Balance laws on omains with moving interfaces. The enthalpy metho for the ice melting problem. Gowin Kakuba 12th March 2008
Outline 1 Balance equations on omains with moving bounaries Introuction Rankine-Hugoniot relations Balance of enthalpy for ice-water melting case, the Stefan conition 2 Weak formulation Existence
Outline 1 Balance equations on omains with moving bounaries Introuction Rankine-Hugoniot relations Balance of enthalpy for ice-water melting case, the Stefan conition 2 Weak formulation Existence
Introuction Motivation for the enthalpy metho: enthalpy escribes the state of the system completely as oppose to temperature Consier a omain Ω(t) moving with a velocity U. On Ω(t) we want to compute C(x, t) x. t Ω(t) C(x, t)x = t Ω(t) Ω(t) C + iv(cu) x
Introuction Motivation for the enthalpy metho: enthalpy escribes the state of the system completely as oppose to temperature Consier a omain Ω(t) moving with a velocity U. On Ω(t) we want to compute C(x, t) x. t Ω(t) C(x, t)x = t Ω(t) Ω(t) C + iv(cu) x
Introuction Motivation for the enthalpy metho: enthalpy escribes the state of the system completely as oppose to temperature Consier a omain Ω(t) moving with a velocity U. On Ω(t) we want to compute C(x, t) x. t Ω(t) C(x, t)x = t Ω(t) Ω(t) C + iv(cu) x
So if we consier a omain with an interface Γ(t) Ω1(t) Ω1(t) Ω2(t) Ω2(t) Γ(t) Ω(t) = Ω1(t) Ω2(t) C(x, t) x = t Ω i (t) C(x, t) x = t Ω(t) Ω i (t) Ω 1 (t) Ω 2 (t) C x + CU nσ + CW n σ Ω i (t) Γ(t) C + iv(cu) x+ [CV ] n σ Γ(t) V := U W is the relative velocity of the flui with respect to Γ(t).
So if we consier a omain with an interface Γ(t) Ω1(t) Ω1(t) Ω2(t) Ω2(t) Γ(t) Ω(t) = Ω1(t) Ω2(t) C(x, t) x = t Ω i (t) C(x, t) x = t Ω(t) Ω i (t) Ω 1 (t) Ω 2 (t) C x + CU nσ + CW n σ Ω i (t) Γ(t) C + iv(cu) x+ [CV ] n σ Γ(t) V := U W is the relative velocity of the flui with respect to Γ(t).
Rankine-Hugoniot relations Suppose t Ω i (t) C x = Ω i (t) f x for i = 1, 2; C + iv(cu) = f in Ω(t) [CV ] n σ = 0 in Γ(t) Γ(t) [CV ] n = 0 on Γ(t) Example: conservation of mass C = ρ ρ 2 v 2 = ρ 1 v 1, v = V n.
Balance of enthalpy C is the enthalpy H; t Ω i (t) H i (x, t)x = Ω i (t) f x + make use of Fourier s law, Ω i (t) Γ(t) q Hi n σ we consier U = 0, an arrive at 8 >: H 9 8 9 (x, t) iv(k T ) f >; x + :[HW ] n [q H n ]; σ = 0 Ω(t) Γ(t) Thus H (x, t) iv(k T ) f = 0 [HW ] n [q H n ] = 0, the Stefan conition [H] = H 2 H 1 = L is the latent heat of melting
Outline 1 Balance equations on omains with moving bounaries Introuction Rankine-Hugoniot relations Balance of enthalpy for ice-water melting case, the Stefan conition 2 Weak formulation Existence
Consier H iv(k T ) = 0. We introuce the function θ as T T m k l (τ) τ, T T m θ = T T m k s (τ) τ, T < T m So we have the fiel equation H θ = 0 in Q s Q l
Definition of the omain: Q T = Ω (0, T ) Q l = Q T {H > L} Q s = Q T {H < 0} Ω l (t) = Q l {t} Ω s(t) = Q s {t} Γ l external bounary of Q l Γ s external bounary of Q s Γ 0 interface in R n+1 Γ(t) = Γ 0 {t} n = normal to Γ(t) pointing towars Ω l (t) H(x, 0) = H 0 (x) Ω l0 Ω s0 T Γl f l (t) > T m, LV n = θ l n + θs θ n T Γs = f s(t) < T m
