Conservation Laws of Surfactant Transport Equationsshevyakov/publ/talks/shev_CMS_Dec2011_v01.pdf · Conservation Laws of Surfactant Transport Equations Alexei Cheviakov Department

Conservation Laws of Surfactant Transport Equations

Alexei Cheviakov

Department of Mathematics and Statistics,University of Saskatchewan, Saskatoon, Canada

Winter 2011 CMS MeetingDec. 10, 2011

A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 1 / 22

Outline

1 SurfactantsBrief OverviewApplicationsSurfactant Transport Equations

2 Conservation LawsDefinition and ExamplesApplications

3 Direct Construction Method for Conservation LawsThe IdeaCompleteness

4 Conservation Laws of Examples of Surfactant Dynamics EquationsThe Convection CaseThe Convection-Diffusion Case

5 Conclusions


Outline





5 Conclusions


Surfactants - Brief Overview

Surfactants

”Surfactant”=”Surface active agent”.

Act as detergents, wetting agents, emulsifiers, foaming agents, and dispersants.

Consist of a hydrophobic group (tail) and a hydrophilic group (head).



Surfactants




Sodium lauryl sulfate:

Saponification: e.g., triglyceride (fat) + base → sodium stearate (soap)



Surfactants




Hydrophilic groups (heads) can have various properties:


Surfactants - Applications

Surfactant molecules adsorb at phase separation interfaces.Stabilization of growth of bubbles / droplets.Creation of emulsions of insoluble substances.Multiple industrial and medical applications.



Can form micelles, double layers, etc.



Soap bubbles...


Surfactant Transport Equations

Derivation:

Can be derived as a special case of multiphase flows with moving interfaces and contactlines:[Y.Wang, M. Oberlack, 2011]

Illustration:Y. Wang, M. Oberlack

Fig. 1 The investigated material domain B with its outer boundary ∂B consisting of three-phase subdomains B(i)(i = 1, 2, 3)surrounded by their outer boundaries ∂B(i) and phase interfaces S(i)

The interface or phase boundary may be not a surface of sudden change of mass, momentum, energy, andentropy; it rather constitutes a thin layer across which mass, momentum, energy, etc. change smoothly butrapidly between the densities of the adjacent phase materials. The reason for this is the necessity of moleculararrays of the adjacent bulk phases. Such an interface layer possesses a thickness of only a few molecular diam-eters or molecular mean free path. Compared to the dimensions of the adjacent bulk materials, the interface isalmost infinitely thin and can therefore be described as two-dimensional continua, representing mathematicallysingular surface with its own thermomechanical properties. The interface can carry mass, momentum, energy,and entropy. The interface is considered to be material in the sense if we assume its tangential velocity is thetangential material velocity of fluid particles that lie on the interface at some instant of time. The singularsurface as a whole, however, may be non-material, because matter may cross it when it represents a phaseboundary. A thermodynamic theory of phase boundaries can thus be founded on the postulation of conservationlaws for surface mass, momentum and energy and a balance law for surface entropy together with constitutiveassumptions for surface dependent fields and jump relations of the bulk fields at the phase boundary. It is alsothe similar case for the three-phase contact line, which represents a three-phase interaction region and hencemay also possess mass, momentum, energy, and entropy, for which corresponding conservation laws can alsobe postulated.

For the material domain B, the quantities arising in the Eq. (1) express the corresponding entities for thewhole domain; they are equivalent to a sum of individual contributions for all particles contained in the domainB or on the boundary ∂B. Thus, we have

Γ =∫

B

γ dv =3∑

i=1

∫

B(i)

γ (i) dv +3∑

i=1

∫

S(i)

γ (si ) da +∫

C

γ (c) dl,

F = −∫

∂B

φ · n da = −3∑

i=1

∫

∂B(i)

φ(i) · n da −3∑

i=1

∫

C(i)

φ(si ) · s(i) dl

−[(φ(c) · λ(c))II − (φ(c) · λ(c))I

], (2)

P =∫

B

π dv =3∑

i=1

∫

B(i)

π(i) dv +3∑

i=1

∫

S(i)

π(si ) da +∫

C

π(c) dl,


Surfactant Transport Equations (ctd.)

