The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection
Chaotic advection in fluids: analytical, numerical, experimental and practical aspects
S.V. Prants
Laboratory of Nonlinear Dynamical Systems Pacific Oceanological Institute of the Russian Academy of Sciences, Vladivostok, Russia URL: dynalab.poi.dvo.ru
Advanced Computational and Experimental Techniques in Nonlinear Dynamics Cusco, Peru, 6–17 May, 2013 The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Contents
1 The toy model of chaotic advection
2 Chaotic advection in laboratory experiments
3 Chaotic cross-jet transport in a model geophysical flow
4 Practical applications of chaotic advection The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Historical background
The term advection refers to the transport of something by a fluid from one region to another due to the fluid’s horizontal bulk mo- tion. It may be a transport of a substance (pollutants) or conserved property such salinity, vorticity, radioactivity, etc. Chaotic advection means chaotic motion of passive particles in a deterministic velocity field. By passive particles one means particles which take on the velocity of a flow very rapidly and do not influence the flow. The term has been coined by Hasan Aref in 1984 to name chaotic advection of passive particles in 2D time periodic flows. Before that in 1965, V. Arnold proved the existence of chaos of streamlines in a steady 3D ABC flow that has been found numerically by M. Henon in 1966. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Advection equations
Motion of a fluid particle in a two-dimensional flow is the trajectory of a dynamical system with given initial conditions governed by the velocity field d~r = ~v(~r, t). (1) dt where ~r = (x, y, z) is the location of the particle, ~v = (u, v, w) are the corresponding components of its velocity. The velocity field is given analytically or computed as the output of a numerical model or derived from a measurement. Even if the Eulerian velocity field is fully deterministic, the particle’s trajectories may be very complicated and practically unpredictable. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Hamiltonian form
For 2D flows and some special cases of 3D flows, the incompressibility condition, div~v = 0, implies the existence of a streamfunction. The advection equations in a 2D flow have now the Hamiltonian form (Aref)
dx ∂Ψ dy ∂Ψ = u(x, y, t) = − , = v(x, y, t) = , (2) dt ∂y dt ∂x
with the phase space being the position space for advected particles, the streamfunction, Ψ, being a Hamiltonian and x and y conjugated variables. The Hamiltonian form takes place with viscid flows too because it is a consequence of incompressibility. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection The toy model of chaotic advection
A point fixed vortex embedded in a planar flow of an ideal incompress- ible fluid with a stationary and periodic components. The respective dimensionless streamfunction p Ψ = ln x2 + y 2 + x( + ξ sin τ) (3)
generates the advection equations dx y dy x = − , = + + ξ sin τ, (4) dt x2 + y 2 dt x2 + y 2 where ξ and are velocities of the stationary and periodic compo- nents of the current flowing in the y direction from the south to the north. M. Uleysky, M. Budyansky, S.P. JTP Letters. 27 (2001) 508. Physics D 195 (2004) 369. JETP 99 (2004) 1018. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Onset of chaos
2
1
y 0
-1
-2 -4 -3 -2 x -1 0 1
Sketch of the phase portrait of the unperturbed flow and transversal intersections of stable and unstable manifolds of the saddle trajectory (solid lines) with the perturbation parameters = 0.5 and ξ = 0.01. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Stable and unstable manifolds
(fig. borrowed from a Legras presentation) Stable (Ws ) and unstable (Wu) manifolds of a hyperbolic trajectory γ(t) are material lines consisting of a set of points through which at time moment t pass trajectories asymptotical to γ(t) at t → ∞ (Ws ) and t → −∞ (Wu). The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Image of the unstable manifold
6 τ=7,5×2π 4,5
3 y 1,5
0
-1,5
-4,5 -3 -1,5 0 x
Snapshot of the unstable manifold of the saddle (hyperbolic) trajectory obtained by injection of a dye. The curve periodically changes in time. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Poincar´e section
