology and Reynolds Averages | Chaotic and unpredictable nature of turbulence. Reynolds decomposition: $u=\bar{u}+u’$. RANS equations. Reynolds stresses: closure problem. Turbulent kinetic energy. | Ch. 6, Sec. 6.1-6.1.1 | L&L Sec. 33-34; Kundu Sec. 12.1-12.3 |
| 25 | Turbulence II: Kolmogorov Cascade and the 5/3 Law | Kolmogorov hypotheses (K41). Energy cascade from large to small scales. Kolmogorov scales ($\eta$, $\tau_\eta$, $v_\eta$). Law $E(k)\sim k^{-5/3}$. Dimensional analysis. Limits of K41. | Ch. 6, Sec. 6.2-6.3.4 | L&L Sec. 34; Kundu Sec. 12.4-12.6 |
Bibliographic References
| Abbrev. | Type | Author(s) | Title | Publisher / Note |
|---|---|---|---|---|
| [T] | Main | Tong, D. | Lectures on Fluid Mechanics | Cambridge (online notes): https://damtp.cam.ac.uk/user/tong/fluids.html |
| [LL] | Secondary | Landau, L.D. and Lifshitz, E.M. | Fluid Mechanics (Vol. 6) | Butterworth-Heinemann, 2nd ed., 1987 |
| [A] | Secondary | Acheson, D.J. | Elementary Fluid Dynamics | Oxford University Press, 1990 |
| [K] | Secondary | Kundu, P.K., Cohen, I.M. and Dowling, D.R. | Fluid Mechanics | Academic Press, 6th ed., 2015 |
| [B] | Classic reference | Batchelor, G.K. | An Introduction to Fluid Dynamics | Cambridge University Press, 2000 |