ຫຼັກສູດ · Graduate

Fluid Dynamics

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

8 ​អប្បបរមា 22 ບົດຮຽນ 131 ຄໍາຖາມ 180 ​នាទី​ចុងក្រោយ

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.

ຫຼັງ: Partial-Differential-Equations, Multivariable-Calculus, Linear-Algebra

ຫຼັກສູດ

  1. Describing a flow

    3 ບົດຮຽນ
    The continuum hypothesis and fieldsEulerian and Lagrangian descriptions, the material derivativeKinematics: streamlines, pathlines and streaklines
    ເຂົ້າ​ສູ່​ລະບົບ​ເພື່ອ​ສອບ​ຖາມ

  2. Mass, the stream function and vorticity

    3 ບົດຮຽນ
    Conservation of mass, the continuity equation and incompressibilityThe stream function and two-dimensional incompressible flowVorticity, circulation and Kelvin's circulation theorem
    ເຂົ້າ​ສູ່​ລະບົບ​ເພື່ອ​ສອບ​ຖາມ

  3. Stress, pressure and ideal flow

    4 ບົດຮຽນ
    The stress tensor and Cauchy's momentum equationHydrostatics and pressureThe Euler equations and Bernoulli's theoremPotential flow and the Laplace equation
    ເຂົ້າ​ສູ່​ລະບົບ​ເພື່ອ​ສອບ​ຖາມ

  4. The Navier-Stokes equations

    3 ບົດຮຽນ
    Viscosity, Newtonian fluids and the Navier-Stokes equationsExact solutions: Couette, Poiseuille and Stokes' first problemDimensional analysis, the Reynolds number and the scaling symmetry
    ເຂົ້າ​ສູ່​ລະບົບ​ເພື່ອ​ສອບ​ຖາມ

  5. Slow flow and thin layers

    2 ບົດຮຽນ
    Stokes flow at low Reynolds numberBoundary layers
    ເຂົ້າ​ສູ່​ລະບົບ​ເພື່ອ​ສອບ​ຖາມ

  6. Vorticity, energy and turbulence

    3 ບົດຮຽນ
    The vorticity equation and vortex stretchingEnergy: the energy equality, dissipation and enstrophyTurbulence and Kolmogorov 1941 (a heuristic theory)
    ເຂົ້າ​ສູ່​ລະບົບ​ເພື່ອ​ສອບ​ຖາມ

  7. The Cauchy problem and weak solutions

    2 ບົດຮຽນ
    The Cauchy problem: the Leray projection and local existenceWeak solutions (Leray-Hopf) and Sobolev spaces for fluids
    ເຂົ້າ​ສູ່​ລະບົບ​ເພື່ອ​ສອບ​ຖາມ

  8. Regularity and criticality

    2 ບົດຮຽນ
    Regularity criteria: Ladyzhenskaya-Prodi-Serrin and Beale-Kato-MajdaPartial regularity (Caffarelli-Kohn-Nirenberg) and criticality: 2D versus 3D
    ເຂົ້າ​ສູ່​ລະບົບ​ເພື່ອ​ສອບ​ຖາມ

ການສອບເສັງສຸດທ້າຍ

ຄໍາຖາມຈາກທຸກໆ ໜ່ວຍ, ສິ່ງທີ່ຍາກທີ່ສຸດແມ່ນສຸດທ້າຍ. ຄໍາຕອບທີ່ພິມລົງແມ່ນຖືກລະບຸໂດຍລະບົບ algebra; ຫຼັກຖານແລະຮູບພາບເຮັດວຽກແມ່ນອ່ານໂດຍແບບ; ຫຼັງຈາກນັ້ນຄົນຫນຶ່ງທົບທວນບົດຂຽນທັງ ໝົດ ກ່ອນທີ່ຜົນໄດ້ຮັບຂອງທ່ານຈະຖືກປ່ອຍອອກມາ. ຜ່ານແລະທ່ານໄດ້ຮັບໃບຢັ້ງຢືນທີ່ມີການເຊື່ອມຕໍ່ຂອງຕົນເອງ. ​ຕີ​ຄະແນນ​ຜ່ານ: 70%.

ເຂົ້າ​ສູ່​ລະບົບ

ລາຄາ

ຜ່ານ​ສວນ

$39

ທຸກໆ ໜ່ວຍ ສອບຖາມ, ການສອບເສັງສຸດທ້າຍກັບ retake ຫນຶ່ງ, ແລະ 20 ກວດເບິ່ງຂອງຕົນເອງເຮັດວຽກ, ສໍາລັບຫນຶ່ງຫຼັກສູດ.

ສອບເສັງສຸດທ້າຍ

$15

ຊຸດ​ຮຽນ​ສຸດ​ທ້າຍ​ຂອງ​ຫຼັກສູດ​ໜຶ່ງ, ຖືກ​ຕີ​ລາຄາ ແລະ ທົບທວນຄືນ. ສຳ​ລັບ​ຜູ້​ທີ່​ຮູ້​ວັດຖຸ​ແລ້ວ ແລະ ຕ້ອງການ​ທົດສອບ​ມັນ.

ທຸກ​ສວນ

$19/ເດືອນ

ຄໍາຖາມທັງຫມົດແລະສຸດທ້າຍໃນທຸກໆຫຼັກສູດ (ສອງສຸດທ້າຍຕໍ່ຫຼັກສູດເດືອນ) ແລະ 100 ກວດວຽກເດືອນ. ຍົກເລີກຈາກບັນຊີຂອງທ່ານ.

ກວດເບິ່ງ​ການ​ເຮັດວຽກ

$9

ກວດສອບສິບຢ່າງ: ສົ່ງຮູບຂອງສິ່ງໃດກໍ່ຕາມທີ່ເຮັດວຽກ ຫຼື ພິມມັນ ແລະ ເບິ່ງວ່າມັນຜິດຫຍັງ

ຫຼັກສູດ​ເປີດ​ໃຫ້​ຊື້​ໃນ​ໄວໆ​ນີ້. ບົດຮຽນ ແລະ ການທົດສອບ​ຄັ້ງ​ທຳອິດ​ຂອງ​ຫຼັກສູດ​ແຕ່ລະ​ອັນ​ເປີດ​ຢູ່​ດຽວ​ນີ້.

ກວດເບິ່ງສ່ວນຂອງວຽກຂອງທ່ານເອງ