3 a) 0.3 b)
2 0.2 0.07 y 1 y 0.1
0 0 -0.025 -0.829 -0.823 -1 -0.1 -3.5 -2.5 -1.5 -0.5 0 0.5 -0.89 -0.84 -0.79 x x
Poincar´e section of the mixing region (a), magnification of the northern part of the long island (b), and magnification of the region between the vortex core and the long island in the inset. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Chaotic invariant set
CIS is a set of all trajectories (except for KAM tori and cantori) that never leave the mixing region. It consists of an infinite number of unstable periodic and aperiodic (chaotic) trajectories. The particles belonging to that set remain on it forever. However, their measure is zero. Each trajectory in the CIS has Ws and Wu. Following trajectories in Ws , particles, advected by the incoming flow, enter the mixing region and remain there forever. Particles that are initially close to those in Ws follow the corresponding trajectories, then deviate from them and eventually leave the mixing region along the unstable manifold Wu. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Trapping and escaping of particles
< y0 7 x;0
100000
10000 N 1000
100
10 100 1000 Te
Top: Escape map showing dependence of the exit time T of particles from the mixing region on their initial positions. The inset shows a zoom of the fragment. Bottom: Histogram of the escape times. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Scattering fractal function
10000 а)
1000 e T
100
-4,65 -4,6 -4,55 -4,5 -4,45 -4,4 -4,35 x0 10000 б)
1000 e T
100
-4,42 -4,41 -4,4 x0
Scattering fractal function showing dependence of the exit time on initial position x0. Bottom: 15-fold magnification of a fragment of T (x0). The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Fractal set
8
6
4
n ν µ α β 2 b c d e g 0 ABCDEFG
-4.65 -4.6 -4.55 -4.5 -4.45 -4.4 -4.35 x0 g 10-2 e d 10-4 c AG j l
10-6
10-8 0 3 6 9 j
Top: Number of particle’s rotations n around the point eddy before reaching a fixed line in the outcoming flow vs their initial positions x0 in the incoming flow. Bottom: Epistrophe lengths, lj , at n = 0 (AG) and n = 1 (c, d, e, g) decrease with their counting number j at the geometric progression. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection How lobes are formed from the epistrophes and strophes
3 2 τ=4.0×2π τ=4.5×2π 2 1 D C
y E y E 1 G D G F 0 0 A F C B A B -1 -1 -3 -2 -1 0 -2 -1 0 x x 2 2 τ=5.0×2π e,g τ=5.5×2π 1 C B 1 B
y y α, β D F C 0 E 0 ν, µ
-1 -1 -2 -1 0 -2 -1 0 x x
Evolution of a material line illustrating the developments of lobes from elements of epistrophes and strophes in the fractal. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection The hierarchy of epistrophes
A segment with equal number of turns around the vortex, n, is called “epistrophe” [Mitchell 2003]. These epistrophes make up a hierarchy. Each epistrophe converges to a limit point on the corresponding material-line segment. The endpoints of each segment of the nth- level epistrophe are the limit points of an n + 1th-level epistrophe. The lengths of segments in an epistrophe decreases in geometric progression. The common ratio of all the progressions q is related to the maximal Lyapunov exponent for the saddle trajectory as follows: λ = − ln q/2π. The hierarchy of epistrophes determines transport of particles, and its fractal properties are generated by the infinite sequences of in- tersections of the advected material line with stable and unstable manifolds of the chaotic invariant set. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection The transient chaos
In open flows, there is typically a set of advected particles that never leave the mixing region, and get trapped in the chaotic invariant set, which is the set of all particle orbits trapped permanently in the mixing region. The measure of this set is zero, and particles in its vicinity will almost always deviate from the chaotic set and leave the mixing region. Despite this, the chaotic set governs the long-time dynamics of the system; particles that happen to come close to the chaotic set wander in the vicinity of the trapped orbits for a long time, and eventually leave them along the unstable manifold. This mechanism gives rise to very large residence times for some particles. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Cavity glycerine flow with moving wells
