Wave fronts with cross-diffusion · Gabriel Morgado1;2, Bogdan Nowakowski1, Annie Lemarchand2 1. Institute of Physical Chemistry, Polish Academy of Sciences 2. Laboratoire de Physique

Combinatorics and Arithmetic for Physics

Wave fronts with cross-diffusion

Gabriel Morgado1,2, Bogdan Nowakowski1, Annie Lemarchand2

1. Institute of Physical Chemistry, Polish Academy of Sciences2. Laboratoire de Physique Théorique de la Matiere Condensee, Sorbonne Université

November 7th, 2019

G. Morgado (IPC, PAS/ LPTMC, SU) November 7th, 2019 1 / 20

Fisher-KPP equation:

∂tu = ∂2xu+ ru(1− u)

with u(x, t) and a parameter r

Belongs to the class of reaction-diffusion equationsAdmits two equilibrium states u = 0 and u = 1Existence of a travelling wave between the two states



∂tu = ∂2xu+ ru(1− u)

with u(x, t) and a parameter rBelongs to the class of reaction-diffusion equations

Admits two equilibrium states u = 0 and u = 1Existence of a travelling wave between the two states



∂tu = ∂2xu+ ru(1− u)

with u(x, t) and a parameter rBelongs to the class of reaction-diffusion equationsAdmits two equilibrium states u = 0 and u = 1

Existence of a travelling wave between the two states



∂tu = ∂2xu+ ru(1− u)

with u(x, t) and a parameter rBelongs to the class of reaction-diffusion equationsAdmits two equilibrium states u = 0 and u = 1Existence of a travelling wave between the two states


We consider the following autocatalytic reaction, in solvent S:

A+ Bk−−−→ 2A

The reaction-diffusion equations are given by:

∂tA = DA∂2xA+ kAB

∂tB = DB∂2xB − kAB

where A(x, t) and B(x, t) are the concentrations of A and B, DAand DB are diffusion coefficients and k is the kinetic constant

When DA = DB, the RD equation of A(x, t) is a Fisher-KPPequation.


In a concentrated system, diffusion is perturbed:

∂tA = DA∂x

[(1− A

C

)∂xA

]−DB∂x

(A

C∂xB

)+ kAB

∂tB = DB∂x

[(1− B

C

)∂xB

]−DA∂x

(B

C∂xA

)− kAB

where C = A+B + S = constant

Questions

How does DA 6= DB affect the velocity and shape of the wavefront ?

How does confinement (i.e. S → 0) affect the velocity andshape of the wave front ?

Can we use these effects to detect perturbed diffusion in aconcentrated Fisher-KPP front ?


In a concentrated system, diffusion is perturbed:

∂tA = DA∂x

[(1− A

C

)∂xA

]−DB∂x

(A

C∂xB

)+ kAB

∂tB = DB∂x

[(1− B

C

)∂xB

]−DA∂x

(B

C∂xA

)− kAB

where C = A+B + S = constantQuestions

How does DA 6= DB affect the velocity and shape of the wavefront ?

How does confinement (i.e. S → 0) affect the velocity andshape of the wave front ?

Can we use these effects to detect perturbed diffusion in aconcentrated Fisher-KPP front ?


Velocity v of a Fisher front

Independent of DB and S

0

2

4

6

8

10

16000 18000 20000 22000 24000 26000 28000 30000

A V0

✲v

B

x

∆x

v = 2√kV0DA


Velocity v of a Fisher front Independent of DB and S

0

2

4

6

8

10

16000 18000 20000 22000 24000 26000 28000 30000

A V0

✲v

B

x

∆x

v = 2√kV0DA


Shape of the front, 2 parameters:

Height defined as the difference of concentration between

A(x0) and B(x0) where A(x0) =V02

⇒ h = B(x0)−V02

Width of the front defined as the inverse of the slope of the A

profile where A(x0) =V02

⇒W = V0A′(x0)

0

2

4

6

8

10

15000 17500 20000 22500 25000 27500 30000

✻

❄

h

AB

x

∆x

0

2

4

6

8

10

15000 17500 20000 22500 25000 27500 30000

✻

❄

❇❇❇❇❇❇❇❇❇❇❇

❇❇❇❇❇❇❇❇❇❇❇

✲✛

h

W

AB

x

∆x


Shape of the front, 2 parameters:

