Kursi · Graduate

Fluid Dynamics

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

8 %s: Mitir 22 Waxbarasho 131 su'aalo 180 daqiiqo

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.

Ka dib: Partial-Differential-Equations, Multivariable-Calculus, Linear-Algebra

Kursiga

  1. Describing a flow

    3 Waxbarasho
    The continuum hypothesis and fieldsEulerian and Lagrangian descriptions, the material derivativeKinematics: streamlines, pathlines and streaklines
    Ku soo gal si aad u qaadato imtixaankan

  2. Mass, the stream function and vorticity

    3 Waxbarasho
    Conservation of mass, the continuity equation and incompressibilityThe stream function and two-dimensional incompressible flowVorticity, circulation and Kelvin's circulation theorem
    Ku soo gal si aad u qaadato imtixaankan

  3. Stress, pressure and ideal flow

    4 Waxbarasho
    The stress tensor and Cauchy's momentum equationHydrostatics and pressureThe Euler equations and Bernoulli's theoremPotential flow and the Laplace equation
    Ku soo gal si aad u qaadato imtixaankan

  4. The Navier-Stokes equations

    3 Waxbarasho
    Viscosity, Newtonian fluids and the Navier-Stokes equationsExact solutions: Couette, Poiseuille and Stokes' first problemDimensional analysis, the Reynolds number and the scaling symmetry
    Ku soo gal si aad u qaadato imtixaankan

  5. Slow flow and thin layers

    2 Waxbarasho
    Stokes flow at low Reynolds numberBoundary layers
    Ku soo gal si aad u qaadato imtixaankan

  6. Vorticity, energy and turbulence

    3 Waxbarasho
    The vorticity equation and vortex stretchingEnergy: the energy equality, dissipation and enstrophyTurbulence and Kolmogorov 1941 (a heuristic theory)
    Ku soo gal si aad u qaadato imtixaankan

  7. The Cauchy problem and weak solutions

    2 Waxbarasho
    The Cauchy problem: the Leray projection and local existenceWeak solutions (Leray-Hopf) and Sobolev spaces for fluids
    Ku soo gal si aad u qaadato imtixaankan

  8. Regularity and criticality

    2 Waxbarasho
    Regularity criteria: Ladyzhenskaya-Prodi-Serrin and Beale-Kato-MajdaPartial regularity (Caffarelli-Kohn-Nirenberg) and criticality: 2D versus 3D
    Ku soo gal si aad u qaadato imtixaankan

Imtixaankii ugu dambeeyay

Su'aalaha ka mid ah mid kasta oo ka mid ah, kuwa ugu adag ee ugu dambeeyay. jawaabaha la daabacay waxaa lagu calaamadeeyay nidaamka algebra; caddayn iyo sawirka shaqada waxaa akhriya qaab; markaas qof ayaa dib u eegista warbixinta oo dhan ka hor inta aad natiijooyinka la sii daayay. Calaamada ka gudubka: 70%.

Ku soo gal

Qiimaha

Kursi

$39

Mid kasta oo iskuulka, imtixaanka ugu dambeeyay oo leh mid ka mid ah inta kale, iyo 20 tijaabooyinka shaqadaada, si ay u mid ka mid ah.

Imtixaan dhamaadka ah

$15

A fadhi kaliya ee dhamaadka mid ka mid ah fasalka, calaamadeeyay oo dib loo eego. Si qof ka horeeya u yaqaan waxyaabaha iyo doonista in ay la tijaabiyay.

Kursi kasta

$19/mi

Dhammaan su'aalaha iyo finals in fasal kasta (laba finals fasal kasta bishii) iyo 100 shahaadooyinka shaqada bishii. Cancel ka xisaabtaada.

Shaqada baaritaan

$9

Toban tijaabooyin: soo deji sawir kasta oo shaqada ah, ama ku qor, oo eeg meesha ay ku dhacay.

Kursiyada furan iibka soon. Waxbarashada iyo su'aalaha ugu horeysay ee fasal kasta oo hadda waa la furay.

Ka hubi qayb ka mid ah shaqadaada