Evolution of two passive blobs in a time periodic cavity flow (Re=1.2)(Ottino, 1990). The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Cavity glycerine flow with moving wells
Islands of regular motion with periods 3 (circle), 2 (cross) and 2 (triangle). T= a) 10, b) 20, c) 20.25, d) 20.5, e) 20.75, f) 21 periods. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Rotor-oscillator cavity flow
Slow viscous flow between long parallel plates driven by the rotation of a slender cylinder (the rotor) and the longitudinal oscillation of one of the plates (the oscillator). Points A, B and C and significant streamlines of the steady flow. The position of the line rotlet is indicated by + (Hackborn, 1997). The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Visualizing unstable manifolds
Advection patterns for the dye injected at point A: (a) trial 1 after 19 periods, (b) trial 3 after 17 periods. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Visualizing unstable manifolds
Advection patterns for the dye injected at point B: (a) trial 2 after 2 periods, (b) trial 4 after 5 periods, (c) trial 6 after 3 periods. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Chaotic scattering
Schematic diagram of experimental setup (Sommerer, 1996). The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Experimental and computed streaklines
а)
б)
a) Experimental and b) numerically computed streaklines at Re = 100. Cylinder is moving to the left. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Unstable periodic orbits
а)
б)
Unstable periodic orbits in wake revealed by particle tracking. Particle coordinates were digitized and assembled into tracks. Portions of orbit shadowed by different particles are indicated by different plot symbols. (a) Period-1 orbit at Re = 100. (b) Period-1 orbit at Re = 250. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Exit time ditribution
PDF of exit times for particles leaving the mixing region. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Dynamical model of a shear flow with two Rossby waves
Motion of two-dimensional incompressible fluid on the rotating Earth is governed by the equation for conserving potential vorticity (∂/∂t + ~v · ∇~ )Π = 0. In the quasigeostrophic approximation, one gets Π = ∇2Ψ + βy, where β is the Coriolis parameter. The x axis is chosen along the zonal flow, from the west to the east and y along the gradient from the south to the north. Barotropic perturba- tions of zonal flows produce planetary Rossby waves propagating to the west and producing an essential impact on transport and mixing in the ocean and the atmosphere. It is possible to find in a lin- ear approximation an exact solution for the stream function obeying to the equation for conserving potential vorticity and consisting of steady zonal flow with the velocity profile of a Bickley jet and two propagating Rossby waves. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Stream function
The stream function is sought in the form
X ikj (x−cj t) Ψ = Ψ0 + Ψint = Ψ0(y) + Φj (y)e , (5) j
where Ψ0 describes a zonal flow and Ψint is its perturbation which is supposed to be a superposition of zonal running Rossby waves. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection
After substituting (5) in the equation for the potential vorticity and a linearization, one gets the Rayleigh-Kuo equation (Kuo 1949)
d 2Φ d 2u (u − c ) j − k2Φ + β − 0 Φ = 0, (6) 0 j dy 2 j j dy 2 j
where the zonal velocity u0 = −dΨ0/dy has a single extremum at y = 0. If one takes the following zonal-velocity profile (Bickley): y u (y) = U sech2 , (7) 0 0 D then Eq. (6) admits two neutrally stable solutions y Φ (y) = A U D sech2 , j = 1, 2, (8) j j 0 D
where U0 is the maximal velocity in the flow, D is a measure of its width, and Aj are the wave amplitudes. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection
It is easy to check that (7) and (8) are compatible with (6) if there is the following condition for the phase velocities:
2 U0 p ∗ ∗ 3D β c1,2 = (1 ± α), α ≡ 1 − β , β ≡ , (9) 3 2U0 which are connected with the wave numbers by the dispersion rela- 2 2 tion c1,2 = U0D k1,2/6. Two neutrally stable Rossby waves exist if 2 βD /U0 < 2/3. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection
