S PECTRAL ANALYSIS FOR STEADY STATES OF COMPRESSIBLE TWO - PHASE FLUIDS OF N AVIER -S TOKES -A LLEN -C AHN TYPE IN BOUNDED DOMAINS Matthias Kotschote Department Mathematics and Statistics, University Konstanz MathFlows 2015, Porquerolles
O UTLINE 1. modelling NSAC/NSCH 2. 2 nd law of thermodynamics and classes of fluids 3. steady states 4. spectral analysis
D IFFUSE INTERFACE MODELS Consider the flow of a binary mixture of macroscopically immiscible, viscous Newtonian fluids filling a domain G ⊂ R 3 . In classical models: both fluids are separated by a sharp interface Γ( t ) , across which certain jump conditions are prescribed. Problem: Topological transitions (e.g. due to droplet formation or coalescence) cannot be described. This motivated the development of diffuse interface models: replace the sharp interface by a narrow transition layer across which the fluids may mix.
NSAC: MODELLING ◮ fluid moves with velocity u : J × G → R 3 ◮ phase field (mass fraction) c : J × G → [ 0 , 1 ] corresponding to concentration of one of two phases c 1 δ x ∈ G phase 1 phase 2 interface ◮ different apparent densities ρ 1 = c ρ , ρ 2 = ( 1 − c ) ρ , ρ - total mass density, ρ j satisfy mass balance equation ∂ t ρ j + ∇ · ( ρ j u ) + J j = 0 , J 1 + J 2 = 0 , (1) J j - transition rates
NSAC: MODELLING ◮ fluid moves with velocity u : J × G → R 3 ◮ phase field (mass fraction) c : J × G → [ 0 , 1 ] corresponding to concentration of one of two phases ◮ different apparent densities ρ 1 = c ρ , ρ 2 = ( 1 − c ) ρ , ρ - total mass density, ρ j satisfy mass balance equation ∂ t ρ j + ∇ · ( ρ j u ) + J j = 0 , J 1 + J 2 = 0 , (1) J j - transition rates ◮ conservation of total mass ρ : ∂ t ρ + ∇ · ( ρ u ) = 0 ◮ equation for c : ∂ t ( ρ c ) + ∇ · ( ρ cu ) + J 1 = 0
NSAC: CONSTITUTIVE EQUATION FOR TRANSITION RATE assume that J AC := J 1 = θδ Ψ � ρ δ c , Ψ := θ F dx G F – Helmholtz energy density θ – temperature δ Ψ δ c - generalised chemical potential: originally proposed F = cF 1 + ( 1 − c ) F 2 + θ ( W ( c ) + δ 2 | φ | 2 ) , φ := |∇ c | 2 , (2) F i – Helmholtz energy densities of phase i = 1 , 2 W ( c ) – double-well potential
NSAC: CONSTITUTIVE EQUATION FOR TRANSITION RATE assume that J AC := J 1 = θδ Ψ � ρ δ c , Ψ := θ F dx G F – Helmholtz energy density θ – temperature δ Ψ δ c - generalised chemical potential: originally proposed F = cF 1 + ( 1 − c ) F 2 + θ ( W ( c ) + δ 2 | φ | 2 ) , φ := |∇ c | 2 , (2) F i – Helmholtz energy densities of phase i = 1 , 2 W ( c ) – double-well potential H ELMHOLTZ ENERGY The restriction ( 1 ) is not necessary! Consider Helmholtz energy density φ := |∇ c | 2 , F = F ( ρ, θ, c , φ ) , F smooth enough.
