Izifundo · Graduate

Fluid Dynamics

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

8 Ama-units 22 Izifundo 131 imibuzo 180 imizuzu edlule

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.

Ivela ngemuva: Partial-Differential-Equations, Multivariable-Calculus, Linear-Algebra

Ibanga

  1. Describing a flow

    3 Izifundo
    The continuum hypothesis and fieldsEulerian and Lagrangian descriptions, the material derivativeKinematics: streamlines, pathlines and streaklines
    Ngena ukuze uthathe isifundo

  2. Mass, the stream function and vorticity

    3 Izifundo
    Conservation of mass, the continuity equation and incompressibilityThe stream function and two-dimensional incompressible flowVorticity, circulation and Kelvin's circulation theorem
    Ngena ukuze uthathe isifundo

  3. Stress, pressure and ideal flow

    4 Izifundo
    The stress tensor and Cauchy's momentum equationHydrostatics and pressureThe Euler equations and Bernoulli's theoremPotential flow and the Laplace equation
    Ngena ukuze uthathe isifundo

  4. The Navier-Stokes equations

    3 Izifundo
    Viscosity, Newtonian fluids and the Navier-Stokes equationsExact solutions: Couette, Poiseuille and Stokes' first problemDimensional analysis, the Reynolds number and the scaling symmetry
    Ngena ukuze uthathe isifundo

  5. Slow flow and thin layers

    2 Izifundo
    Stokes flow at low Reynolds numberBoundary layers
    Ngena ukuze uthathe isifundo

  6. Vorticity, energy and turbulence

    3 Izifundo
    The vorticity equation and vortex stretchingEnergy: the energy equality, dissipation and enstrophyTurbulence and Kolmogorov 1941 (a heuristic theory)
    Ngena ukuze uthathe isifundo

  7. The Cauchy problem and weak solutions

    2 Izifundo
    The Cauchy problem: the Leray projection and local existenceWeak solutions (Leray-Hopf) and Sobolev spaces for fluids
    Ngena ukuze uthathe isifundo

  8. Regularity and criticality

    2 Izifundo
    Regularity criteria: Ladyzhenskaya-Prodi-Serrin and Beale-Kato-MajdaPartial regularity (Caffarelli-Kohn-Nirenberg) and criticality: 2D versus 3D
    Ngena ukuze uthathe isifundo

Ukuhlolwa kokuphela

Imibuzo evela kunoma iyiphi iyunithi, enzima kakhulu ihlala. Izixazululo ezishicilelwe ziphawulwa nge-algorithm system; iziqinisekiso kanye nokusebenza okuphothiwe kufundeka yimodeli; bese umuntu ubheka iphepha lonke ngaphambi kokuba imiphumela yakho ikhishwa. Udlula futhi uthola isitifiketi esixhunywe kuyo. Iphutha lokuphawula: 70%.

Ngena

Amantengo

I-course pass

$39

Iyunithi ngayinye yokuzibuza, isifundo sokugcina esinomqulu owodwa, kanye nezibuyekezo ezingu-20 zomsebenzi wakho, kuhlelo olulodwa.

Ukuhlolwa kokuphela

$15

Ukuhlala okufanayo kwesigaba esisodwa sokugcina, okuphawulwe futhi okubuyekezwe. Kumuntu owaziyo ngaphambili izinto futhi ofuna ukuzihlola.

Isigaba

$19/ihora

Zonke izifundo neziqu eziphezulu ezikuzo zonke izifundo (iziqu eziphezulu ezimbili ngasinye isifundo ngasinye ngenyanga) kanye ne-100 yokubheka umsebenzi ngenyanga. Cisha kusuka ku-akhawunti yakho.

Ukuhlolwa kokusebenzela

$9

Ukuhlolwa okungu-10: khipha isithombe somsebenzi, noma ubhale, bese ubona lapho kudlule khona.

Izifundo zivuliwe ukuthengwa ngokushesha. Izifundo kanye nemibuzo yokuqala yesifundo ngasinye ivuliwe manje.

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