Definition of the omain: Q T = Ω (0, T ) Q l = Q T {H > L} Q s = Q T {H < 0} Ω l (t) = Q l {t} Ω s(t) = Q s {t} Γ l external bounary of Q l Γ s external bounary of Q s Γ 0 interface in R n+1 Γ(t) = Γ 0 {t} n = normal to Γ(t) pointing towars Ω l (t) H(x, 0) = H 0 (x) Ω l0 Ω s0 T Γl f l (t) > T m, LV n = θ l n + θs θ n T Γs = f s(t) < T m
Weak formulation Introuce the set of test functions V = {ϕ W 2,1 (Q T ) ϕ = 0 on Γ l Γ s Ω {T }} multiply by ϕ an use partial integration In Q s Q s 8 >: H 9 θ>; ϕxt = (H ϕ Q s + + θ ϕ)xt Γ s θ(f s) ϕ n s σt H 0 ϕ(x, 0)x Ω 0s T 0 Γ(t) ϕ θs n σt In Q l Q l 8 >: H 9 θ>; ϕxt = (H ϕ Q l Γ l θ(f l ) ϕ n l σt + θ ϕ)xt T 0 Γ(t) + ϕ H 0 ϕ(x, 0)x+ Ω 0l θl n + VnL σt
Weak formulation Introuce the set of test functions V = {ϕ W 2,1 (Q T ) ϕ = 0 on Γ l Γ s Ω {T }} multiply by ϕ an use partial integration In Q s Q s 8 >: H 9 θ>; ϕxt = (H ϕ Q s + + θ ϕ)xt Γ s θ(f s) ϕ n s σt H 0 ϕ(x, 0)x Ω 0s T 0 Γ(t) ϕ θs n σt In Q l Q l 8 >: H 9 θ>; ϕxt = (H ϕ Q l Γ l θ(f l ) ϕ n l σt + θ ϕ)xt T 0 Γ(t) + ϕ H 0 ϕ(x, 0)x+ Ω 0l θl n + VnL σt
Weak formulation A an use the Stefan conition to get: Q T 8 >:H ϕ 9 + θ ϕ>; xt = θ(f ϕ s) σt+ θ(f l ) ϕ σt H 0 ϕ(x, 0)x Γ s n s Γ l n l Ω 0 (1) Definition A weak solution of the Stefan problem is a function H L (Q T ) satisfying this integral for all ϕ V.
Existence assumptions Suppose the bounary of Ω 0, an consequently Γ s, Γ l, are smooth f s, f l are continuously ifferentiable an boune away from zero in Γ s, Γ l for t = 0, f s an f l match the initial ata an H 0 C 1 ( Ω 0 ) k is twice continuously ifferentiable Uner these assumptions, there is at least one weak solution. Proof. Regularization Consier the problems H n(θ n ) θn θ n = 0 in Q T θ n Γs = θ(f s ), θ n Γl = θ(f l ) θ n (x, 0) = Θ n (H 0 (x))
Existence From classical literature, there is a unique solution to these problems for each n θ n n s Γs, θ n Γl n l are uniformly boune Lemma There is a constant C inepenent of n an etermine by the ata such that ω θ 2 n 1 xτ + Ω {t} 2 θ n 2 x Cx, t (0, T ) Q t Consequently the sequence {θ n } is weakly compact in H 1 (Q T ). there is a subsequence {θ n } converging almost uniformly to θ in Q T
Existence Consier {H n (θ n )}, select a subsequence which is weakly convergent in L 2 (Q T ) to H(x, t) L We may interpret H(x, t) as H(θ(x, t)) a.e. in Q T So, take the weak version: {H n (θ n ) ϕ + θ n ϕ}xt = Q T θ(f s ) ϕ σt + θ(f l ) ϕ σt H 0 ϕ(x)x, ϕ V Γ s n s Γ l n l Ω 0 L Exploit the limits θ 2 n θ, H n (θ n ) L2 H an the properties of H to conclue the existence of the weak solution. (2)
Summary R. Teaman & A. Miranville, Mathematical Moeling in Continuum Mechanics, Cambrige University Press, 2000 Antonio Fasano, Mathematical Moels of some Diffusive Processes with Free Bounaries S.J.L. van Einhoven, Mathematical Moels base on free bounary problems