Height defined as the difference of concentration between

A(x0) and B(x0) where A(x0) =V02⇒ h = B(x0)−

V02

Width of the front defined as the inverse of the slope of the A

profile where A(x0) =V02⇒W = V0

A′(x0)

0

2

4

6

8

10

15000 17500 20000 22500 25000 27500 30000

✻

❄

h

AB

x

∆x

0

2

4

6

8

10

15000 17500 20000 22500 25000 27500 30000

✻

❄

❇❇❇❇❇❇❇❇❇❇❇

❇❇❇❇❇❇❇❇❇❇❇

✲✛

h

W

AB

x

∆x


To study the RD equations, we define the moving frame:

(x, t)→ (ζ)

ζ =x

v− t

ζ0 =x0v− t0 ≡ 0

A(x, t)→ f(ζ)B(x, t)→ g(ζ)

� = 1/v2

d = dilute (regular diffusion)

c = concentrated (perturbed diffusion)

Examples:

fd = A concentration in the dilute case

gc = B concentration in the concentrated case


Dilute case RD equations in the moving frame:

0 = kfdgd + f′d + �DAf

′′d

0 = −kfdgd + g′d + �DBg′′d

Concentrated case RD equations in the moving frame:

0 = kfcgc + f′c + �

(DA

[(1− f

′c

C

)f ′′c −

(f ′c)2

C

]−DB

[fcg′′c

C+f ′cg′c

C

])0 = −kfcgc + g′c + �

(DB

[(1− g

′c

C

)g′′c −

(g′c)2

C

]−DA

[gcf′′c

C+f ′cg′c

C

])


We consider k, V0 and DA such as:

�� 1Diffusion can be considered as a perturbation of reaction (do notmix up with perturbed diffusion !).We write f and g as a perturbation series:

f = f0 + �f1 + �2f2 + ...

g = g0 + �g1 + �2g2 + ...

Zero-th order solutions (no diffusion) are straightforwardly obtained:

fd,0 = fc,0 =V0

1 + ekV0ζ

gd,0 = gc,0 =V0

1 + e−kV0ζ


Now, instead of deriving the solutions for the higher-order termsf1,2,... and g1,2,..., we focus on the point ζ0 = 0.

The expressions for the height and the width, up to thesecond-order, are given by:

hd =V016

(1− DB

DA

)[1 +

1

8

(1− DB

DA

)]hc =

V016

(1− DB

DA

)(1− V0

C

)[1 +

1

8

(1− DB

DA

)(1− 2V0

C

)]Wd = 8

√DAkV0

[1 +

1

8

(1− DB

DA

)− 1

64

DBDA

(3− DB

DA

)]−1Wc = 8

√DAkV0

[1 +

1

8

(1− DB

DA

)(1− 3V0

2C

)− 1

64

[DBDA

(3− DB

DA

)+

(9

2− 8DB

DA+

7

2

D2BD2A

)V0C−(7

2− 7DB

DA+

7

2

D2BD2A

)V 20C2

]]−1


Now, instead of deriving the solutions for the higher-order termsf1,2,... and g1,2,..., we focus on the point ζ0 = 0.The expressions for the height and the width, up to thesecond-order, are given by:

hd =V016

(1− DB

DA

)[1 +

1

8

(1− DB

DA

)]hc =

V016

(1− DB

DA

)(1− V0

C

)[1 +

1

8

(1− DB

DA

)(1− 2V0

C

)]Wd = 8

√DAkV0

[1 +

1

8

(1− DB

DA

)− 1

64

DBDA

(3− DB

DA

)]−1Wc = 8

√DAkV0

[1 +

1

8

(1− DB

DA

)(1− 3V0

2C

)− 1

64

[DBDA

(3− DB

DA

)+

(9

2− 8DB

DA+

7

2

D2BD2A

)V0C−(7

2− 7DB

DA+

7

2

D2BD2A

)V 20C2

]]−1G. Morgado (IPC, PAS/ LPTMC, SU) November 7th, 2019 10 / 20

Comparison between analytic and numerical results:

Height in thediluted case

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.1 1 10

hd

V0

DBDA

a)


Comparison between analytic and numerical results: Height in thediluted case

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.1 1 10

hd

V0

DBDA

a)


