Natuurlik · Graduate

Fluid Dynamics

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

8 eenhede 22 lesse 131 vrae 180 minuut finale

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.

Kome na: Partial-Differential-Equations, Multivariable-Calculus, Linear-Algebra

Die kursus

  1. Describing a flow

    3 lesse
    The continuum hypothesis and fieldsEulerian and Lagrangian descriptions, the material derivativeKinematics: streamlines, pathlines and streaklines
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  2. Mass, the stream function and vorticity

    3 lesse
    Conservation of mass, the continuity equation and incompressibilityThe stream function and two-dimensional incompressible flowVorticity, circulation and Kelvin's circulation theorem
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  3. Stress, pressure and ideal flow

    4 lesse
    The stress tensor and Cauchy's momentum equationHydrostatics and pressureThe Euler equations and Bernoulli's theoremPotential flow and the Laplace equation
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  4. The Navier-Stokes equations

    3 lesse
    Viscosity, Newtonian fluids and the Navier-Stokes equationsExact solutions: Couette, Poiseuille and Stokes' first problemDimensional analysis, the Reynolds number and the scaling symmetry
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  5. Slow flow and thin layers

    2 lesse
    Stokes flow at low Reynolds numberBoundary layers
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  6. Vorticity, energy and turbulence

    3 lesse
    The vorticity equation and vortex stretchingEnergy: the energy equality, dissipation and enstrophyTurbulence and Kolmogorov 1941 (a heuristic theory)
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  7. The Cauchy problem and weak solutions

    2 lesse
    The Cauchy problem: the Leray projection and local existenceWeak solutions (Leray-Hopf) and Sobolev spaces for fluids
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  8. Regularity and criticality

    2 lesse
    Regularity criteria: Ladyzhenskaya-Prodi-Serrin and Beale-Kato-MajdaPartial regularity (Caffarelli-Kohn-Nirenberg) and criticality: 2D versus 3D
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Die finale eksamen

Vrae van elke eenheid, die moeilikstes laaste. Getipe antwoorde word deur die algebra-stelsel aangedui; bewyse en gefotografeerde werk word deur 'n model gelees; dan hersien 'n persoon die hele papier voordat jou resultaat vrygestel word. Gee en jy kry' n sertifikaat met sy eie skakel. Herhaling merk: 70%.

Meld aan

Pryse

Natuurlik verby

$39

Elke eenheid vra, die finale eksamen met een herhaal en 20 tjeks van jou eie werk, vir een kursus.

Een finale eksamen

$15

Vir iemand wat reeds die materiaal ken en wil hê dat dit getoets moet word, is dit'n teken en hersiening.

Elke kursus

$19/mo

Alle vasvrae en finale in elke kursus (twee punte per kursus per maand) en 100 werkskontroles per maand. Kanselleer van jou rekening.

Werkskontroles

$9

Tien tjeks: Laai'n foto op van enige werk, of tik dit, en kyk waar dit verkeerd geloop het.

Die lesse en die eerste vasvra van elke kursus is nou oop.

Gaan 'n stuk van jou eie werk na