Iikhosi · Graduate

Fluid Dynamics

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

8 Iiyunithiunit description in lists 22 Iinkqubo 131 iimibuzo 180 Imini

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 emva: Partial-Differential-Equations, Multivariable-Calculus, Linear-Algebra

Inkqubo

  1. Describing a flow

    3 Iinkqubo
    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 Iinkqubo
    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 Iinkqubo
    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 Iinkqubo
    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 Iinkqubo
    Stokes flow at low Reynolds numberBoundary layers
    Ngena ukuze uthathe isifundo

  6. Vorticity, energy and turbulence

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

I-Exam

Imibuzo evela kwiyunithi nganye, ezinzima ziphela. Iimpendulo ezishicilelwe ziphawulwe yinkqubo ye algebra; iimvavanyo kunye nomsebenzi othathwe ngefoto ufundwa ngumzobo; ngoko umntu uvavanya iphepha lonke phambi kokuba impumelelo yakho iphume. Ugqitha kwaye ufumana isiqinisekiso kunye nekhonkco layo. Inqaku le-Pass: 70%.

Ngena

Ixabiso

Iinkqubo

$39

Iiyunithi zonke zovavanyo, uvavanyo oluphelileyo olunokuphinda luthathelwe kwakhona, kunye neentsuku ezili-20 zokuhlola umsebenzi wakho, kwinkqubo enye.

Iinkqubo zolawulo

$15

Ukuhlala olunye kwesinye isifundo esiphelelwe lixesha, esiphawulwe kwaye sihlaziywe. Kulowo sele eyaziyo i-material kwaye ufuna ukuba ivavanywe.

Iinkqubo

$19/Iiyure

Zonke iimvavanyo kunye neengxelo zesiphelo kwinkqubo nganye (iingxelo ezimbini zesiphelo kwinkqubo nganye ngenyanga) kunye neengxelo ezili-100 zomsebenzi ngenyanga.

Iinkqubo zolawulo

$9

Uvavanyo lweshumi: thumela umfanekiso wento esebenza, okanye ubhale, kwaye ubone apho ihambe khona kakubi.

Iikhosi zivuliwe ukuthengwa ngoku. Iikhosi kunye nemibuzo yokuqala yekhosi nganye ivuliwe ngoku.

Uvavanyo lweenkcukacha