= 0

n

u

Parameters

Surfactant concentration c = c(x, t).

Flow velocity u(x, t).

Two-phase interface: phase separation surface Φ(x, t) = 0.

Unit normal: n = − ∇Φ

|∇Φ| .



= 0

n

u

Surface gradient

Surface projection tensor: pij = δij − ninj .

Surface gradient operator: ∇s = p · ∇ = (δij − ninj)∂

∂x j.

Surface Laplacian:

∆sF = (δij − ninj)∂

∂x j

((δik − nink)

∂F

∂xk

).



= 0

n

u

Governing equations

Incompressibility condition: ∇ · u = 0.

Fluid dynamics equations: Euler or Navier-Stokes.

Interface transport by the flow: Φt + u · ∇Φ = 0.

Surfactant transport equation:

ct + ui ∂c

∂x i− cninj

∂ui

∂x j− α(δij − ninj)

∂

∂x j

((δik − nink)

∂c

∂xk

)= 0.



= 0

n

u

Fully conserved form

Specific numerical methods (e.g., discontinuous Galerkin) require the system to bewritten in a fully conserved form.

Straightforward for continuity, momentum, and interface transport equations.

Can the surfactant transport equation be written in the conserved form?

ct + ui ∂c

∂x i− cninj

∂ui


∂

∂x j

((δik − nink)

∂c

∂xk

)= 0.


Outline





5 Conclusions


Conservation Laws: Some Definitions

Conservation law:

Given: some model.Independent variables: x = (t, x , y , ...); dependent variables: u = (u, v , ...).

A conservation law: a divergence expression equal to zero

DtΘ(x,u, ...) + DxΨ1(x,u, ...) + DyΨ2(x,u, ...) + · · · = 0.

Time-independent:

DiΨi ≡ divx,y,...Ψ = 0.

Time-dependent:

DtΘ + divx,y,...Ψ = 0.

A conserved quantity:

Dt

∫V

Θ dV = 0.


A Simple PDE Example

Example: small string oscillations, 1D wave equation

Independent variables: x , t; dependent: u(x , t).

utt = c2uxx , c2 = T/ρ.

Conservation of momentum:

Divergence expression: Dt(ρut)−Dx(Tux) = 0.

Arises from a multiplier:

Dt(ρut)−Dx(Tux) = Λ(utt − c2uxx) = ρ(utt − c2uxx) = 0;

Conserved quantity: total momentum M =∫ρutdx = const.


A Simple PDE Example

Example: small string oscillations, 1D wave equation

Independent variables: x , t; dependent: u(x , t).

utt = c2uxx , c2 = T/ρ.

Conservation of Energy:

Divergence expression: Dt

(ρu2

t

2+

Tu2x

2

)−Dx(Tutux) = 0.

Arises from a multiplier:

Dt

(ρu2

t

2+

Tu2x

2

)−Dx(Tutux) = ρut(utt − c2uxx) = 0;

Conserved quantity: total ebergy E =∫ (ρu2

t

2+

Tu2x

2

)dx = const.


Applications of Conservation Laws

ODEs

Constants of motion.

Integration.

PDEs

Direct physical meaning.

Constants of motion.

Differential constraints (divB = 0, etc.).

Analysis: existence, uniqueness, stability.

Nonlocally related PDE systems, exact solutions. Potentials, stream functions, etc.:

V = (u, v), divV = ux + vy = 0,

{u = Φy ,v = −Φx .

An infinite number of conservation laws can indicate integrability / linearization.

Numerical simulation.


Outline





5 Conclusions


The Idea of the Direct Construction Method

Definition

The Euler operator with respect to U j :

EU j =∂

∂U j−Di

∂

∂U ji

+ · · ·+ (−1)sDi1 . . .Dis

∂

∂U ji1...is

+ · · · , j = 1, . . . ,m.

