Mga Kurso · Graduate

Fluid Dynamics

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

8 unit 22 Mga aralin 131 Mga katanungan 180 minuto huling

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.

Nagmumula pagkatapos: Partial-Differential-Equations, Multivariable-Calculus, Linear-Algebra

Ang kurso

  1. Describing a flow

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

    3 Mga aralin
    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 Mga aralin
    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 Mga aralin
    Regularity criteria: Ladyzhenskaya-Prodi-Serrin and Beale-Kato-MajdaPartial regularity (Caffarelli-Kohn-Nirenberg) and criticality: 2D versus 3D
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Ang huling pagsusulit

Mga katanungan mula sa bawat yunit, ang pinaka-mahirap na mga huling. Typed sagot ay minarkahan ng algebra system; proofs at photographed nagtatrabaho ay basahin ng isang modelo; pagkatapos ay isang tao review ang buong papel bago ang iyong resulta ay inilabas. Pumasa at makakuha ka ng isang sertipiko na may sariling link. Pass marka: 70%.

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Mga Presyo

Kurso pass

$39

Ang bawat yunit ng kuwento, ang huling pagsusulit na may isang retake, at 20 check ng iyong sariling trabaho, para sa isang kurso.

Isang huling pagsusulit

$15

Isang solong pag-upo ng isang kurso ng huling, minarkahan at nasuri. Para sa isang tao na alam na ang materyal at nais ito na sinubukan.

Ang bawat kurso

$19/Ang Mo

Lahat ng mga quiz at finals sa bawat kurso (dalawang finals bawat kurso sa isang buwan) at 100 mga check na nagtatrabaho sa isang buwan. I-cancel mula sa iyong account.

Paggawa ng mga check

$9

Sampung checks: mag-upload ng isang larawan ng anumang gumagana, o i-type ito, at makita kung saan ito ay nagpunta mali.

Ang mga aralin at ang unang quiz ng bawat kurso ay bukas na ngayon.

Tingnan ang isang piraso ng iyong sariling trabaho