NSAC: CONSTITUTIVE EQUATION FOR TRANSITION RATE ◮ Helmholtz energy density F = F ( ρ, θ, c , φ ) , φ := |∇ c | 2 ◮ computing δ Ψ δ c yields δ Ψ δ c = ∂ c ( ρ θ F ) − ∇ · ( ∂ ∇ c ( ρ θ F )) = ∂ c ( ρ θ F ) − ∇ · ( 2 ∂ φ ( ρ θ F ) ∇ c ) ◮ equation for c (Allen-Cahn equation): � � ∇ · ( 2 ∂ φ ( ρ θ F ) ∇ c ) − ∂ c ( ρ ∂ t ( ρ c ) + ∇ · ( ρ cu ) − θ θ F ) = 0
NSAC: BALANCE OF MOMENTUM ◮ So far we have ∂ t ρ + ∇ · ( ρ u ) = 0 , ∂ t ( ρ c ) + ∇ · ( c ρ u ) + J AC = 0 , with � � − ∇ · ( 2 ∂ φ ( ρ θ F ) ∇ c ) + ∂ c ( ρ J AC = θ θ F ) . ◮ balance of momentum ∂ t ( ρ u ) + ∇ · ( ρ u ⊗ u ) − ∇ · T = ρ f ext T - Cauchy stress f ext - external force
NSAC: CONSTITUTIVE EQUATION FOR T ◮ assume T = S + P S - Newtonian viscous stress, P - pressure tensor ◮ viscous stress: D ( u ) = 1 2 ( ∇ u + ∇ u T ) S = 2 η D ( u ) + λ ∇ · u I , η = η ( ρ, c , θ ) - shear viscosity, λ = λ ( ρ, c , θ ) - bulk viscosity, 2 η + λ > 0 ◮ pressure tensor P : P = − π I − ∇ c ⊗ ∂ ∇ c ( ρ F ) = − π I − 2 ρ∂ φ F ∇ c ⊗ ∇ c π = ρ 2 ∂ ρ F thermodynamic pressure, ∇ c ⊗ ∂ ∇ c ( ρ F ) - Ericksen’s stress represents capillarity PRESSURE TENSOR In order that the 2 nd law of thermodynamics is satisfied the tensor P must have this form.
NSAC: ENERGY EQUATION ∂ t ( ρ E ) + ∇ · ( ρ E u ) − ∇ · ( β ∇ θ + [ S + P ] · u ) = ρ f ext · u , ◮ total energy density E = E + 1 2 | u | 2 ◮ internal energy E = E ( ρ, S , c , |∇ c | 2 ) , S -Entropy Laws of thermodynamics relate E and F through the Legendre transform, E = F + θ S , S = − ∂ θ F , Θ = ∂ S E . ◮ β = β ( ρ, c , θ ) - heat conductivity
NSAC: MATHEMATICAL PROBLEM Let J = [ 0 , T ] and G ⊂ R 3 be a domain (with C 2 boundary Γ ). Consider the Navier-Stokes-Allen-Cahn system for compressible fluids ∂ t ρ + ∇ · ( ρ u ) = 0 , J × G , ∂ t ( ρ u ) + ∇ · ( ρ u ⊗ u ) − ∇ · ( S + P ) = ρ f ext , J × G , (3) ∂ t ( ρ E ) + ∇ · ( ρ E u ) − ∇ · ( β ∇ θ + [ S + P ] · u ) = ρ f ext · u , J × G , ∂ t ( c ρ ) + ∇ · ( c ρ u ) + J AC = 0 , J × G , with initial data ρ ( 0 ) = ρ 0 , u ( 0 ) = u 0 , θ ( 0 ) = θ 0 , c ( 0 ) = c 0 (4) and boundary conditions u = 0 non-slip: ( u | ν ) = 0 , Q ( ν ) S · ν = 0 , pure slip: (5) Q ( ν ) := I − ν ⊗ ν and (inhomogeneous) Dirichlet or Neumann b.c. for c and θ .