Consider a general DE system Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N withvariables x = (x1, ..., xn), u = u(x) = (u1, ..., um).

Theorem

The equationsEU j F (x ,U, ∂U, . . . , ∂sU) ≡ 0, j = 1, . . . ,m

hold for arbitrary U(x) if and only if

F (x ,U, ∂U, . . . , ∂sU) ≡ DiΨi

for some functions Ψi (x ,U, ...).


The Idea of the Direct Construction Method

Consider a general DE system Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N withvariables x = (x1, ..., xn), u = u(x) = (u1, ..., um).

Direct Construction Method [Anco, Bluman (1997,2002)]

Specify dependence of multipliers: Λσ = Λσ(x ,U, ...), σ = 1, ...,N.

Solve the set of determining equations

EU j (ΛσRσ) ≡ 0, j = 1, . . . ,m,

for arbitrary U(x) (off of solution set!) to find all such sets of multipliers.

Find the corresponding fluxes Φi (x ,U, ...) satisfying the identity

ΛσRσ ≡ DiΦi .

Each set of fluxes and multipliers yields a local conservation law

DiΦi (x , u, ...) = 0,

holding for all solutions u(x) of the given PDE system.


Completeness

Completeness of the Direct Construction Method

For the majority of physical DE systems (in particular, all systems in solved form),all conservation laws follow from linear combinations of equations!

ΛσRσ ≡ DiΦi .


Outline





5 Conclusions


CLs of the Surfactant Dynamics Equations: The Convection Case

Governing equations (α = 0)

R1 =∂ui

∂x i= 0,

R2 = Φt +∂(uiΦ)

∂x i= 0,

R3 = ct + ui ∂c

∂x i− cninj

∂ui

∂x j= 0.

Multiplier ansatz

Λi = Λi (t,x,Φ, c,u, ∂Φ, ∂c, ∂u, ∂2Φ, ∂2c, ∂2u).

Conservation Law Determining Equations

Euj (ΛσRσ) = 0, j = 1, ..., 3; EΦ(ΛσRσ) = 0; Ec(ΛσRσ) = 0.




R1 =∂ui

∂x i= 0,

R2 = Φt +∂(uiΦ)

∂x i= 0,

R3 = ct + ui ∂c

∂x i− cninj

∂ui

∂x j= 0.

Multiplier ansatz







R1 =∂ui

∂x i= 0,

R2 = Φt +∂(uiΦ)

∂x i= 0,

R3 = ct + ui ∂c

∂x i− cninj

∂ui

∂x j= 0.

Multiplier ansatz





CLs of the Surfactant Dynamics Equations: The Convection Case (ctd.)


R1 =∂ui

∂x i= 0,

R2 = Φt +∂(uiΦ)

∂x i= 0,

R3 = ct + ui ∂c

∂x i− cninj

∂ui

∂x j= 0.

Principal Result 1 (multipliers)

There exist an infinite family of multiplier sets with Λ3 6= 0, i.e., essentiallyinvolving c.

Family of conservation laws with

Λ3 = |∇Φ| K(Φ, c|∇Φ|).




R1 =∂ui

∂x i= 0,

R2 = Φt +∂(uiΦ)

∂x i= 0,

R3 = ct + ui ∂c

∂x i− cninj

∂ui

∂x j= 0.

Principal Result 1 (divergence expressions)

Usual form:∂

∂tG(Φ, c|∇Φ|) +

∂

∂x i

(uiG(Φ, c|∇Φ|)

)= 0.

Material form:d

dtG(Φ, c|∇Φ|) = 0.




R1 =∂ui

∂x i= 0,

R2 = Φt +∂(uiΦ)

∂x i= 0,

R3 = ct + ui ∂c

∂x i− cninj

∂ui

∂x j= 0.

Simplest conservation law with c-dependence

Can take G(Φ, c|∇Φ|) = c|∇Φ|.

∂

∂t(c|∇Φ|) +

∂

∂x i

(uic|∇Φ|

)= 0.


