Masomo · Graduate

Fluid Dynamics

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

8 ↓ 22 masomo 131 maswali 180 mwisho wa dakika

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.

Nyotamkia baada: Partial-Differential-Equations, Multivariable-Calculus, Linear-Algebra

Mwendo

  1. Describing a flow

    3 masomo
    The continuum hypothesis and fieldsEulerian and Lagrangian descriptions, the material derivativeKinematics: streamlines, pathlines and streaklines
    Ishara ya kutumia mtihani

  2. Mass, the stream function and vorticity

    3 masomo
    Conservation of mass, the continuity equation and incompressibilityThe stream function and two-dimensional incompressible flowVorticity, circulation and Kelvin's circulation theorem
    Ishara ya kutumia mtihani

  3. Stress, pressure and ideal flow

    4 masomo
    The stress tensor and Cauchy's momentum equationHydrostatics and pressureThe Euler equations and Bernoulli's theoremPotential flow and the Laplace equation
    Ishara ya kutumia mtihani

  4. The Navier-Stokes equations

    3 masomo
    Viscosity, Newtonian fluids and the Navier-Stokes equationsExact solutions: Couette, Poiseuille and Stokes' first problemDimensional analysis, the Reynolds number and the scaling symmetry
    Ishara ya kutumia mtihani

  5. Slow flow and thin layers

    2 masomo
    Stokes flow at low Reynolds numberBoundary layers
    Ishara ya kutumia mtihani

  6. Vorticity, energy and turbulence

    3 masomo
    The vorticity equation and vortex stretchingEnergy: the energy equality, dissipation and enstrophyTurbulence and Kolmogorov 1941 (a heuristic theory)
    Ishara ya kutumia mtihani

  7. The Cauchy problem and weak solutions

    2 masomo
    The Cauchy problem: the Leray projection and local existenceWeak solutions (Leray-Hopf) and Sobolev spaces for fluids
    Ishara ya kutumia mtihani

  8. Regularity and criticality

    2 masomo
    Regularity criteria: Ladyzhenskaya-Prodi-Serrin and Beale-Kato-MajdaPartial regularity (Caffarelli-Kohn-Nirenberg) and criticality: 2D versus 3D
    Ishara ya kutumia mtihani

Mtihani wa mwisho

Questions from every unit, the hardest ones last. Typed answers are marked by the algebra system; proofs and photographed working are read by a model; then a person reviews the whole paper before your result is released. Pass and you get a certificate with its own link. Njia za kupita: 70%.

Ishara Katika

Bei

Mwendo unapita

$39

Kila tafiti, mtihani wa mwisho wenye kufanyiwa tena jambo moja, na uchunguzi 20 wa kazi yako mwenyewe, kwa mtaala mmoja.

Mtihani mmoja wa mwisho

$15

Kiti kimoja cha mwisho cha mtaala mmoja, kilitia alama na kupitiwa.

Kila jambo

$19/mo

Kila mmoja huulizwa maswali na kumaliza katika kila mtaala (humaliza vipindi viwili kwa mwezi) na hundi 100 za kazi kwa mwezi.

Uchunguzi wa kazi

$9

Uchunguzi kumi: Pakua picha ya mtu yeyote anayefanya kazi, au chora aina yake, uone mahali ilipoharibika.

Masomo na maswali ya kwanza ya kila mtaala yanafunguliwa sasa.

Chunguza kipande cha kazi yako mwenyewe