PhD defense presentation

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, 2012
Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 1 / 27

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. 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. 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. 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. Motivation: Ozone depletion in the winter-time
stratospheric 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. Motivation: Ozone depletion in the winter-time
stratospheric 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. Motivation: Ozone depletion in the winter-time
stratospheric 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. 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. 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. 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. 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. An infinitely fast chemical reaction.
We study the bimolecular reaction: A + B −→ C
Eulerian 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. An infinitely fast chemical reaction.
We study the bimolecular reaction: A + B −→ C
Eulerian 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
φ = CA − CB is a passive tracer:
Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 6 / 27

15. An infinitely fast chemical reaction.
We study the bimolecular reaction: A + B −→ C
Eulerian 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
φ = 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. An infinitely fast chemical reaction.
A and B react instantaneously: they cannot coexist. The passive tracer
φ = 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. An infinitely fast chemical reaction.
A and B react instantaneously: they cannot coexist. The passive tracer
φ = 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
The 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. Results: general approach.
We want to understand the behavior of d|φ| , how it depends on diffusion,
dt
what 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. Results: general approach.
We want to understand the behavior of d|φ| , how it depends on diffusion,
dt
what 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 of
the domain).
Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 8 / 27

20. Results: general approach.
We want to understand the behavior of d|φ| , how it depends on diffusion,
dt
what 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 of
the 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. 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. 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. 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. 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 where
this maximum is reached.
Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 11 / 27

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. 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)
2π
FTLE 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. 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)
2π
FTLE 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. 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)
2π
FTLE 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 0
Asymptotically 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. 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. Gradients along the contact line:
probability density function.
Using the expression for the gradient advected along each contact line
element, we can show that the probability density function of
√
πκ
G = 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. Gradients along the contact line:
probability density function.
Using the expression for the gradient advected along each contact line
element, we can show that the probability density function of
√
πκ
G = 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
For 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 lt
If τ and λ were independent the pdf of G along the contact line would be
1
equal to the pdf of √τ .
Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 20 / 27

41. Gradients along the contact line
probability 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. 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. Probability distribution function of |φ|.
As long as there is separation of scale between the contact zone and the flow, we can
find 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. 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 initial
Decay 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Thanks!
Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 27 / 27