Course 1 : basic concepts. Pierre Simon de Laplace (1814), in his book Essai Philosophique sur les Probabilités (Philosophical Essay on Probability):
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1 Course 1 : basic concepts Pierre Simon de Laplace (1814), in his book Essai Philosophique sur les Probabilités (Philosophical Essay on Probability): We must consider the present state of Universe as the effect of its past state and the cause of its future state. An intelligence that would know all forces of nature and the respective situation of all its elements, if furthermore it was large enough to be able to analyze all these data, would embrace in the same expression the motions of the largest bodies of Universe as well as those of the slightest atom: nothing would be uncertain for this intelligence, all future and all past would be as known as present. Poincaré map points-periodic closed figure quasi periodic
2 Course 1 : basic concepts Fixed point Limit cycle Strange attractor
3 Perrin experiment particle 0.52 µm grid 3,2 µm Course 1 : basic concepts
4 Perrin experiment particle 0.52 µm grid 3,2 µm Course 1 : basic concepts Henri Poincaré Science et Méthode (1908): Avery small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of the same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still knowt we require, and we should say that the phenomenon had been predicted, that it is governed by the laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible and we have the fortuitous phenomenon.he situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all
5 Course 1 : basic concepts The variable x versus t of the Lorenz system, for σ = 10, b = 8/3 and r = 28.
6 Course 1 : chaos and ergodicity logistic map
7 Course 1 : chaos and mixing Cat map Evolution of the cat map. Going from left to right and from top to bottom, the evolutions are plotted with points, at times t = 0, 2, 4, 6.
8 Course II: Probability basic concepts algebra-probability Random variable - pdf Independence Conditional probability-bayes formula Let s make a deal
9 Course II: Probability basic concepts
10 Course II: Probability limit theorems
11 Course II: Probability limit theorems Chebyshev inequality Strong law large numbers CLT: (i)σ x < (ii)x i independent (iii) Lindberg cond
12 Bernoulli Course II: LDT very briefly
13 Course III: Stochastic processes Def Markov Process Forward Kolmogorov Fokker-Planck
14 Gaussian process Wiener Course III: Stochastic processes
15 Course III: Stochastic calculus SDE Ito integral Ito formula Wiener properties
16 Course III: coarse-graining
17 Course III: coarse-graining
18 Course III: coarse-graining
19 Course III: coarse-graining
20 Course IV: Fluid Mechanics basic definition Flow visualization
21 Course IV: Fluid Mechanics viscosity
22 Course IV: Fluid Mechanics basic definition
23 Course IV: Fluid Mechanics ideal and incompressible flows Euler Crocco-Bernoulli Kelvin s Theorem Potential flows
24 Course IV: Fluid Mechanics: potential flows Flow around an obstacle Any flow for which the fluid is initially at rest must be a potential flow. In fact, however, all these conclusions are of only very limited validity. The reason is that the proof given above that curl v = 0 all along a streamline is, strictly speaking, invalid for a line which lies in the surface of a solid body past which the flow takes place, since the presence of this surface makes it impossible to draw a closed contour in the fluid encircling such a streamline. The equations of motion of an ideal fluid therefore admit solutions for which separation occurs at the surface of the body: the streamlines, having followed the surface for some distance, become separated from it at some point and continue into the fluid. The resulting flow pattern is characterized by the presence of a "surface of tangential discontinuity" proceeding from the body; on this surface the fluid velocity, which is everywhere tangential to the surface, has a discontinuity. In other words, at this surface one layer of fluid "slides" on another. Figure 1 shows a surface of discontinuity which separates moving fluid from a region of stationary fluid behind the body. From a mathematical point of view, the discontinuity in the tangential velocity component corresponds to a surface on which the curl of the velocity is non-zero.
25 Course IV: Fluid Mechanics: Viscous fluids NS Vorticity Low Reynolds number: Stokes formula Navier-Stokes from Boltzmann eq.
26 Course IV: Fluid Mechanics similarity ncfmf.html
27 Course IV: Fluid Mechanics Present research in low reynolds number complex flows Antkowiak, Audoly, Josserand, Neukirch and Rivetti, Instant fabrication and selection of folded structures using drop impact, PNAS, 108 (2011).
28 Drops on elastic fibers 0 g 2d 0 top view 2d z g (t) z 0 g side view H (t) L z Flexible fibers: glass, r=0.145 mm, B=7 x Nm 2 Totally wetting liquid: silicone oil
29 Drops between parallel flexible fibers side L=4.5 cm top side L=4 cm top
30 Phase diagram NO SPREADING PARTIAL SPREADING V ƫ/ 2 3$57,$/ 635($',1* NO SPREADING $/ 635($',1* 727$/ 635($',1* L (cm) 4 5 Three distinct regimes Ref: Wetting of flexible fiber array (Nature 2012) Final state depends on d/r, L, V, B and γ Two critical drop sizes: One critical volume trigger coalescence/spreading One optimal volume induces maximum spreading
31 BOUNCING DROPS Drop bouncing at the surface of a bath of the same fluid (oscillating). Fast camera: 1000 fpss D=1mm, µ= Pa.s., f=80hz, acc=3g Ref: From Bouncing to Floating: non-coalesence of drops on a fluid bath (PRL 2004)
32 WALKING DROP View from top of a walking drop (occurs close to the threshold of the Faraday instability) Walker velocity: V W =18 mm.s -1 Drop bounces on the slope of the wave created at its previous bounce giving it this radial impulsion. Ref: Dynamical phenomena: walking and orbiting droplets (Nature 2005) Particle-wave association on a fluid interface (JFM 2006) 32
33 3 scenarios Ruelle-Takes Feigenbaum Pomeau-Manneville
34 ncfmf.html Course V: Turbulence visualization
35 Course V: Turbulence Richardson cascade KO Spectrum
36 Course V: Turbulence
37 Course V: Turbulence 2D
38 Course V: Turbulence multifractal
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