Kudzidza · Graduate

Fluid Dynamics

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

8 unit description in lists 22 Zvidzidzo 131 mibvunzo 180 minute final

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.

Inobva mushure me: Partial-Differential-Equations, Multivariable-Calculus, Linear-Algebra

Chikoro

  1. Describing a flow

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

    3 Zvidzidzo
    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 Zvidzidzo
    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 Zvidzidzo
    Regularity criteria: Ladyzhenskaya-Prodi-Serrin and Beale-Kato-MajdaPartial regularity (Caffarelli-Kohn-Nirenberg) and criticality: 2D versus 3D
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The final exam

Questions from every unit, the hardest ones last. Typed answers are marked by the algebra system; proofs and photographed working are read by a model; then a person reviews the whole paper before your result is released. Pass and you get a certificate with its own link. Kupinda: 70%.

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Mitengo

Course pass

$39

Every unit quiz, the final exam with one retake, and 20 checks of your own working, for one course.

One final exam

$15

A imwe kugara kwechikoro chekupedzisira, akanyorwa uye akaongororwa. For someone who already knows the material and wants it tested.

Yese Course

$19/mo

All quizzes and finals in every course (two finals per course a month) and 100 working checks a month.Cancel from your account.

Kushanda kwezviyeuchidzo

$9

Mamwe machecks: wedzera vhidhiyo yechimwe chinhu chaunoda kuita, kana kuti tinya pano, uye ona kuti chaiva nei.

Zvidzidzo zvedu uye chidzidzo chekutanga chese chezvidzidzo zvedu zvinotanga kuwanikwa nhasi.

Cheka chidimbu chebasa rako