Bimolecular chemical reaction in two-dimensional Navier-Stokes flow

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Bimolecular chemical reaction in two-dimensional Navier-Stokes flow

  1. 1. Bimolecular chemical reaction in two-dimensional Navier-Stokes flow Farid Ait-Chaalal Under the supervision of Prof. Peter Bartello and Prof. Michel Bourqui McGill University Department of Atmospheric and Oceanic Sciences PhD defense April 18, 2012Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 1 / 27
  2. 2. Motivation: mixing of chemicals in the stratosphere.Stratospheric dynamics. Average atmosphere Zonal mean dynamics of the stratosphere from temperature profile. From Haynes (2000). The isolines from 300 to 850 NOAA. indicate the potential temperature of the isentropes in Kelvin. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 2 / 27
  3. 3. Motivation: mixing of chemicals in the stratosphere. Tracer distribution in the stratosphere advected by wind from reanalysis (January 1992) on an isentrope (450K). The tracer are initiated as potential vorticity (PV) contours and the integration is run for 12 days. From Waugh (1994). Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 3 / 27
  4. 4. Motivation: mixing of chemicals in the stratosphere. 1 Exponential lengthening of tracer filaments (e-folding time of about 5 days, which corresponds to a strain of 0.2 day−1 ). Tracer distribution in the stratosphere advected by wind from reanalysis (January 1992) on an isentrope (450K). The tracer are initiated as potential vorticity (PV) contours and the integration is run for 12 days. From Waugh (1994). Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 3 / 27
  5. 5. Motivation: mixing of chemicals in the stratosphere. 1 Exponential lengthening of tracer filaments (e-folding time of about 5 days, which corresponds to a strain of 0.2 day−1 ). 2 Small scales processes can be Tracer distribution in the parametrized by an effective stratosphere advected by wind from diffusivity of about 103 m2 /s reanalysis (January 1992) on an to 104 m2 /s: typical width of isentrope (450K). The tracer are filaments O(10km) initiated as potential vorticity (PV) contours and the integration is run for 12 days. From Waugh (1994). Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 3 / 27
  6. 6. Motivation: Ozone depletion in the winter-timestratospheric surf zone. 1 Depletion controlled by the deactivation of activated chlorine originating from the polar vortex with NOx originating from low latitudes (Tan et al., 1998 ) Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 4 / 27
  7. 7. Motivation: Ozone depletion in the winter-timestratospheric surf zone. 1 Depletion controlled by the deactivation of activated chlorine originating from the polar vortex with NOx originating from low latitudes (Tan et al., 1998 ) 2 Stirring of chemicals into filaments and subsequent mixing essential: strong dependence of the chemical concentration on resolution in climate-chemistry models (Tan et al., 1998 ) Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 4 / 27
  8. 8. Motivation: Ozone depletion in the winter-timestratospheric surf zone. 1 Depletion controlled by the deactivation of activated chlorine originating from the polar vortex with NOx originating from low latitudes (Tan et al., 1998 ) 2 Stirring of chemicals into filaments and subsequent mixing essential: strong dependence of the chemical concentration on resolution in climate-chemistry models (Tan et al., 1998 ) 3 Only a few studies tackle this problem from a theoretical point of view (Thuburn and Tan, 1997; Wonhas an Vassilicos, 2003 ) Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 4 / 27
  9. 9. Objectives and methodology. 1 Our flow: doubly periodic barotropic two-dimensional flow as a simplified model for isentropic stirring in the stratospheric surf zone. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 5 / 27
  10. 10. Objectives and methodology. 1 Our flow: doubly periodic barotropic two-dimensional flow as a simplified model for isentropic stirring in the stratospheric surf zone. 2 Our chemical reaction A + B −→ C is infinitely fast (controlled by diffusion). Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 5 / 27