Perturbed diffusion (V0/C = 0.25) vs. regular diffusion (V0/C = 0):

Height

0

0.05

0.1

0.15

0.2

0.25

0.3

0.1 1 10

hd

−h

c

hd

DBDA

b)


Perturbed diffusion (V0/C = 0.25) vs. regular diffusion (V0/C = 0):Height

0

0.05

0.1

0.15

0.2

0.25

0.3

0.1 1 10

hd

−h

c

hd

DBDA

b)


Comparison between analytic and numerical results:

Width in thediluted case

0.6

0.8

1

1.2

1.4

1.6

1.8

0.01 0.1 1 10

Wd

DBDA

a)


Comparison between analytic and numerical results: Width in thediluted case

0.6

0.8

1

1.2

1.4

1.6

1.8

0.01 0.1 1 10

Wd

DBDA

a)


Perturbed diffusion (V0/C = 0.25) vs. regular diffusion (V0/C → 0):

Width

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.1 1 10

Wd

−W

c

Wd

DBDA

b)


Perturbed diffusion (V0/C = 0.25) vs. regular diffusion (V0/C → 0):Width

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.1 1 10

Wd

−W

c

Wd

DBDA

b)


Concentration effects :

Height Width

0.03

0.04

0.05

0.06

0.07

0 0.1 0.2 0.3 0.4

hc

V0

V0C

a)

0.7

0.72

0.74

0.76

0.78

0.8

0 0.1 0.2 0.3 0.4 0.5

Wc

V0C

for DB/DA = 1/16


Concentration effects :

Height Width

0.03

0.04

0.05

0.06

0.07

0 0.1 0.2 0.3 0.4

hc

V0

V0C

a)

0.7

0.72

0.74

0.76

0.78

0.8

0 0.1 0.2 0.3 0.4 0.5W

c

V0C

for DB/DA = 1/16


Conclusion

The velocity of the front is not affected by the diffusion of Bspecies nor the perturbation of diffusion by solventconcentration.

The shape of the front is affected by both effects. The heightis strongly influenced by the ratio of diffusion coefficientsDB/DA and the deviation from ideal solution V0/C. However,its discriminating property is more efficient for low values ofDB/DA.

The width is also mainly affected by the ratio DB/DA andV0/C, working better for large values of DB/DA.


Thank You

[1] Morgado, Nowakowski, Lemarchand, Phys. Rev. E 99, 022205 (2019)

This project has received funding from the European Union’s Horizon2020 research and innovation programme under the MarieSk lodowska-Curie grant agreement No. 711859.


Cross-diffusion terms derivation from linear irreversiblethermodynamicsThe entropy production per unit mass due to isothermal diffusion isgiven by:

σ =1

T

∑X=A,B,S

~X ·(−~∇µX

)where T is the temperature, ~X the flux of species X, and µX is thechemical potential of species X.We consider the framework of the solvent. The flux of species X inthis framework is defined by:

~∗X = ρX (~uX − ~uS) = ρX (~uX − ~u) + ρX (~u− ~uS)

= ~X −ρXρS~S

where ρX is the concentration of X, and ~uX is the velocity of X.


Prigogine’s theorem: Entropy production does not depend on thechosen framework.Therefore:

σ =1

T

∑X=A,B,S

~X ·(−~∇µX

)=

1

T

∑X=A,B

~∗X ·(−~∇µX

)Assuming that the solution is ideal, we can write:

µX = µ0X +RT ln

ρXρ

Using the expression for ~∗X and µX , we get:(~A~B

)=

1− ρAρ −ρAρ−ρBρ

1− ρBρ

( ~∗A~∗B

)


Using the expression for ~∗X and µX , we get:

(~A~B

)=

1− ρAρ −ρAρ−ρBρ

1− ρBρ

( ~∗A~∗B

)

Hypothesis: ~∗X = −DX ~∇ρX (Fick’s first law in solvent framework)Then:

(~A~B

)= −

1− ρAρ ρAρρBρ

1− ρBρ

(DA~∇ρADB ~∇ρB

)


Wave fronts with cross-diffusion · Gabriel Morgado1;2, Bogdan Nowakowski1, Annie Lemarchand2 1. Institute of Physical Chemistry, Polish Academy of Sciences 2. Laboratoire de Physique

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