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Fluid Dynamics
Continuity, vorticity, the Euler and Navier-Stokes equations, and what is proved about them.
Mësime
integrate 1000*(1 + r)*4*pi*r^2 dr from 0 to 1
Core
Eulerian and Lagrangian descriptions, the material derivative
Following particles versus watching fixed points, and the operator D/Dt that connects them.
material derivative of T = x*t with velocity [y, 0, 0]
Introductory
Kinematics: streamlines, pathlines and streaklines
Three different curves drawn by a flow, why they coincide for steady flow, and how to compute each.
y' = -x/y
Core
Conservation of mass, the continuity equation and incompressibility
From a balance of mass in a box to the continuity equation, and why incompressible means divergence-free.
divergence of [x, -y, 0]
Core
The stream function and two-dimensional incompressible flow
A single scalar that builds every 2D divergence-free velocity, draws the streamlines and measures flux.
gradient of x*y
Core
Vorticity, circulation and Kelvin's circulation theorem
The curl of the velocity as local spin, circulation around loops, and why ideal fluids keep their circulation.
curl of [-y, x, 0]
Core
The stress tensor and Cauchy's momentum equation
Surface forces as a linear map from normals to tractions, the momentum balance, and why the stress is symmetric.
eigenvalues of [[2,1,0],[1,2,0],[0,0,1]]
Introductory
Hydrostatics and pressure
Pressure in a fluid at rest, forces on dams, Archimedes' principle and the exponential atmosphere.
101325 + 1000*9.81*10
Core
The Euler equations and Bernoulli's theorem
The equations of an ideal fluid, the Lamb form of the acceleration, and the quantity conserved along streamlines.
solve 0.5*1000*v^2 = 1000*9.81*5
Core
Potential flow and the Laplace equation
Irrotational incompressible flow as harmonic functions, complex potentials, flow past a cylinder and d'Alembert's paradox.
laplacian of x/(x^2 + y^2)
Core
Viscosity, Newtonian fluids and the Navier-Stokes equations
The constitutive law for viscous stress and the derivation, term by term, of the incompressible Navier-Stokes equations.
laplacian of sin(x)*sin(y)
Core
Exact solutions: Couette, Poiseuille and Stokes' first problem
The flows in which the nonlinear term vanishes, solved exactly and checked against the full equations.
y'' = -2
Core
Dimensional analysis, the Reynolds number and the scaling symmetry
Non-dimensionalising Navier-Stokes, dynamic similarity, and the scaling that leaves the equations unchanged.
1000*2*0.1/0.001
Core
Stokes flow at low Reynolds number
Dropping inertia: the linear Stokes equations, drag on a sphere, reversibility and the scallop theorem.
2*(0.000025)^2*(2650 - 1000)*9.81/(9*0.001)
Advanced
Boundary layers
Prandtl's idea: at high Reynolds number viscosity matters only in thin layers near walls, which is how drag reappears.
5/sqrt(1000000)
Advanced
The vorticity equation and vortex stretching
Taking the curl of Navier-Stokes, the stretching term that exists only in 3D, and why 2D flow is so much tamer.
curl of [y*z, -x*z, 0]
Core
Energy: the energy equality, dissipation and enstrophy
The one estimate every Navier-Stokes solution obeys, what it controls, and why enstrophy saves two dimensions.
integrate (cos(x))^2 dx from 0 to 2*pi
Advanced
Turbulence and Kolmogorov 1941 (a heuristic theory)
The energy cascade, Kolmogorov's dimensional predictions, the one exact law, and what is and is not proved.
(0.000001^3/0.001)^(1/4)
Advanced
The Cauchy problem: the Leray projection and local existence
Stating the initial value problem precisely, eliminating the pressure, the mild formulation and what local well-posedness gives.
derivative of exp(-k^2*t) with respect to t
Advanced
Weak solutions (Leray-Hopf) and Sobolev spaces for fluids
Solutions that exist for all time and all finite-energy data, the function spaces they live in, and what is known about them.
integrate 1/(1+x^2)^2 dx from -oo to oo
Advanced
Regularity criteria: Ladyzhenskaya-Prodi-Serrin and Beale-Kato-Majda
Conditions that force a weak solution to be smooth, why the exponents are exactly the scale-invariant ones, and what a blow-up would have to do.
solve 2/p + 3/4 = 1
Advanced
Partial regularity (Caffarelli-Kohn-Nirenberg) and criticality: 2D versus 3D
Critical, subcritical and supercritical quantities; the size of a possible singular set; and a map of what is proved.
solve 1 - 3/p = 0
Fluid dynamics writes the motion of water and air as partial differential equations and then asks what those equations can do. The course builds the incompressible Navier-Stokes equations term by term from conservation of mass and momentum, solves the flows that can be solved exactly, and then reads the modern theory: energy estimates, vortex stretching, scaling and criticality, Leray-Hopf weak solutions and the criteria that guarantee smoothness.
Pyetja që bëjnë njerëzit
What do I need before starting fluid dynamics?
Multivariable calculus (divergence, curl, the divergence and Stokes theorems), linear algebra (symmetric matrices and eigenvalues) and partial differential equations (the heat and Laplace equations). The last lessons also use Sobolev spaces, which are introduced where they are needed.
What are the Navier-Stokes equations in one sentence?
Newton's second law for each particle of a viscous incompressible fluid: acceleration equals the pressure force plus viscous diffusion of momentum, with the constraint that the velocity field is divergence-free.
Why is two-dimensional flow easier than three-dimensional flow?
In 2D the vorticity is a scalar that is only carried and diffused, so its maximum never grows; in 3D vortex lines can be stretched, which amplifies vorticity, and no known bound rules out unlimited growth.
Is this course physics or mathematics?
Both, in order. The first half derives the equations from physical principles and solves classical flows; the second half treats the equations as mathematical objects and studies which of their properties can be proved.
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