Kursus · Graduate

Fluid Dynamics

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

8 unit description in lists 22 lesson 131 soalan 180 minit terakhir

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.

Datang selepas: Partial-Differential-Equations, Multivariable-Calculus, Linear-Algebra

Kursus

  1. Describing a flow

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

    3 lesson
    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 lesson
    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 lesson
    Regularity criteria: Ladyzhenskaya-Prodi-Serrin and Beale-Kato-MajdaPartial regularity (Caffarelli-Kohn-Nirenberg) and criticality: 2D versus 3D
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Ujian akhir.

Soalan dari setiap unit, yang paling sukar terakhir. Jawapan yang ditandakan oleh sistem algebra; bukti dan kerja yang difoto dibaca oleh model; kemudian seseorang meninjau keseluruhan kertas sebelum hasil anda dikeluarkan. lulus dan anda mendapat sijil dengan pautan sendiri. Tanda lulus: 70%.

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Harga

Laluan

$39

Setiap unit kuiz, peperiksaan akhir dengan satu retake, dan 20 periksa kerja anda sendiri, untuk satu kursus.

Satu peperiksaan akhir

$15

Satu sesi akhir satu kursus, ditanda dan dikaji semula untuk seseorang yang sudah tahu bahan dan mahu diuji.

Setiap kursus

$19/bulan

Semua soalan dan akhir dalam setiap kursus (dua akhir setiap kursus setiap bulan) dan 100 periksa kerja setiap bulan. Batalkan dari akaun anda.

Semak kerja

$9

Sepuluh periksa: muat naik foto apa-apa yang berfungsi, atau taipkannya, dan lihat di mana ia salah.

Kursus dibuka untuk pembelian tidak lama lagi. Pelajaran dan kuiz pertama setiap kursus dibuka sekarang.

Periksa bahagian kerja awak sendiri.