课程 · Graduate

Fluid Dynamics

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

8 单位 22 经验教训 131 问题 180 定时数

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.

之后的来: Partial-Differential-Equations, Multivariable-Calculus, Linear-Algebra

课程

  1. Describing a flow

    3 经验教训
    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 经验教训
    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 经验教训
    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 经验教训
    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 经验教训
    Stokes flow at low Reynolds numberBoundary layers
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  6. Vorticity, energy and turbulence

    3 经验教训
    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 经验教训
    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 经验教训
    Regularity criteria: Ladyzhenskaya-Prodi-Serrin and Beale-Kato-MajdaPartial regularity (Caffarelli-Kohn-Nirenberg) and criticality: 2D versus 3D
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期末考试

每个单位的问题,最难回答的问题最后。输入的答案以代数系统为标志;证据和照片工作用模型读取;然后一个人在结果发布之前审查整份文件。通过并获得带有其链接的证书。 通行证: 70%.

签署日期

价格

课程通过

$39

每个单元测验,期末考试 一次再考, 和20检查 自己工作, 一次课程。

一次期末考试

$15

单次的一门课程的终极、标记和审查。对于已经知道材料并想要测试材料的人来说,这是一次。

每门课程

$19/以 单位

每门课程的所有测验和决赛(每月两次决赛)以及每月100次工作检查。取消您的账户。

工作检查

$9

十次检查:上传任何工作的照片,或输入它,看看它哪里出错。

课程开放后很快开始购买。每个课程的课目和第一个测验现在都开放。

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