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.

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.

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.

Arbejdskontrol

$9

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

Kurser, der snart kan købes. Undervisningen og den første quiz på hvert kursus er åben nu.

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