Korsijiet · Graduate

Fluid Dynamics

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

8 unità 22 lezzjonijiet 131 mistoqsijiet 180 minuta finali

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.

Jiġi wara: Partial-Differential-Equations, Multivariable-Calculus, Linear-Algebra

Il-kors

  1. Describing a flow

    3 lezzjonijiet
    The continuum hypothesis and fieldsEulerian and Lagrangian descriptions, the material derivativeKinematics: streamlines, pathlines and streaklines
    Idħol biex tieħu l-kwizz

  2. Mass, the stream function and vorticity

    3 lezzjonijiet
    Conservation of mass, the continuity equation and incompressibilityThe stream function and two-dimensional incompressible flowVorticity, circulation and Kelvin's circulation theorem
    Idħol biex tieħu l-kwizz

  3. Stress, pressure and ideal flow

    4 lezzjonijiet
    The stress tensor and Cauchy's momentum equationHydrostatics and pressureThe Euler equations and Bernoulli's theoremPotential flow and the Laplace equation
    Idħol biex tieħu l-kwizz

  4. The Navier-Stokes equations

    3 lezzjonijiet
    Viscosity, Newtonian fluids and the Navier-Stokes equationsExact solutions: Couette, Poiseuille and Stokes' first problemDimensional analysis, the Reynolds number and the scaling symmetry
    Idħol biex tieħu l-kwizz

  5. Slow flow and thin layers

    2 lezzjonijiet
    Stokes flow at low Reynolds numberBoundary layers
    Idħol biex tieħu l-kwizz

  6. Vorticity, energy and turbulence

    3 lezzjonijiet
    The vorticity equation and vortex stretchingEnergy: the energy equality, dissipation and enstrophyTurbulence and Kolmogorov 1941 (a heuristic theory)
    Idħol biex tieħu l-kwizz

  7. The Cauchy problem and weak solutions

    2 lezzjonijiet
    The Cauchy problem: the Leray projection and local existenceWeak solutions (Leray-Hopf) and Sobolev spaces for fluids
    Idħol biex tieħu l-kwizz

  8. Regularity and criticality

    2 lezzjonijiet
    Regularity criteria: Ladyzhenskaya-Prodi-Serrin and Beale-Kato-MajdaPartial regularity (Caffarelli-Kohn-Nirenberg) and criticality: 2D versus 3D
    Idħol biex tieħu l-kwizz

L-eżami finali

Mistoqsijiet minn kull unità, l-aktar diffiċli l-aħħar. Tweġibiet tat-tip huma mmarkati mill-sistema algebra; provi u xogħol fotografati huma maqluba minn mudell; imbagħad persuna tirrevedi l-karta kollha qabel ir-riżultat tiegħek huwa rilaxxat. Pass u inti tikseb ċertifikat mal-link tagħha stess. Marka ta’ approvazzjoni: 70%.

Idħol

Prezzijiet

Pass tal-kors

$39

Kull unità kwizz, l-eżami finali ma retake wieħed, u 20 kontrolli tax-xogħol tiegħek stess, għal kors wieħed.

Eżami finali wieħed

$15

Seduta waħda tal-finali ta' kors wieħed, immarkata u riveduta għal xi ħadd li diġà jaf il-materjal u jrid li jiġi ttestjat.

Kull kors

$19/10 mg/ kg

Il-kwizzi u l-finali kollha f’kull kors (żewġ finali għal kull kors fix-xahar) u 100 verifika tax-xogħol fix-xahar.

Kontrolli ta’ ħidma

$9

Għaxar kontrolli: ittella' ritratt ta' xi xogħol, jew ittajpjah, u ara fejn mar ħażin.

Il-korsijiet miftuħa għax-xiri dalwaqt. Il-lezzjonijiet u l-ewwel kwizz ta' kull kors huma miftuħa issa.

Iċċekkja biċċa tax-xogħol tiegħek stess