  11. 11. Objectives and methodology. 1 Our flow: doubly periodic barotropic two-dimensional flow as a simplified model for isentropic stirring in the stratospheric surf zone. 2 Our chemical reaction A + B −→ C is infinitely fast (controlled by diffusion). 3 Numerical simulations: ensemble of simulations for various diffusion coefficients 1 ≤ Pr = diffusion ν ≤ 128 in a doubly periodic box viscosity κ [−π, π]2 . The viscosity ν is kept constant. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 5 / 27
  12. 12. Objectives and methodology. 1 Our flow: doubly periodic barotropic two-dimensional flow as a simplified model for isentropic stirring in the stratospheric surf zone. 2 Our chemical reaction A + B −→ C is infinitely fast (controlled by diffusion). 3 Numerical simulations: ensemble of simulations for various diffusion coefficients 1 ≤ Pr = diffusion ν ≤ 128 in a doubly periodic box viscosity κ [−π, π]2 . The viscosity ν is kept constant. 4 Theoretical approach: local Lagrangian straining theory (LLST, Antonsen, 1996 ). How does the chemical production depend on the tracer diffusion? Can we relate the concentration of the chemicals to the Lagrangian straining properties of the flow? Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 5 / 27
  13. 13. An infinitely fast chemical reaction.We study the bimolecular reaction: A + B −→ CEulerian equations for the concentrations Ci (x, t), i = A, B, C ,in the flow u: ∂CA + u · CA = κ 2 CA − kc CA CB ∂t ∂CB + u · CB = κ 2 CB − kc CA CB ∂t ∂CC + u · CC = κ 2 CC + kc CA CB , ∂t Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 6 / 27
  14. 14. An infinitely fast chemical reaction.We study the bimolecular reaction: A + B −→ CEulerian equations for the concentrations Ci (x, t), i = A, B, C ,in the flow u: ∂CA + u · CA = κ 2 CA − kc CA CB ∂t ∂CB + u · CB = κ 2 CB − kc CA CB ∂t ∂CC + u · CC = κ 2 CC + kc CA CB , ∂tφ = CA − CB is a passive tracer: Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 6 / 27
  15. 15. An infinitely fast chemical reaction.We study the bimolecular reaction: A + B −→ CEulerian equations for the concentrations Ci (x, t), i = A, B, C ,in the flow u: ∂CA + u · CA = κ 2 CA − kc CA CB ∂t ∂CB + u · CB = κ 2 CB − kc CA CB ∂t ∂CC + u · CC = κ 2 CC + kc CA CB , ∂tφ = CA − CB is a passive tracer: ∂φ 2 +u· φ=κ φ ∂t Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 6 / 27
  16. 16. An infinitely fast chemical reaction.A and B react instantaneously: they cannot coexist. The passive tracerφ = CA − CB gives CA and CB through: CA (x, t) = φ(x, t) and CB (x, t) = 0 if φ(x, t) > 0 CB (x, t) = −φ(x, t) and CA (x, t) = 0 if φ(x, t) < 0 Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 7 / 27
  17. 17. An infinitely fast chemical reaction.A and B react instantaneously: they cannot coexist. The passive tracerφ = CA − CB gives CA and CB through: CA (x, t) = φ(x, t) and CB (x, t) = 0 if φ(x, t) > 0 CB (x, t) = −φ(x, t) and CA (x, t) = 0 if φ(x, t) < 0The space average of the concentrations are: |φ| CA = CB = 2 |φ(t = 0)| − |φ| CC = 2 Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 7 / 27
  18. 18. Results: general approach.We want to understand the behavior of d|φ| , how it depends on diffusion, dtwhat determines its time evolution. We call it the chemical speed. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 8 / 27
  19. 19. Results: general approach.We want to understand the behavior of d|φ| , how it depends on diffusion, dtwhat determines its time evolution. We call it the chemical speed.We can write: 1 d|φ| A = −κ φ · n dl = −κ | φ| dl 2 dt L(t) L(t)Where the contact line L is the set {x|φ(x) = 0} (A is the total area ofthe domain). Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 8 / 27
  20. 20. Results: general approach.We want to understand the behavior of d|φ| , how it depends on diffusion, dtwhat determines its time evolution. We call it the chemical speed.We can write: 1 d|φ| A = −κ φ · n dl = −κ | φ| dl 2 dt L(t) L(t)Where the contact line L is the set {x|φ(x) = 0} (A is the total area ofthe domain).Three regimes: Initial regime: L is a clearly defined material line. It does not depend on diffusion. Increase of the chemical speed. Intermediate regime: merging of tracer filaments. The chemical speed reaches a maximum. Long time decay of the tracer fluctuations and of the chemical speed. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 8 / 27
  21. 21. Results: general approach.Three regimes: Initial regime: L is a clearly defined material line. It does not depend on diffusion. Increase of the chemical speed. Intermediate regime: merging of tracer filaments. The chemical speed reaches a maximum. Long time decay of the tracer fluctuations and of the chemical speed. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 9 / 27
