Conservation Laws of Surfactant Transport Equations

Conservation Laws of Surfactant Transport Equations Alexei Cheviakov Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada Winter 2011 CMS Meeting Dec. 10, 2011 A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 1 / 22

Outline 1 Surfactants Brief Overview Applications Surfactant Transport Equations 2 Conservation Laws Definition and Examples Applications 3 Direct Construction Method for Conservation Laws The Idea Completeness 4 Conservation Laws of Examples of Surfactant Dynamics Equations The Convection Case The Convection-Diffusion Case 5 Conclusions A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 2 / 22

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). A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 4 / 22

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). Sodium lauryl sulfate: Saponification: e.g., triglyceride (fat) + base sodium stearate (soap) A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 4 / 22

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). Hydrophilic groups (heads) can have various properties: A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 4 / 22

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. A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 5 / 22

Surfactants - Applications Can form micelles, double layers, etc. A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 5 / 22

Surfactants - Applications Soap bubbles... A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 5 / 22

Surfactant Transport Equations Derivation: Can be derived as a special case of multiphase flows with moving interfaces and contact lines: [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) A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 6 / 22

Surfactant Transport Equations (ctd.) n = 0 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. A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 7 / 22

Surfactant Transport Equations (ctd.) n = 0 u Surface gradient Surface projection tensor: p ij = δ ij n i n j. Surface gradient operator: s = p = (δ ij n i n j ) x. j Surface Laplacian: s F = (δ ij n i n j ) ( (δ x j ik n i n k ) F ). x k A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 7 / 22

Surfactant Transport Equations (ctd.) n = 0 u Governing equations Incompressibility condition: u = 0. Fluid dynamics equations: Euler or Navier-Stokes. Interface transport by the flow: Φ t + u Φ = 0. Surfactant transport equation: c t + u i c x i cn i n j u i x j α(δ ij n i n j ) x j ( (δ ik n i n k ) c ) = 0. x k A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 7 / 22

Surfactant Transport Equations (ctd.) n = 0 u Fully conserved form Specific numerical methods (e.g., discontinuous Galerkin) require the system to be written in a fully conserved form. Straightforward for continuity, momentum, and interface transport equations. Can the surfactant transport equation be written in the conserved form? c t + u i c u i cn x i i n j α(δ x j ij n i n j ) ( (δ x j ik n i n k ) c ) = 0. x k A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 7 / 22

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 D tθ(x, u,...) + D xψ 1 (x, u,...) + D y Ψ 2 (x, u,...) + = 0. Time-independent: D i Ψ i div x,y,...ψ = 0. Time-dependent: A conserved quantity: D tθ + div x,y,...ψ = 0. D t Θ dv = 0. V A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 9 / 22

A Simple PDE Example Example: small string oscillations, 1D wave equation Independent variables: x, t; dependent: u(x, t). u tt = c 2 u xx, c 2 = T /ρ. Conservation of momentum: Divergence expression: D t(ρu t) D x(tu x) = 0. Arises from a multiplier: D t(ρu t) D x(tu x) = Λ(u tt c 2 u xx) = ρ(u tt c 2 u xx) = 0; Conserved quantity: total momentum M = ρu tdx = const. A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 10 / 22

A Simple PDE Example Example: small string oscillations, 1D wave equation Independent variables: x, t; dependent: u(x, t). u tt = c 2 u xx, c 2 = T /ρ. Conservation of Energy: ( ρu 2 Divergence expression: D t t 2 + Tu2 x 2 Arises from a multiplier: ( ρu 2 D t t 2 + Tu2 x 2 ) D x(tu tu x) = 0. ) D x(tu tu x) = ρu t(u tt c 2 u xx) = 0; Conserved quantity: total ebergy E = ( ρut 2 2 + Tu2 x 2 ) dx = const. A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 10 / 22

Applications of Conservation Laws ODEs Constants of motion. Integration. PDEs Direct physical meaning. Constants of motion. Differential constraints (div B = 0, etc.). Analysis: existence, uniqueness, stability. Nonlocally related PDE systems, exact solutions. Potentials, stream functions, etc.: V = (u, v), div V = u x + v y = 0, { u = Φy, v = Φ x. An infinite number of conservation laws can indicate integrability / linearization. Numerical simulation. A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 11 / 22

The Idea of the Direct Construction Method Definition The Euler operator with respect to U j : E U j = D U j i U j i + + ( 1) s D i1... D is U j i 1...i s +, j = 1,..., m. Consider a general DE system R σ [u] = R σ (x, u, u,..., k u) = 0, variables x = (x 1,..., x n ), u = u(x) = (u 1,..., u m ). σ = 1,..., N with Theorem The equations E U j F (x, U, U,..., s U) 0, hold for arbitrary U(x) if and only if j = 1,..., m for some functions Ψ i (x, U,...). F (x, U, U,..., s U) D i Ψ i A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 13 / 22

The Idea of the Direct Construction Method Consider a general DE system R σ [u] = R σ (x, u, u,..., k u) = 0, σ = 1,..., N with variables x = (x 1,..., x n ), u = u(x) = (u 1,..., u m ). Direct Construction Method [Anco, Bluman (1997,2002)] Specify dependence of multipliers: Λ σ = Λ σ(x, U,...), σ = 1,..., N. Solve the set of determining equations E U j (Λ σr σ ) 0, j = 1,..., m, for arbitrary U(x) (off of solution set!) to find all such sets of multipliers. Find the corresponding fluxes Φ i (x, U,...) satisfying the identity Λ σr σ D i Φ i. Each set of fluxes and multipliers yields a local conservation law D i Φ i (x, u,...) = 0, holding for all solutions u(x) of the given PDE system. A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 14 / 22

