Tečaji · Graduate

Fluid Dynamics

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

8 enote 22 učne izkušnje 131 vprašanja 180 minutna končna

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.

Pride po: Partial-Differential-Equations, Multivariable-Calculus, Linear-Algebra

Tečaj

  1. Describing a flow

    3 učne izkušnje
    The continuum hypothesis and fieldsEulerian and Lagrangian descriptions, the material derivativeKinematics: streamlines, pathlines and streaklines
    Prijavi se, da boš vzel kviz.

  2. Mass, the stream function and vorticity

    3 učne izkušnje
    Conservation of mass, the continuity equation and incompressibilityThe stream function and two-dimensional incompressible flowVorticity, circulation and Kelvin's circulation theorem
    Prijavi se, da boš vzel kviz.

  3. Stress, pressure and ideal flow

    4 učne izkušnje
    The stress tensor and Cauchy's momentum equationHydrostatics and pressureThe Euler equations and Bernoulli's theoremPotential flow and the Laplace equation
    Prijavi se, da boš vzel kviz.

  4. The Navier-Stokes equations

    3 učne izkušnje
    Viscosity, Newtonian fluids and the Navier-Stokes equationsExact solutions: Couette, Poiseuille and Stokes' first problemDimensional analysis, the Reynolds number and the scaling symmetry
    Prijavi se, da boš vzel kviz.

  5. Slow flow and thin layers

    2 učne izkušnje
    Stokes flow at low Reynolds numberBoundary layers
    Prijavi se, da boš vzel kviz.

  6. Vorticity, energy and turbulence

    3 učne izkušnje
    The vorticity equation and vortex stretchingEnergy: the energy equality, dissipation and enstrophyTurbulence and Kolmogorov 1941 (a heuristic theory)
    Prijavi se, da boš vzel kviz.

  7. The Cauchy problem and weak solutions

    2 učne izkušnje
    The Cauchy problem: the Leray projection and local existenceWeak solutions (Leray-Hopf) and Sobolev spaces for fluids
    Prijavi se, da boš vzel kviz.

  8. Regularity and criticality

    2 učne izkušnje
    Regularity criteria: Ladyzhenskaya-Prodi-Serrin and Beale-Kato-MajdaPartial regularity (Caffarelli-Kohn-Nirenberg) and criticality: 2D versus 3D
    Prijavi se, da boš vzel kviz.

Končni izpit

Vprašanja vsake enote, najtežji zadnji. Vtisnjeni odgovori so označeni z algebra sistem; dokazi in fotografirani delo se preberejo s modelom; potem oseba pregleda celoten časopis, preden je vaš rezultat objavljen. Prejmite in dobite certifikat z lastno povezavo. Oddaljenost: 70%.

Vpišite se

Cene

Prehod smeri

$39

Vsak kviz, končni izpit z eno ponovno in 20 čekov svojega dela, za en tečaj.

Eden končni izpit.

$15

Eno sejo enega tečaja je končno, označeno in pregledano, za nekoga, ki že pozna material in želi, da se testira.

Vsaka smer

$19/Mo

Vsi kvizi in finale v vsakem tečaju (dva finala na tečaj na mesec) in 100 delovnih pregledov na mesec. Prekliči s vašega računa.

Delovni pregledi

$9

Deset čekov: naložite sliko katerega koli dela, ali jo natipkajte, in poglejte, kam je šla narobe.

Tečaji so odprti za nakup kmalu. Lekcije in prvi kviz vsakega tečaja so odprti zdaj.

Preverite del svojega dela