Local well-posedness of a quasi-incompressible two-phase flow
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Journal of Evolution Equations
Local well-posedness of a quasi-incompressible two-phase flow Helmut Abels
and Josef Weber
Dedicated to Matthias Hieber on the occasion of his 60th birthday Abstract. We show well-posedness of a diffuse interface model for a two-phase flow of two viscous incompressible fluids with different densities locally in time. The model leads to an inhomogeneous Navier– Stokes/Cahn–Hilliard system with a solenoidal velocity field for the mixture, but a variable density of the fluid mixture in the Navier–Stokes type equation. We prove existence of strong solutions locally in time with the aid of a suitable linearization and a contraction mapping argument. To this end, we show maximal L 2 -regularity for the Stokes part of the linearized system and use maximal L p -regularity for the linearized Cahn–Hilliard system.
1. Introduction and main result In this contribution, we study a thermodynamically consistent, diffuse interface model for two-phase flows of two viscous incompressible system with different densities in a bounded domain in two or three space dimensions. The model was derived by Garcke and Grün [6] and leads to the following inhomogeneous Navier–Stokes/Cahn– Hilliard system: ∂t (ρv) + div(ρv ⊗ v) + div v ⊗ ρ˜1 −2 ρ˜2 m(ϕ)∇( 1ε W (ϕ) − εΔϕ) = div(−ε∇ϕ ⊗ ∇ϕ) + div(2η(ϕ)Dv) − ∇q,
(1)
divv = 0,
(2)
∂t ϕ + v · ∇ϕ = div(m(ϕ)∇μ),
(3)
1 μ = −εΔϕ + W (ϕ) ε
(4)
Mathematics Subject Classification: 76T99, 35Q30, 35Q35, 76D03, 76D05, 76D27, 76D45 Keywords: Two-phase flow, Navier–Stokes equation, Diffuse interface model, Mixtures of viscous fluids, Cahn–Hilliard equation. The authors acknowledge support by the SPP 1506 “Transport Processes at Fluidic Interfaces” of the German Science Foundation (DFG) through Grant GA695/6-1 and GA695/6-2. The results are part of the second author’s PhD-thesis [16].
J. Evol. Equ.
H. Abels and J. Weber
in Q T := Ω × (0, T ) together with the initial and boundary values v|∂Ω = ∂n ϕ|∂Ω = ∂n μ|∂Ω = 0 ϕ(0) = ϕ0 , v(0) = v0
on (0, T ) × ∂Ω,
in Ω.
(5) (6)
Here, Ω ⊆ Rd , d = 2, 3, is a bounded domain with C 4 -boundary. In this model, the fluids are assumed to be partly miscible and ϕ : Ω × (0, T ) → R denotes the volume fraction difference of the fluids. v, q, and ρ denote the mean velocity, the pressure and the density of the fluid mixture. It is assumed that the density is a given function of ϕ, more precisely ρ = ρ(ϕ) =
ρ˜2 − ρ˜1 ρ˜1 + ρ˜2 + ϕ 2 2
for all ϕ ∈ R.
where ρ˜1 , ρ˜2 are the specific densities of the (non-mixed) fluids. Moreover, μ is a chemical potential and W (ϕ) is a homogeneous free energy density associated with the fluid mixture, ε > 0 is a constant related to “thickness” of the diffuse interface, which is described by {x ∈ Ω : |ϕ(x, t)| < 1 − δ} for some (small) δ > 0, and m(ϕ) is a mobility coefficient, which controls the strength of the diffusion in the system. Finally, η(ϕ) is a viscosity coefficient and Dv = 21 (∇v + ∇vT ). Existence of weak solution for this system globally in time was shown by Depner and Garcke [4] a
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