The Convection-Diffusion Case

Governing equations (α 6= 0)

R1 =∂ui

∂x i= 0,

R2 = Φt +∂(uiΦ)

∂x i= 0,

R3 = ct + ui ∂c

∂x i− cninj

∂ui


∂

∂x j

((δik − nink)

∂c

∂xk

)= 0.




The Convection-Diffusion Case (ctd.)


R1 =∂ui

∂x i= 0,

R2 = Φt +∂(uiΦ)

∂x i= 0,

R3 = ct + ui ∂c

∂x i− cninj

∂ui


∂

∂x j

((δik − nink)

∂c

∂xk

)= 0.

Principal Result 2 (multipliers)

Λ1 = ΦF(Φ) |∇Φ|−1

(∂

∂x j

(c∂Φ

∂x j

)− cninj

∂2Φ

∂x i∂x j

),

Λ2 = −F(Φ) |∇Φ|−1

(∂

∂x j

(c∂Φ

∂x j

)− cninj

∂2Φ

∂x i∂x j

),

Λ3 = F(Φ)|∇Φ|,




R1 =∂ui

∂x i= 0,

R2 = Φt +∂(uiΦ)

∂x i= 0,

R3 = ct + ui ∂c

∂x i− cninj

∂ui


∂

∂x j

((δik − nink)

∂c

∂xk

)= 0.

Principal Result 2 (divergence expressions)

An infinite family of conservation laws:

∂

∂t(c F(Φ) |∇Φ|) +

∂

∂x i

(Ai F(Φ) |∇Φ|

)= 0,

where

Ai = cui − α(

(δik − nink)∂c

∂xk

), i = 1, 2, 3,

and F is an arbitrary sufficiently smooth function.




R1 =∂ui

∂x i= 0,

R2 = Φt +∂(uiΦ)

∂x i= 0,

R3 = ct + ui ∂c

∂x i− cninj

∂ui


∂

∂x j

((δik − nink)

∂c

∂xk

)= 0.

Simplest conservation law with c-dependence

Can take F(Φ) = 1:

∂

∂t(c |∇Φ|) +

∂

∂x i

(Ai |∇Φ|

)= 0.


Conclusions

Some conclusions

Conservation laws can be obtained systematically through the Direct ConstructionMethod, which employs multipliers and Euler operators.

The method is implemented in a symbolic package GeM for Maple.

Surfactant dynamics equations can be written in a fully conserved form.

Infinite families of c-dependent conservation laws for cases with and withoutdiffusion.

Open problems

Higher order conservation laws?

Can we get more by including fluid dynamics equations?


Conclusions

Some conclusions

Conservation laws can be obtained systematically through the Direct ConstructionMethod, which employs multipliers and Euler operators.

The method is implemented in a symbolic package GeM for Maple.

Surfactant dynamics equations can be written in a fully conserved form.

Infinite families of c-dependent conservation laws for cases with and withoutdiffusion.

Open problems

Higher order conservation laws?

Can we get more by including fluid dynamics equations?


Some references

Anco, S.C. and Bluman, G.W. (1997).Direct construction of conservation laws from field equations. Phys. Rev. Lett. 78,2869–2873.

Anco, S.C. and Bluman, G.W. (2002).Direct construction method for conservation laws of partial differential equations.Part I: Examples of conservation law classifications. Eur. J. Appl. Math. 13,545–566.

Bluman, G.W., Cheviakov, A.F., and Anco, S.C. (2010).Applications of Symmetry Methods to Partial Differential Equations.Springer: Applied Mathematical Sciences, Vol. 168.

Cheviakov, A.F. (2007).GeM software package for computation of symmetries and conservation laws ofdifferential equations. Comput. Phys. Commun. 176, 48–61.

Kallendorf, C., Cheviakov, A.F., Oberlack, M., and Wang, Y.Conservation Laws of Surfactant Transport Equations. Submitted, 2011.


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Conservation Laws of Surfactant Transport Equationsshevyakov/publ/talks/shev_CMS_Dec2011_v01.pdf · Conservation Laws of Surfactant Transport Equations Alexei Cheviakov Department