The dynamically-consistent stream function has the form (del Castillo, Phys. Fl. 1993) y y h Ψ(x, y, t) = −U D tanh − sech2 A cos k (x − c t)+ 0 D D 1 1 1 i + A2 cos k2(x − c2t) , (10)
U0 maximal velocity, D measure of its width, and Aj are the wave amplitudes. Let the wave numbers be represented as n1 = mN1 and n2 = mN2, where m 6= 1 is the greatest common divisor and N1/N2 is an irreducible fraction. In new coordinates x0, y 0, and t0 we rewrite the stream function (10) in the frame moving with the phase velocity of the first wave
0 0 0 0 0 2 0 0 Ψ (x , y , t ) = − tanh y + A1 sech y cos(N1x )+ 2 0 0 0 0 + A2 sech y cos(N2x + ω2t ) + C2y , (11) where 2 2 2 2N2(N1 − N2 ) 2N1 ω2 ≡ 2 2 , C2 ≡ 2 2 . (12) 3(N1 + N2 ) 3(N1 + N2 ) The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Chaotic cross-jet transport at odd-odd wavenumber values
If both the reduced wave numbers N1,2 are odd, then the advection equations have two symmetries ( ( x˜ = π + x, x˜ = −x, Sˆ : ˆI0 : (13) y˜ = −y, y˜ = y,
ˆ2 ˆ2 which are involutions, i. e., S = 1 and I0 = 1. Solving the equation ˆI0(xj , yj ) = Sˆ(xj , yj ), j = 1, 2, one gets indicator points [Shinohara and Aizawa, 1998]: (x1 = π/2, y1 = 0) and (x2 = 3π/2, y2 = 0). Iterating them, we construct a CIC (Uleysky, Budyansky and SP, PRE 2010) in the central part of the jet which is the last transport barrier in the sense that the CIC breaks down and is replaced by a stochastic layer with variation of the wave amplitudes. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Poincar´e sections
Poincar´e sections with N1 = 5 and N2 = 1 in the moving frame. (a) A2 = 0. Streamlines of the steady flow. (b) A2 = 0.09. CIC (the bold curve) is a barrier to transport across the jet. (c) A2 = 0.095. Destruction of CIC and onset of local cross-jet transport. (d) A2 = 0.2. Global chaotic cross-jet transport. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Central invariant curve
We define a CIC as a curve that is invariant under the operator Sˆ = 1 and the evolution operator GˆT (xi , yi ) = (xi+1, yi+1), where T ≡ 2π/ω2 perturbation period. The CIC separates the northern and southern parts of the flow. When CIC breaks down, the cros-jet transport becomes possible. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Iterations form CIC, periodic orbit or stochastic layer
The three cases are possible in dependence on the dimension d of the iteration set:
1 The iterations lie on a curve with d = 1 which is a CIC. 2 The iterations is an organized set of points with d = 0. It means that they constitute either a central periodic orbit or a central almost periodic orbit. 3 The iterations form a central stochastic layer with d = 2. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Global chaotic transport
If the iterations are not confined by any invariant curves in a bounded region, i. e. they occupy all the accessible phase plane to the south and north from the central jet, then there exists global chaotic trans- port. To illustrate the mechanism of destruction of CIC we fix A1 = 0.2418 and gradually increase A2. In the range 0 < A2 < 0.088, there exists a smooth CIC or a central almost periodic orbit which together with neighboring invariant curves form a transport barrier (Fig. a and b). At A2 ' 0.088, invariant manifolds of hyperbolic orbits of the resonance 7 : 3 cross each other, the CIC breaks down, and there appears at its place a narrow stochastic layer locked between remained invariant curves (Fig. c). When A2 increases further islands of the resonance 7 : 3 diverge, the stochastic layer becomes larger and we observe eventually global chaotic cross-jet transport (Fig. d). The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Amplitude mechanism for onset of cross-jet transport
Iterations of the indicator point form a CIC (a) and a central almost periodic orbit (b) which provide a transport barrier. CIC is broken and the iterations form stochastic layers with local (c) and global chaotic cross-jet transport (d). The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Ballistic resonance
It is reasonable to suppose that destruction of CIC is caused by a ballistic resonance between the maximal frequency of the particle motion in the central jet and the perturbation frequency ω2.