  22. 22. Results: general approach.Initial regime. We follow individual contact line elements, calculate the evolution of their length in a Lagrangian framework, and of the corresponding advected gradient φ. This is justified by a separation of scale: the contact line is clearly defined and the contact zone is small compared to the flow scale as long as t Tmix ≈ T ln Pe = T ln RePr , where Tmix is the mixing time scale from the large scale to the diffusive scale. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 10 / 27
  23. 23. Lagrangian straining properties (LSP) of the flow:finite time Lyapunov exponent (FTLE).Definition: 1 |δl(t)| λ(x, t) = max lim ln t α |δl0 |→0 |δl0 |The maximum is calculated over all the possible orientations α of |δl0 |. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 11 / 27
  24. 24. Lagrangian straining properties (LSP) of the flow:finite time Lyapunov exponent (FTLE).Definition: 1 |δl(t)| λ(x, t) = max lim ln t α |δl0 |→0 |δl0 |The maximum is calculated over all the possible orientations α of |δl0 |.We define a singular vector ψ+ (x, t) corresponding to the direction wherethis maximum is reached. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 11 / 27
  25. 25. Lagrangian straining properties (LSP) of the flow:FTLE probability density function (pdf).FTLE PDFs shown at different times Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 12 / 27
  26. 26. Lagrangian straining properties (LSP) of the flow:FTLE probability density function (pdf). In ergodic chaotic flow, the pdf of the FTLE is given, for sufficiently large times, by the large deviation theory result (e.g. Balkovsky(1999), Ott(2002)): s tG (λ0 ) Pλ (t, λ) = e exp(−tG (λ)), (3) 2πFTLE PDFs shown at different times where G (λ) is the Cramer function. It is concave, minimum at λ0 with G (λ0 ) = G (λ0 ) = 0. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 12 / 27
  27. 27. Lagrangian straining properties (LSP) of the flow:FTLE probability density function (pdf). In ergodic chaotic flow, the pdf of the FTLE is given, for sufficiently large times, by the large deviation theory result (e.g. Balkovsky(1999), Ott(2002)): s tG (λ0 ) Pλ (t, λ) = e exp(−tG (λ)), (3) 2πFTLE PDFs shown at different times where G (λ) is the Cramer function. It is concave, minimum at λ0 with G (λ0 ) = G (λ0 ) = 0. λ0 is the infinite time Lyapunov exponent (slow algebraic convergence). Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 12 / 27
  28. 28. Lagrangian straining properties (LSP) of the flow:FTLE probability density function (pdf). In ergodic chaotic flow, the pdf of the FTLE is given, for sufficiently large times, by the large deviation theory result (e.g. Balkovsky(1999), Ott(2002)): s tG (λ0 ) Pλ (t, λ) = e exp(−tG (λ)), (3) 2πFTLE PDFs shown at different times where G (λ) is the Cramer function. It is concave, minimum at λ0 with G (λ0 ) = G (λ0 ) = 0. λ0 is the infinite time Lyapunov exponent (slow algebraic convergence). Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 12 / 27
  29. 29. Example of strain map (FTLE as time goes to 0).Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 13 / 27
  30. 30. Time evolution of the FTLE maps (length of the animation: 20T ), plotted at the initial positions of the trajectories. This is the map of future stretching rates of Lagrangian parcels. The singular vectors converge to the (forward) Lyapunov vector exponentially fast in time: it will be assumed constant (when needed).Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 14 / 27
  31. 31. Time evolution of the FTLE maps (length of the simulation: 20T ), plotted at the initial positions of the trajectories. This is the map of future stretching rates of Lagrangian parcels. The singular vectors converge to the (forward) Lyapunov vector exponentially fast in time: it will be assumed constant (when needed).Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 15 / 27
  32. 32. Lagrangian straining properties of the flow:equivalent times.We define two equivalent times: t e 2uλ(u) du t τ= 0 and τ = e −2uλ(u) du. e 2tλ(t) 0 Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 16 / 27