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! Λ σr σ D i Φ i. A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 15 / 22

CLs of the Surfactant Dynamics Equations: The Convection Case Governing equations (α = 0) R 1 = ui x i = 0, R 2 = Φ t + (ui Φ) x i = 0, R 3 = c t + u i c x i cn i n j u i x j = 0. A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 17 / 22

CLs of the Surfactant Dynamics Equations: The Convection Case Governing equations (α = 0) Multiplier ansatz R 1 = ui x i = 0, R 2 = Φ t + (ui Φ) x i = 0, R 3 = c t + u i c x i cn i n j u i x j = 0. Λ i = Λ i (t, x, Φ, c, u, Φ, c, u, 2 Φ, 2 c, 2 u). A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 17 / 22

CLs of the Surfactant Dynamics Equations: The Convection Case Governing equations (α = 0) Multiplier ansatz R 1 = ui x i = 0, R 2 = Φ t + (ui Φ) x i = 0, R 3 = c t + u i c x i cn i n j u i x j = 0. Λ i = Λ i (t, x, Φ, c, u, Φ, c, u, 2 Φ, 2 c, 2 u). Conservation Law Determining Equations E u j (Λ σ R σ ) = 0, j = 1,..., 3; E Φ (Λ σ R σ ) = 0; E c(λ σ R σ ) = 0. A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 17 / 22

CLs of the Surfactant Dynamics Equations: The Convection Case (ctd.) Governing equations (α = 0) R 1 = ui x i = 0, R 2 = Φ t + (ui Φ) x i = 0, R 3 = c t + u i c x i cn i n j u i x j = 0. Principal Result 1 (multipliers) There exist an infinite family of multiplier sets with Λ 3 0, i.e., essentially involving c. Family of conservation laws with Λ 3 = Φ K(Φ, c Φ ). A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 18 / 22

CLs of the Surfactant Dynamics Equations: The Convection Case (ctd.) Governing equations (α = 0) R 1 = ui x i = 0, R 2 = Φ t + (ui Φ) x i = 0, R 3 = c t + u i c x i cn i n j u i x j = 0. Principal Result 1 (divergence expressions) Usual form: Material form: G(Φ, c Φ ) + ( ) u i G(Φ, c Φ ) = 0. t x i d G(Φ, c Φ ) = 0. dt A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 18 / 22

CLs of the Surfactant Dynamics Equations: The Convection Case (ctd.) Governing equations (α = 0) R 1 = ui x i = 0, R 2 = Φ t + (ui Φ) x i = 0, R 3 = c t + u i c x i cn i n j u i x j = 0. Simplest conservation law with c-dependence Can take G(Φ, c Φ ) = c Φ. t (c Φ ) + ( ) u i c Φ = 0. x i A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 18 / 22

The Convection-Diffusion Case Governing equations (α 0) R 1 = ui x i = 0, R 2 = Φ t + (ui Φ) = 0, x i R 3 = c t + u i c x i cn i n j u i x j α(δ ij n i n j ) x j ( (δ ik n i n k ) c ) = 0. x k Conservation Law Determining Equations E u j (Λ σ R σ ) = 0, j = 1,..., 3; E Φ (Λ σ R σ ) = 0; E c(λ σ R σ ) = 0. A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 19 / 22

The Convection-Diffusion Case (ctd.) Governing equations (α = 0) R 1 = ui x i = 0, R 2 = Φ t + (ui Φ) = 0, x i R 3 = c t + u i c x i cn i n j u i x j α(δ ij n i n j ) x j ( (δ ik n i n k ) c ) = 0. x k Principal Result 2 (multipliers) Λ 1 = ΦF(Φ) Φ 1 ( x j Λ 2 = F(Φ) Φ 1 ( x j Λ 3 = F(Φ) Φ, ( c Φ ) ) 2 Φ cn x j i n j, x i x ( j c Φ ) ) 2 Φ cn x j i n j, x i x j A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 20 / 22

The Convection-Diffusion Case (ctd.) Governing equations (α = 0) R 1 = ui x i = 0, R 2 = Φ t + (ui Φ) = 0, x i R 3 = c t + u i c x i cn i n j u i x j α(δ ij n i n j ) x j ( (δ ik n i n k ) c ) = 0. x k Principal Result 2 (divergence expressions) An infinite family of conservation laws: (c F(Φ) Φ ) + ( ) A i F(Φ) Φ = 0, t x i where A i = cu i α ( (δ ik n i n k ) c ), i = 1, 2, 3, x k and F is an arbitrary sufficiently smooth function. A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 20 / 22

The Convection-Diffusion Case (ctd.) Governing equations (α = 0) R 1 = ui x i = 0, R 2 = Φ t + (ui Φ) = 0, x i R 3 = c t + u i c x i cn i n j u i x j α(δ ij n i n j ) x j ( (δ ik n i n k ) c ) = 0. x k Simplest conservation law with c-dependence Can take F(Φ) = 1: (c Φ ) + ( ) A i Φ = 0. t x i A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 20 / 22

Conclusions Some conclusions Conservation laws can be obtained systematically through the Direct Construction Method, 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 without diffusion. A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 21 / 22

Conclusions Some conclusions Conservation laws can be obtained systematically through the Direct Construction Method, 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 without diffusion. Open problems Higher order conservation laws? Can we get more by including fluid dynamics equations? A. Cheviakov (Math&Stat) Cons. laws for Surfactant Transport Equations 10 Dec. 2011 21 / 22

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