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.

Gulani kuphunzira kwa kuphunzira

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.

Gulani imodzi yachiwiri

Mfundo zonse

$19/mphindi

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

Kulembetsa

Kugwira ntchito zoyeserera

$9

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

Gulani ma cheque asanu

Onani gawo la ntchito yanu yokha