  33. 33. Lagrangian straining properties of the flow:equivalent times.We define two equivalent times: t e 2uλ(u) du t τ= 0 and τ = e −2uλ(u) du. e 2tλ(t) 0 The time τ measures the inverse of the Lagrangian stretching over a short time before t on a chaotic orbit (Antonsen (1996), Haynes and Vanneste (2004)), for a long enough integration time t. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 16 / 27
  34. 34. Lagrangian straining properties of the flow:equivalent times.We define two equivalent times: t e 2uλ(u) du t τ= 0 and τ = e −2uλ(u) du. e 2tλ(t) 0 The time τ measures the inverse of the Lagrangian stretching over a short time before t on a chaotic orbit (Antonsen (1996), Haynes and Vanneste (2004)), for a long enough integration time t. The time τ measures the inverse of the Lagrangian stretching over a short after t = 0, for a long enough integration time t. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 16 / 27
  35. 35. Lagrangian straining properties of the flow:equivalent times.Probability density function of 1/τ : Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 17 / 27
  36. 36. Lagrangian straining properties of the flow:equivalent times.Probability density function of 1/τ :Joint probability density function of λ and 1/τ : Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 17 / 27
  37. 37. Contact line lengthening. Ensemble average L of the length of the contact line. With γ the angle between the contact line and the singular vector at the initial time, we have: ∞ 2π 1 dγ Pλ (t, λ) e 2λt cos2 γ + e −2λt sin2 γ 2 L = L0 dλ λ=0 γ=0 2π ∞ 2L0 ∼ LE = Pλ (t, λ)e λt dλ t T 2 π 0Asymptotically LE ≈ e λ1 t with λ1 = maxλ [λ − G (λ)]. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 18 / 27
  38. 38. Advection of gradients along the contact line. Ensemble average of the gradients advected with the contact line, √ κπ multiplied by A0 . In the limit of infinite initial gradients, we can relate them to the LSP: A0 L 0 e 2λt cos2 γ + e −2λt sin2 γ dγ ZZZZ | φL | = √ p P(t, λ, τ, τ ) dλ dτ d τ e e e Calculation from the LSP πκ L τ e 2λt cos2 γ + τ sin2 γ e 2π 2A0 L0 e λt ZZ ∼ √ √ Pλ,τ (t, λ, τ ) dλ dτ Calculation from the LSP t T π 3 κ LE τ contact line equilibrated with the flow √With the statistical independence between λ and τ , we could approximate, at large times (t T ), | φL | by A0 / πκτ . Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 19 / 27
  39. 39. Gradients along the contact line:probability density function.Using the expression for the gradient advected along each contact lineelement, we can show that the probability density function of √ πκG = A0 | φL | is: dγ 2lt cos2 γ + e −2lt sin2 γ π dlPG ,λ (t, g , l) e PG ,L (t, g ) = dγ 2lt cos2 γ + e −2lt sin2 γ π dlPλ (t, l) e Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 20 / 27
  40. 40. Gradients along the contact line:probability density function.Using the expression for the gradient advected along each contact lineelement, we can show that the probability density function of √ πκG = A0 | φL | is: dγ 2lt cos2 γ + e −2lt sin2 γ π dlPG ,λ (t, g , l) e PG ,L (t, g ) = dγ 2lt cos2 γ + e −2lt sin2 γ π dlPλ (t, l) eFor a contact line equilibrated with the flow: dlP √ lt 1 τ ,λ (t, g , l)e PG ,L (t, g ) ∼ PG ,L,∞ (t, g ) = dlPλ (t, l)e ltIf τ and λ were independent the pdf of G along the contact line would be 1equal to the pdf of √τ . Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 20 / 27
  41. 41. Gradients along the contact lineprobability density function. Comparison between the pdf PG ,Pr of G along the contact line approximated from the direct numerical simulations and the calculation from the LSP PG ,L , 1 PG ,L,∞ and the pdf of √τ . We have only plotted the curves PG ,Pr corresponding to direct numerical simulations consistent with the infinite initial gradient hypothesis. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 21 / 27
  42. 42. Chemical speed. Ensemble average of the chemical speed divided by the diffusion κ. d|φ| L 0 A0 √ e 2λt cos2 γ + e −2λt sin2 γ dγ ZZZZ − = √ κ p P(t, λ, τ, τ ) dλ dτ d τ e e e Calculation from the LSP dt πA τ e 2λt cos2 γ + τ sin2 γ e 2π λt 2L0 A0 √ e ZZ ∼ √ κ √ Pλ,τ (t, λ, τ ) dλ dτ Calculation from the LSP t T π3 A τ contact line equilibrated with the flow Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 22 / 27