2 2 f1 N1 + 3N2 = 2 2 . (14) ω2 2N2(N1 − N2 ) At small amplitudes, this ratio gives an approximate estimate for the CIC winding number w. Equating the right-hand side of Eq. (14) to a rational number, one finds those values of the wave numbers N1 and N2 for which the CIC is strongly influenced by the corresponding resonance, and, therefore, cross-jet transport becomes possible. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Resonance mechanism of CIC destruction
0.64 x 0.66 0.64 x 0.66 y y
-0.72 a) b) -0.72
y y
-0.72 c) d) -0.72
0.64 x 0.66 0.64 x 0.66
A1 = 0.243 and gradual increasing A2. (a) Smooth CIC and neighboring invariant curves form a transport barrier. (b) After saddle-center bifurcation islands of the resonance 151 : 64 appear around the meandering CIC with 151 meanders. (c) Merging of invariant manifolds of hyperbolic orbits, breakdown of the CIC and appearence of a local stochastic layer. (d) New CIC appears. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Chaotic cross-jet transport diagrams
h 0 2 4 6
0.5
0.3
0.1 3−1 5−1 0.5
0.3 A2
0.1 5−3 7−1 0.5
0.3
0.1 7−3 9−1 0.2 0.4 0.2 0.4 A1
Color codes the width of the stochastic layer h at the place of a broken CIC. White – no transport, grey and black – local and global transport. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Chaotic cross-jet transport at even-odd wavenumber values
In this case advection equations (7) have no symmetry Sˆ (Eq. (13)) that allowed us to elaborate the CIC methodology. We develop an- other method for finding chaotic cross-jet transport based not on the CIC construction but on computing overlapping north and south stochastic layers. Let F1 be a fraction of particles from the first layer belonging to the second one. F2 a fraction of particles from the second layer belonging to the first one.
1 F1 = F2 = 0. No overlapping of the stochastic layers.
2 F1 ' n ' F2. Stochastic layers overlap entirely.
3 F1 ' F2 1. Stochastic layers do not overlap but touch each other. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Chaotic cross-jet transport diagrams
A1 A1 0.6 0.8 1.0 1.2 h 1.2 1.4 1.6 1.8 h 6 1.2 6 0.8
1.0 4 4 0.6 2 2
A A 0.8 0.4 2 2
0.6 0.2 3−2 5−4 0 0 A1 1.7 1.9 h 1.2 6
4 1.1 2 A
2 1.0 6−5 0
Global chaotic cross-jet transport vs A1,2 for even-odd wavenumbers : 3 − 2, 5 − 4, 6 − 5. Color codes the width of the united stochastic layer. White color no transport. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Laboratory experiment
A dye photograph showing the barrier to mixing between the inner and outer parts of the flow in the rotating tank with two dominant waves with wavenumbers 5 − 4 (Behringer et al, 1991). The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Comparison with laboratory experiments
Comparison with laboratory experiments with rotating fluid imitating geostrophical geophysical flows in the ocean and the atmosphere. Azimuthal jet with Rossby waves was produced by the action of the Coriolis force on radially pumped fluid in a rotating tank with a slope imitating the β-effect on the rotating Earth. Rapid mixing on either side of the jet was observed for a quasiperiodic flow, but no significant transport was observed across the jet. In our opinion the reason is that the experiments have been carried out under conditions that were far away from the resonances which are capable of destroying the transport barrier at the values of the Rossby wave numbers realized in the experiment. Our recommendation to observe cross-jet transport in such experiments is to produce an azimuthal jet with Rossby waves with odd wave numbers, say 3 : 1, 5 : 1 or 7 : 3. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Chaotic advection in blood flow
Blood flows in our veins and arteries follow a pulsating pattern, driven by the heart. This means that we can model blood flow as a time- periodic flow. It is only the fluid motion that governs the behavior of the transported particles, and the effects of diffusion, inertia and feedback are negligible (Schelin et al. PRE v.80, 016213 2009). Simple 2D model to mimic blood which is considered to be incom- pressible and Newtonian with a constant dynamic viscosity. The Reynolds number is 1600, with peak value 2200 close to turbulent threshold of around 2300 during the cardiac cycle. Vessel wall irreg- ularity, being either a narrowing or expansion of the vessel, generates time-dependent flow patterns which can result in very complex mo- tion by the advected particles. It has been shown (Schelin et al), using numeric models with realistic parameters, that the dynamics of particles transported by the blood flow in vessels with wall irreg- ularities is typically chaotic. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Numerical results
