für Mathematik in den Naturwissenschaften Leipzig

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1 ŠܹÈÐ Ò ¹ÁÒ Ø ØÙØ für Mathematik in den Naturwissenschaften Leipzig Diffuse Interface Models for Two-Phase Flows of Viscous Incompressible Fluids (revised version: February 008) by Helmut Abels Lecture note no.:

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3 Diffuse Interface Models for Two-Phase Flows of Viscous Incompressible Fluids Helmut Abels February 1, 008

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5 Contents 1 Introduction Classical Sharp Interface Model Diffuse Interface Model for Matched Densities Diffuse Interface Model for General Densities Overview of the Analytical Results Derivation of the Diffuse Interface Model 15.1 Fundamental Quantities and Equations Local Dissipation Inequality Reformulation of the System Simple Mixtures and Consequences Mathematical Background Notation Some Results from Functional Analysis Interpolation Spaces Function Spaces on Domains Some Inequalities and Product Estimates Neumann Laplace Equation and Helmholtz Decomposition Vector-Valued Function Spaces and Real Interpolation Some Results Related to Weak and Strong Convergence Linear Evolution Equation Monotone Operators and Subgradients Evolution Equations for Subgradients Subgradients Related to the Free Energy Cahn-Hilliard Equation with Convection Existence and Regularity Theory Double Obstacle Limit Lojasiewicz-Simon Inequality Navier-Stokes System with Variable Viscosity Weak Solutions of the Non-Stationary Stokes System

6 CONTENTS i 5. Stationary Stokes System and the Stokes Operator Maximal Regularity for the Instationary Stokes System Strong Solutions to the Navier-Stokes System Matched Densities Main Results Existence of Weak Solutions Uniqueness and Regularity of Weak Solutions Asymptotic Behavior in Time Double Obstacle Limit General Densities Introduction and Overview Weak Solutions and the Main Results Approximate System and Implicit Time Discretization Existence of Weak Solutions for the General System Weak Solutions for a Quasi-Stationary Approximation

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8 Chapter 1 Introduction In the present contribution we study the flow of two incompressible, viscous and immiscible Newtonian fluids like oil and water inside a bounded domain R d, d =, 3. In classical models the interface between both fluids is assumed to be a (d 1)-dimensional sufficiently smooth surface and capillary phenomena are related to contact angle conditions and a jump condition for the stress tensor across the interface. But the classical description fails when several parts of the interface merge or reconnect (develop singularities) due droplet formation or coalescence of several droplets. In order to describe flows beyond these kind of singularities, one needs either a suitable weak formulation of the classical model or a suitable description by alternative models. The present work is devoted to the mathematical analysis of one class of alternative models. The models, we are considering, are so-called diffuse interface models where the classical sharp interface, described by a lower dimensional surface, is replaced by a thin interfacial region of finite thickness, measured by a small parameter ε > 0, where a partial mixing of the macroscopically immiscible fluids are taken into account. Moreover, diffusional effects are included in the model. Before describing the diffuse interface models, we will shortly discuss the classical sharp interface model in order to see some analogies when presenting the diffuse interface models. Moreover, we will give some comments on known results and difficulties concerning the construction of weak solutions globally in time for the classical model. 1.1 Classical Sharp Interface Model In the classical model the fluids fill two disjoint domains + (t), (t) for all times t 0, and are separated by a (d 1)-dimensional surface Γ(t) = ± (t). Hence = + (t) (t) Γ(t). The flow is described using the velocity field v: (0, ) R d and the pressure p: (0, ) R in both fluids in Eulerian coordinates. For simplicity we assume that the densities of both fluids are the same and equal to one. Since we assume that the fluids are of Newtonian type, the stress tensors are of the form T ± (v, p) = ν ± Dv pi with viscosity constants ν ± > 0 and 1

