Kurser · Graduate

Fluid Dynamics

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

8 enheder 22 lektioner 131 spørgsmål 180 minut endelig

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.

Kommer efter: Partial-Differential-Equations, Multivariable-Calculus, Linear-Algebra

Kurset

  1. Describing a flow

    3 lektioner
    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 lektioner
    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 lektioner
    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 lektioner
    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 lektioner
    Stokes flow at low Reynolds numberBoundary layers
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  6. Vorticity, energy and turbulence

    3 lektioner
    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 lektioner
    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 lektioner
    Regularity criteria: Ladyzhenskaya-Prodi-Serrin and Beale-Kato-MajdaPartial regularity (Caffarelli-Kohn-Nirenberg) and criticality: 2D versus 3D
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Den afsluttende eksamen

Spørgsmål fra hver enhed, de hårdeste sidste. Indtastede svar er markeret med algebra system; beviser og fotograferet arbejde læses af en model; så en person gennemgår hele papiret, før dit resultat er frigivet. Pass og du får et certifikat med sit eget link. Adgangskode: 70%.

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Priser

Kursuskort

$39

Hver enhed quiz, den endelige eksamen med en retake, og 20 checks af din egen arbejde, for et kursus.

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En afsluttende eksamen

$15

Et enkelt møde i et kursuss endelige, markerede og gennemgik. For en person, der allerede kender materialet og ønsker det testet.

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Hvert kursus

$19/m

Alle quizzer og finaler i hvert kursus (to finaler pr. kursus om måneden) og 100 arbejdschecks om måneden. Annuller fra din konto.

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Arbejdskontrol

$9

Ti checks: uploade et billede af enhver arbejder, eller skrive det, og se, hvor det gik galt.

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