  43. 43. Probability distribution function of |φ|.As long as there is separation of scale between the contact zone and the flow, we canfind a small such that the pdf PΦ of φ is given by: 4 √ 1 `φ´ PΦ (φ) = κL Erf −1 for φ ∈ [0, A0 (1 − )]. (4) AA0 G A0(It is normalized when considering values of Φ larger than A0 (1 − )). Time evolution of the pdf of A0 − φ, √ 1 multiplied by Pr 1/G L , calculated from trajectories. The red curve (theoretical √ 4 prediction) corresponds to AAν Erf −1 A0 , `φ´ 0 where Erf is the Gauss error function. Log-log scale. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 23 / 27
  44. 44. Long-term decay.Decay of φ2 for various Prandtl numbers: Two successive exponential decays: In the first regime, the decay seems globally controlled, and not predicted by local Lagrangian straining theories (LLST). However LLST are successful in predicting the shape of the pdf of φ, away from the tails, and some features of the variance spectrum. In the second regime, the system keeps a memory of the initialDecay of φ2 at Pr = 128 for different members: conditions and the decay is more sensitive to them than in regime first regime. However the decay of φ2 might not be controlled by any mechanism described in the literature (global or local). Tracer captured in vortices and ejected with tracer filaments seem to be an important process for the control of the decay. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 24 / 27
  45. 45. Conclusions LLST adapted to the study of an initial regime (5 to 10 T). d|φ| √ dt scales like κ. Rare events in the FTLE pdf determine the global chemical production. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 25 / 27
  46. 46. Conclusions LLST adapted to the study of an initial regime (5 to 10 T). d|φ| √ dt scales like κ. Rare events in the FTLE pdf determine the global chemical production. LLST also give how gradients pdf along the contact line and chemicals pdf scale with diffusion. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 25 / 27
  47. 47. Conclusions LLST adapted to the study of an initial regime (5 to 10 T). d|φ| √ dt scales like κ. Rare events in the FTLE pdf determine the global chemical production. LLST also give how gradients pdf along the contact line and chemicals pdf scale with diffusion. In the long-term decay, previous theories or frameworks (LLST, strange eigenmode) do not seem directly applicable to flows solution of the Navier-Stokes equation, in particular because they do not capture the rˆle of coherent transient structures. o Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 25 / 27
  48. 48. What next? What about the assumption of a stationary singular vector? Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 26 / 27
  49. 49. What next? What about the assumption of a stationary singular vector? Intermediate regime. Use of LLST and of the fractal structure of the contact line? Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 26 / 27
  50. 50. What next? What about the assumption of a stationary singular vector? Intermediate regime. Use of LLST and of the fractal structure of the contact line? Slower chemistry, other kinds of reactions. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 26 / 27
  51. 51. What next? What about the assumption of a stationary singular vector? Intermediate regime. Use of LLST and of the fractal structure of the contact line? Slower chemistry, other kinds of reactions. More realistic flows: critical layers, vertical structure. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 26 / 27
  52. 52. What next? What about the assumption of a stationary singular vector? Intermediate regime. Use of LLST and of the fractal structure of the contact line? Slower chemistry, other kinds of reactions. More realistic flows: critical layers, vertical structure. LSP: time evolution of the FTLE pdf, existence of a Cram`r function, e statistical dependence between FTLE and equivalent times, equivalent times pdf. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 26 / 27
  53. 53. What next? What about the assumption of a stationary singular vector? Intermediate regime. Use of LLST and of the fractal structure of the contact line? Slower chemistry, other kinds of reactions. More realistic flows: critical layers, vertical structure. LSP: time evolution of the FTLE pdf, existence of a Cram`r function, e statistical dependence between FTLE and equivalent times, equivalent times pdf. Mechanisms for the decay of tracer fluctuations in two-dimensional Navier-Stokes flows. Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 26 / 27
  54. 54. Thanks!Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 27 / 27

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