Fluid Dynamics

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

8 Ìṣàmúlò-ètò 22 Àwọn ìwé-ìwé 131 Àwọn àtòjọ-ètò 180 Àwọn àkóónú àtòjọ-ẹ̀yàn

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.

Tí a bá ṣẹ̀dà: Partial-Differential-Equations, Multivariable-Calculus, Linear-Algebra

Àwọn Ìjádè

Describing a flow
3 Àwọn ìwé-ìwé

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

Fi wọlé láti mú ìṣàmúlò-ètò náà
Mass, the stream function and vorticity
3 Àwọn ìwé-ìwé

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

Fi wọlé láti mú ìṣàmúlò-ètò náà
Stress, pressure and ideal flow
4 Àwọn ìwé-ìwé

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

Fi wọlé láti mú ìṣàmúlò-ètò náà
The Navier-Stokes equations
3 Àwọn ìwé-ìwé

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

Fi wọlé láti mú ìṣàmúlò-ètò náà
Slow flow and thin layers
2 Àwọn ìwé-ìwé

Stokes flow at low Reynolds number Boundary layers

Fi wọlé láti mú ìṣàmúlò-ètò náà
Vorticity, energy and turbulence
3 Àwọn ìwé-ìwé

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

Fi wọlé láti mú ìṣàmúlò-ètò náà
The Cauchy problem and weak solutions
2 Àwọn ìwé-ìwé

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

Fi wọlé láti mú ìṣàmúlò-ètò náà
Regularity and criticality
2 Àwọn ìwé-ìwé

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

Fi wọlé láti mú ìṣàmúlò-ètò náà

Àwọn ìwé-ìwé ìparí

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. Àwọn àmì-ìwé ìṣàmúlò-ètò: 70%.

Fi wọlé

Àwọn ìṣàmúlò-ètò

$39

Àwọn ààyè-iṣẹ́ gbogbo, àwọn ìwé-ìwé ìparí nípa ìṣàfarawé, àti àwọn ìṣàmúlò-ètò 20 tí o ń ṣiṣẹ́, fún ìṣàmúlò-ètò kan.

Àwọn ìwé-ìwé àìpẹ̀

$15

A single sitting of one course's final, marked and reviewed. For someone who already knows the material and wants it tested.

Àwọn Ìjánu-ìsún

$19/Àwọn àwọn ààyè-iṣẹ́

Àwọn àwọn ìṣàmúlò-ètò àti àwọn ìṣàmúlò-ètò gbogbo nínú àwọn ìwé-ìwé (ìwé-ìwé meji nínú àwọn ìwé-ìwé kan nínú oṣù) àti àwọn ìṣàmúlò-ètò iṣẹ́ 100 nínú oṣù. Gbàyàn láti inú àwọn

Àwọn ìṣàmúlò-ètò ìṣàmúlò-ètò

Àwọn ìṣàmúlò-ètò mẹ́tà: fi àwòrán tí a tí n ṣiṣẹ́ pamọ́, tàbí tẹ̀ rẹ̀, àti wò ibi tí a tì fi ṣẹ́.

Àwọn ìwé-ìwé tí a ṣì fún ìrànwọ́. Àwọn ìwé-ìwé àti àwọn ìwé-ìwé ìṣàmúlò-ètò tí a ṣì ní bayii.

Ṣayẹwo àwọn ààyè-iṣẹ́ rẹ̀

Fluid Dynamics

Àwọn Ìjádè

Describing a flow

Mass, the stream function and vorticity

Stress, pressure and ideal flow

The Navier-Stokes equations

Slow flow and thin layers

Vorticity, energy and turbulence

The Cauchy problem and weak solutions

Regularity and criticality

Àwọn ìwé-ìwé ìparí

Àwọn ìṣàmúlò-ètò

Àwọn ìṣàmúlò-ètò

Àwọn ìwé-ìwé àìpẹ̀

Àwọn Ìjánu-ìsún

Àwọn ìṣàmúlò-ètò ìṣàmúlò-ètò