Maphunziro · Graduate

Fluid Dynamics

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

8 Maolaunit description in lists 22 Maphunziro 131 mafunso 180 minute final

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.

Koma pambuyo: Partial-Differential-Equations, Multivariable-Calculus, Linear-Algebra

Mfundo

  1. Describing a flow

    3 Maphunziro
    The continuum hypothesis and fieldsEulerian and Lagrangian descriptions, the material derivativeKinematics: streamlines, pathlines and streaklines
    Kulowa kuti atenge chidziwitso

  2. Mass, the stream function and vorticity

    3 Maphunziro
    Conservation of mass, the continuity equation and incompressibilityThe stream function and two-dimensional incompressible flowVorticity, circulation and Kelvin's circulation theorem
    Kulowa kuti atenge chidziwitso

  3. Stress, pressure and ideal flow

    4 Maphunziro
    The stress tensor and Cauchy's momentum equationHydrostatics and pressureThe Euler equations and Bernoulli's theoremPotential flow and the Laplace equation
    Kulowa kuti atenge chidziwitso

  4. The Navier-Stokes equations

    3 Maphunziro
    Viscosity, Newtonian fluids and the Navier-Stokes equationsExact solutions: Couette, Poiseuille and Stokes' first problemDimensional analysis, the Reynolds number and the scaling symmetry
    Kulowa kuti atenge chidziwitso

  5. Slow flow and thin layers

    2 Maphunziro
    Stokes flow at low Reynolds numberBoundary layers
    Kulowa kuti atenge chidziwitso

  6. Vorticity, energy and turbulence

    3 Maphunziro
    The vorticity equation and vortex stretchingEnergy: the energy equality, dissipation and enstrophyTurbulence and Kolmogorov 1941 (a heuristic theory)
    Kulowa kuti atenge chidziwitso

  7. The Cauchy problem and weak solutions

    2 Maphunziro
    The Cauchy problem: the Leray projection and local existenceWeak solutions (Leray-Hopf) and Sobolev spaces for fluids
    Kulowa kuti atenge chidziwitso

  8. Regularity and criticality

    2 Maphunziro
    Regularity criteria: Ladyzhenskaya-Prodi-Serrin and Beale-Kato-MajdaPartial regularity (Caffarelli-Kohn-Nirenberg) and criticality: 2D versus 3D
    Kulowa kuti atenge chidziwitso

The final exam

Mafunso a m’gulu lililonse, ovuta kwambiri ndi omwe amamaliza. Mafunso olemba amadziwika ndi dongosolo la algebra; zitsimikiziro ndi ntchito zojambula zithunzi zimawerengedwa ndi model; kenako munthu amafufuza nkhani yonseyo m’mbuyomu kuti zotsatira zanu zituluke. Kusintha: 70%.

Kulowa

Mitengo

Course pass

$39

M’chilankhulo chilichonse pali mafunso a unit, chiyeso cha finali ndi mfundo imodzi yopitanso, ndi 20 zoyeserera za ntchito yanu, kwa m’chilankhulo chimodzi.

One final exam

$15

Kukhala ndi nthawi yokha ya kumaliza kwa phunziro limodzi, kumarked ndi kufufuzidwa. Kwa munthu amene kale amadziwa zinthuzo ndipo akufuna kuti ayesedwe.

Mfundo zonse

$19/mphindi

Zomwe zimachitika pazokambirana zonse ndi zomaliza pazokambirana zonse (zomaliza ziwiri pazokambirana mwezi uliwonse) ndi 100 zoyeserera zogwira ntchito mwezi uliwonse.

Kugwira ntchito zoyeserera

$9

Kuyesa kwa makumi asanu: kutsitsa chithunzi cha ntchito iliyonse, kapena kuyika, ndikuwona komwe kunali koipa.

Maphunziro atsegulidwa kuti agwiritse ntchito. Maphunziro ndi chidziwitso choyamba cha maphunziro onse atsegulidwa pano.

Onani gawo la ntchito yanu yokha