Courses · Graduate

Fluid Dynamics

From the continuum hypothesis to the regularity theory of the incompressible Navier-Stokes equations.

8 units 22 lessons 131 questions 180 minute final

A mathematical course in fluid dynamics. The first half derives the equations of motion: fields and the material derivative, conservation of mass, vorticity and circulation, the stress tensor, the Euler equations, potential flow, and the incompressible Navier-Stokes equations with their exact solutions and scaling. The second half reads the equations as mathematics: Stokes flow and boundary layers, the vorticity equation and vortex stretching, the energy equality, the Kolmogorov theory of turbulence, the Cauchy problem and the Leray projection, Leray-Hopf weak solutions, regularity criteria and partial regularity.

It is for readers who know multivariable calculus, linear algebra and the basic theory of partial differential equations. Each unit ends with a quiz of computations, conceptual questions and short proofs; the final exam covers the whole course.

Comes after: Partial-Differential-Equations, Multivariable-Calculus, Linear-Algebra

The course

  1. Describing a flow

    3 lessons
    The continuum hypothesis and fieldsEulerian and Lagrangian descriptions, the material derivativeKinematics: streamlines, pathlines and streaklines
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  2. Mass, the stream function and vorticity

    3 lessons
    Conservation of mass, the continuity equation and incompressibilityThe stream function and two-dimensional incompressible flowVorticity, circulation and Kelvin's circulation theorem
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  3. Stress, pressure and ideal flow

    4 lessons
    The stress tensor and Cauchy's momentum equationHydrostatics and pressureThe Euler equations and Bernoulli's theoremPotential flow and the Laplace equation
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  4. The Navier-Stokes equations

    3 lessons
    Viscosity, Newtonian fluids and the Navier-Stokes equationsExact solutions: Couette, Poiseuille and Stokes' first problemDimensional analysis, the Reynolds number and the scaling symmetry
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  5. Slow flow and thin layers

    2 lessons
    Stokes flow at low Reynolds numberBoundary layers
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  6. Vorticity, energy and turbulence

    3 lessons
    The vorticity equation and vortex stretchingEnergy: the energy equality, dissipation and enstrophyTurbulence and Kolmogorov 1941 (a heuristic theory)
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  7. The Cauchy problem and weak solutions

    2 lessons
    The Cauchy problem: the Leray projection and local existenceWeak solutions (Leray-Hopf) and Sobolev spaces for fluids
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  8. Regularity and criticality

    2 lessons
    Regularity criteria: Ladyzhenskaya-Prodi-Serrin and Beale-Kato-MajdaPartial regularity (Caffarelli-Kohn-Nirenberg) and criticality: 2D versus 3D
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The final exam

Questions from every unit, the hardest ones last. Typed answers are marked by the algebra system; proofs and photographed working are read by a model; then a person reviews the whole paper before your result is released. Pass and you get a certificate with its own link. Pass mark: 70%.

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Prices

Course pass

$39

Every unit quiz, the final exam with one retake, and 20 checks of your own working, for one course.

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One final exam

$15

A single sitting of one course's final, marked and reviewed. For someone who already knows the material and wants it tested.

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Every course

$19/mo

All quizzes and finals in every course (two finals per course a month) and 100 working checks a month. Cancel from your account.

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Working checks

$9

Ten checks: upload a photo of any working, or type it, and see where it went wrong.

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Check a piece of your own working