Top: Pattern traced by initial blob of particles in aorta with aneurysm. Bottom: Time spent in the region vs initial position. Lighter colors indicate long residence time, 100 heartbeat cycles. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Numerical results: Residence time
Residence time of particles started from the x=9 cm line in the region of observation before reaching the x=13 cm line as a function of the initial position in exercise conditions. Residence time is measured in heartbeats as time units. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Chaotic advection in blood flow
A consequence of this chaotic advection is the appearance of a char- acteristic filamentary distribution of advected particles. The particles transported by the blood which spend a long time around a distur- bance either stick to the vessel wall or reside on fractal filaments. The effects of irregular chaotic motion of particles transported by blood can play a major role in the development of serious circulatory diseases. Vessel wall irregularities modify the flow field, changing in a nontrivial way the transport and activation of biochemically active particles. Blood particle transport is often chaotic in realistic phys- iological conditions. This chaotic behavior of the flow has crucial consequences for the dynamics of important processes in the blood, such as the activation of platelets (the blood particles responsible for thrombus formation) which are involved in the thrombus formation. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Some applications
Chaotic advection has seen numerous applications in diverse areas of fluid mechanics and the list of applications continues to grow. Some of the most important may be those addressing the stirring of fluids on geophysical or planetary scales. Numerical simulations of stirring due to convective motions in Earth’s mantle show every sign of chaotic advection, in the sense of folds and layers on smaller and smaller scales. This idea can be pursued down to the scale of the striations seen in individual rocks. Two new application areas in the engineering sciencesmicrofluidic devices and materials processing. Mixing plays a vital role in mi- crofluidic devices. The flow in microfluidic devices are predominantly laminar and producing turbulence is almost impractical. Chaotic ad- vection is the way to improve mixing in microfluidics. The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Materials processing by chaotic advection
a) Multilayer films in an extruded 2.5 mm filament consisting of two thermoplastics. b) Thin parallel striations among carbon black particles in polystyrene (Zumbrunnen 2001, 1998). The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection Film breakup in material processing by chaotic advection
a) Fracture surface of a fiber with a polypropylene-polystyrene microstructure. b) SEM micrograph of interconnected polyethylene structures derived as chaotic advection in films with polyethylene and polystyrene components (Zumbrunnen 2000, 2001). The toy model of chaotic advection Chaotic advection in laboratory experiments Chaotic cross-jet transport in a model geophysical flow Practical applications of chaotic advection What to read on chaotic advection
Aref H. The development of chaotic advection. Phys. Fluids. 2002. Vol. 14. P.1315. Ottino J.M. The kinematics of mixing: stretching, chaos, and transport. Cambridge: Cambridge University Press. 1989. Wiggins S., Ottino J.M. Foundation of chaotic mixing. Phil. Trans. R. Soc. Lond. A. 2004. Vol. 362. P. 937. Koshel K.V., Prants S.V. Chaotic advection in the ocean. Physics – Uspekhi. 2006. Vol.49. P.1151. Samelson R.M., Wiggins S. Lagrangian transport in geophysical jets and waves. New York: Springer. 2006. Lai Y.-C., Tel T. Transient Chaos: Complex Dynamics on Finite-time Scales. New York: Springer. 2011. Prants S.V. Dynamical systems theory methods for styding mixing and transport in the ocean. Physica Scripta. 2013. March issue.