Kurslar · Graduate

Fluid Dynamics

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

8 %s göterimunit-format 22 _Öňki 131 soraglar 180 minut

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.

Comes after: Partial-Differential-Equations, Multivariable-Calculus, Linear-Algebra

Kurs

  1. Describing a flow

    3 _Öňki
    The continuum hypothesis and fieldsEulerian and Lagrangian descriptions, the material derivativeKinematics: streamlines, pathlines and streaklines
    _Gün tertibini görkez

  2. Mass, the stream function and vorticity

    3 _Öňki
    Conservation of mass, the continuity equation and incompressibilityThe stream function and two-dimensional incompressible flowVorticity, circulation and Kelvin's circulation theorem
    _Gün tertibini görkez

  3. Stress, pressure and ideal flow

    4 _Öňki
    The stress tensor and Cauchy's momentum equationHydrostatics and pressureThe Euler equations and Bernoulli's theoremPotential flow and the Laplace equation
    _Gün tertibini görkez

  4. The Navier-Stokes equations

    3 _Öňki
    Viscosity, Newtonian fluids and the Navier-Stokes equationsExact solutions: Couette, Poiseuille and Stokes' first problemDimensional analysis, the Reynolds number and the scaling symmetry
    _Gün tertibini görkez

  5. Slow flow and thin layers

    2 _Öňki
    Stokes flow at low Reynolds numberBoundary layers
    _Gün tertibini görkez

  6. Vorticity, energy and turbulence

    3 _Öňki
    The vorticity equation and vortex stretchingEnergy: the energy equality, dissipation and enstrophyTurbulence and Kolmogorov 1941 (a heuristic theory)
    _Gün tertibini görkez

  7. The Cauchy problem and weak solutions

    2 _Öňki
    The Cauchy problem: the Leray projection and local existenceWeak solutions (Leray-Hopf) and Sobolev spaces for fluids
    _Gün tertibini görkez

  8. Regularity and criticality

    2 _Öňki
    Regularity criteria: Ladyzhenskaya-Prodi-Serrin and Beale-Kato-MajdaPartial regularity (Caffarelli-Kohn-Nirenberg) and criticality: 2D versus 3D
    _Gün tertibini görkez

The final exam

Her bir bölümden soraglar, iň kynlary soňuna çenli. Ýazan jogaplar aljebra sistemasy tarapyndan nyşanlanýar; subutnama we surata alnan iş bir model tarapyndan okalýar; soňra bir adam siziň netijeňizi çap etmekden öň ähli kagyzy synlaýar. Geçiň we öz baglanyşygy bilen bir şahadatnama alyň. Geçir: 70%.

_Gün

Prices

Kursy geç

$39

Her bir bölüm soragy, bir tekrar bilen sonky synag, we 20 öz işiň barlagy, bir kurs üçin.

Bir soňky synag

$15

Bir kursyň soňky bir sessiýasy, nyşanlan we gözden geçirilmiş. Maddeyi örän gowy bilýän we ony synap görmek isleýän biri üçin.

Her bir kurs

$19/10 minut

Her kursda ähli soraglar we finallar (ýolda iki final) we aýda 100 iş barlaglary. Siziň hasabyňyzdan kes.

Iş barlaglary

$9

On barlag: her bir işleýäniň suratyny ýükle, ýa-da ony ýaz, we nädip hata düşendigini gör.

Kurslar çaltdan satyn almak üçin açyldy. Her bir kursyň leksiýalary we ilkinji soragy indi açyldy.

Özüň işleýän bir parçany barla