Fluid Dynamics

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

8 ሴኮንዶችunit-format 22 ማስተማሪያ 131 ጥያቄዎች 180 ደቂቃ

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.

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

የኮርሱ

Describing a flow
3 ማስተማሪያ

The continuum hypothesis and fields Eulerian and Lagrangian descriptions, the material derivative Kinematics: streamlines, pathlines and streaklines

የምርመራውን ፋይል አስቀምጥ
Mass, the stream function and vorticity
3 ማስተማሪያ

Conservation of mass, the continuity equation and incompressibility The stream function and two-dimensional incompressible flow Vorticity, circulation and Kelvin's circulation theorem

የምርመራውን ፋይል አስቀምጥ
Stress, pressure and ideal flow
4 ማስተማሪያ

The stress tensor and Cauchy's momentum equation Hydrostatics and pressure The Euler equations and Bernoulli's theorem Potential flow and the Laplace equation

የምርመራውን ፋይል አስቀምጥ
The Navier-Stokes equations
3 ማስተማሪያ

Viscosity, Newtonian fluids and the Navier-Stokes equations Exact solutions: Couette, Poiseuille and Stokes' first problem Dimensional analysis, the Reynolds number and the scaling symmetry

የምርመራውን ፋይል አስቀምጥ
Slow flow and thin layers
2 ማስተማሪያ

Stokes flow at low Reynolds number Boundary layers

የምርመራውን ፋይል አስቀምጥ
Vorticity, energy and turbulence
3 ማስተማሪያ

The vorticity equation and vortex stretching Energy: the energy equality, dissipation and enstrophy Turbulence and Kolmogorov 1941 (a heuristic theory)

የምርመራውን ፋይል አስቀምጥ
The Cauchy problem and weak solutions
2 ማስተማሪያ

The Cauchy problem: the Leray projection and local existence Weak solutions (Leray-Hopf) and Sobolev spaces for fluids

የምርመራውን ፋይል አስቀምጥ
Regularity and criticality
2 ማስተማሪያ

Regularity criteria: Ladyzhenskaya-Prodi-Serrin and Beale-Kato-Majda Partial regularity (Caffarelli-Kohn-Nirenberg) and criticality: 2D versus 3D

የምርመራውን ፋይል አስቀምጥ

የመጨረሻው ምርመራ

ጥያቄዎች ከየአካባቢው ጥያቄዎች, በጣም አስቸጋሪዎቹ መጨረሻ. የተጻፉ መልሶች በአልጀርባ ስርዓት ምልክት; ማስረጃዎች እና ፎቶግራፍ ስራዎች በሞዴል ይነበባሉ; ከዚያ ሰው ሁሉንም ጽሑፍ ውጤትዎ ከመለቀቁ በፊት ይመልከቱ. ተቀበሉ እና የራሱን አገናኝ ያለውን ማስረጃ ያገኛሉ. ተቀባይ: 70%.

መዝገብ ቤትን አስገባ