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.

Iibso fasalka

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.

Iibso mid dhamaadka ah

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.

Ku biir

Shaqada baaritaan

$9

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

Iibso toban warqad

Ka hubi qayb ka mid ah shaqadaada