NSCH ◮ phase field (mass fraction) c : J × G → [ 0 , 1 ] ◮ fluid moves with velocity u : J × G → R 3 ◮ Helmholtz energy density F = F ( ρ, θ, c , φ ) , φ := |∇ c | 2 ◮ ρ 1 = c ρ , ρ 2 = ( 1 − c ) ρ , ρ - total mass density, ρ j satisfy the conservation law ∂ t ρ j + ∇ · ( ρ j u ) + ∇ · J j = 0 with J 1 + J 2 = 0 ⇒ ∂ t ρ + ∇ · ( ρ u ) = 0 ◮ suppose that J CH := J 1 is given by (Fick’s law) J CH = γ ∇ ( 1 θ µ ) with mobility γ and generalised chemical potential µ , θ µ := δ Ψ � ρ ρ δ c , Ψ = θ F dx Ω
NSCH: MATHEMATICAL PROBLEM Let J = [ 0 , T ] and G ⊂ R 3 be a domain (with C 4 boundary Γ ). Consider the compressible Navier-Stokes-Cahn-Hilliard system ∂ t ρ + ∇ · ( ρ u ) = 0 , J × G , ∂ t ( ρ u ) + ∇ · ( ρ u ⊗ u ) − ∇ · ( S + P ) = ρ f ext , J × G , ∂ t ( ρ E ) + ∇ · ( ρ E u ) − ∇ · ( β ∇ θ + [ S + P ] · u ) = ρ f ext · u , J × G , (6) � 1 � ∂ t ( c ρ ) + ∇ · ( c ρ u ) − ∇ · ( γ ∇ θ µ ) = 0 , J × G , ∂ c ( ρ 2 ∂ φ ( ρ = ρ � � θ F ) − ∇ · θ F ) ∇ c θ µ, J × G , with initial data ρ ( 0 ) = ρ 0 , u ( 0 ) = u 0 , c ( 0 ) = c 0 (7) and boundary conditions u = 0 non-slip: ( u | ν ) = 0 , Q ( ν ) S · ν = 0 , Q := I − ν ⊗ ν pure-slip: (8) ( ∇ µ | ν ) = 0 , ( ∇ c | ν ) = 0 Neumann:
S OME FACTS ◮ type of equations system of hyperbolic-parabolic equations quasilinear partial differential equations, e.g. ρ∂ t u , ∇ · ( 2 ∂ φ ( ρ F ) ∇ c ) quasilinearity of highest order: ∇ · ( 2 ∂ φ ( ρ F ) ∇ c ⊗ ∇ c ) , similar to quasilinear elliptic operators: ∇ · ( a ( ∇ v ) ∇ v ) avoid vacuum, e.g. in ρ∂ t u + ρ ∇ u · u − ∇ · S ( u ) + ∇ · ( π I + 2 ρ∂ φ F ∇ c ⊗ ∇ c ) = ρ f ext ◮ existence and uniqueness local and global well-posedness in 1 D (strong and classical solutions) local well-posedness in any dimension existence of weak solutions (for specific Helmholtz energy and modified Allen-Cahn/Cahn-Hilliard equation) ◮ existence of travelling waves ◮ 2 nd law of thermodynamics ◮ relation to Korteweg
2 nd LAW OF THERMODYNAMICS For any Helmholtz energy F ( ρ, θ, c , |∇ c | 2 ) the transformation rate ( J AC / J CH ) is defined in such a way that both NSAC and NSCH induce an entropy balance ∂ t ( ρ S ) + ∇ · ( ρ Su ) = ∇ · Σ + σ (9) with a net entropy production rate σ ≥ 0 and entropy exchange ∇ · Σ . T HEOREM The thermodynamically closed systems of NSAC and NSCH are thermodynamically and mechanically consistent and ρθ |J AC | 2 , 1 Σ AC = 1 � � σ AC = σ 0 + β ∇ θ + J AC ∂ ∇ c F , θ γ |J CH | 2 , σ CH = σ 0 + 1 Σ CH = 1 � � β ∇ θ + ∇ · J CH ∂ ∇ c F − µ J CH , θ � 2 . σ 0 := 1 � � ∇ θ � θ S : D + β θ
C LASSES OF FLUIDS How can we construct a Helmholtz energy for mixture of fluids? Prototypical approach (for “real applications”): F ( τ, θ, c , φ ) = cF 1 + ( 1 − c ) F 2 + F mix ( θ, c , |∇ c | 2 ) , τ = 1 /ρ, F mix = W ( θ, c ) + δ 2 |∇ c | 2 convex combination of energies F 1 and F 2 of the separate phases F mix - mixing entropy, δ - a measure of thickness for the interface typically: W - double-well potential, e.g. W ( c ) = k 1 [ c ln ( c ) + ( 1 − c ) ln ( 1 − c )] + k 2 c ( 1 − c ) , k 2 < 0 - interactions between different phases are more favorable k 2 > 0 - separation of the phases Question: What are the variables in the equation of state F i ?
C LASSES OF FLUIDS Helmholtz energy: F ( τ, θ, c , φ ) = cF 1 + ( 1 − c ) F 2 + F mix ( θ, c , |∇ c | 2 ) , τ = 1 /ρ, F mix = W ( θ, c ) + δ 2 |∇ c | 2 Question: What are the variables in the equation of state F i ? Answer: As F 1 and F 2 are the Helmholtz energies of the separate phases, the variables are the specific ones, i.e. 1. take a “energy-law” of fluid j = 1 , 2, τ j , ˜ F j = F j (˜ θ j ) with ˜ τ j = 1 / ˜ ρ j denoting the specific volume of phase j ;
Recommend
More recommend