9 CHAPTER 1. INTRODUCTION Dv = 1 ( v + vt ). Moreover, we assume for simplicity that the interface does not intersect with the boundary of the domain. We consider the cases with and without surface tension at the interface. Under suitable smoothness assumptions, the flow is obtained as solution of the system t v + v v div T ± (v, p) = 0 in ± (t), t > 0, (1.1.1) div v = 0 in ± (t), t > 0, (1.1.) n T + (v, p) n T (v, p) = κhn on Γ(t), t > 0, (1.1.3) V = n v on Γ(t), t > 0, (1.1.4) v = 0 on, t > 0, (1.1.5) v t=0 = v 0 in, (1.1.6) together with an initial condition + (0) = + 0. Here V and H denote the normal velocity and mean curvature of Γ(t), resp., taken with respect to the exterior normal n of + (t), and κ 0 is the surface tension constant (κ = 0 means no surface tension present). Equations (1.1.1)-(1.1.) describe the conservation of linear momentum and mass in both fluids, (1.1.3) is the so-called Young-Laplace law, which describes the balance of the inner forces of the fluids and the capillary forces at the interface, (1.1.4) is the kinematic condition that the interface is transported with the flow of the mass particles, and (1.1.5) is the no-slip condition at the boundary of. Moreover, it is assumed that the velocity field v is continuous along the interface. Most publications on the mathematical analysis of free boundary value problems for viscous incompressible fluids study quite regular solutions and often deal with well-posedness locally in time or global existence close to equilibrium states, cf. e.g. Solonnikov [67, 68], Beale [10, 11], Denisova and Solonnikov [], Tani and Tanaka [73], Shibata and Shimizu [61] or Abels []. These approaches are a priori limited to flows, in which the interface does not develop singularities and the domain filled by the fluid does not change its topology. There are only few works on global existence of weak/generalized solutions, which will be commented later. In order to derive a suitable weak formulation of the system above, we multiply (1.1.1) with a divergence free vector field ϕ, integrate in space and use in particular the jump relation (1.1.4). Then we obtain (v, t ϕ) Q (v 0, ϕ t=0 ) + (v v, ϕ) Q +(ν(χ)dv, Dϕ) Q = κ HΓ(t), ϕ(t) dt (1.1.7) for all ϕ C(0) ( [0, ))d with div ϕ = 0, where (.,.) M denotes the L (M)-inner product, Q = (0, ), χ = χ +, ν(1) = ν +, ν(0) = ν, and HΓ(t), ϕ(t) := Hn ϕ(x, t) dh d 1 (x). (1.1.8) Γ(t) Here H d 1 denotes the (d 1)-dimensional Hausdorff measure and χ A denotes the characteristic function of a set A. 0

10 1.1. CLASSICAL SHARP INTERFACE MODEL 3 The aim would be to construct generalized solutions in a class of functions determined by the energy estimate. If v and Γ(t) are sufficiently smooth, then, choosing ϕ = vχ [0,T] in (1.1.7), one obtains the energy inequality 1 v(t) L () + κhd 1 (Γ(T)) T + S(χ, Dv) : Dv dxdt 1 v 0 L () + κhd 1 (Γ 0 ) (1.1.9) 0 for all T > 0 (even with equality), where Γ 0 = + 0. Note that d dt Hd 1 (Γ(t)) = HV dh d 1 = H Γ(t), v(t) (1.1.10) Γ(t) due to (1.1.4), cf. [34, Equation 10.1]. Hence the equality above gives a uniform bound of v L (0, ; L σ ()) and Dv L ((0, ) ), (1.1.11) which is the usual a priori estimate for the Navier-Stokes equation. Here L p (M), 1 p, denotes the usual Lebesgue space and L p (M; X) its vector-valued variant for a given Banach space X. Moreover, we note that, if E is a set of finite perimeter, then χ E BV () and χ E M() = H d 1 ( E), where E is the reduced boundary of E, cf. Evans and Gariepy [30, Chapter 5]. Here BV () = {f L 1 () : f M()} is the space of functions with bounded variation and M() = C 0 () is the space of Radon measures on a domain with finite total variation. In particular, if Γ(t) = + (t) is a compact C 1 -surface, then + (t) = + (t) and χ + (t) M() = H d 1 (Γ(t)), which implies that in the presence of surface tension (σ > 0) χ L (0, ; BV ()) due to (1.1.9) and since χ(t) L 1 () = + (t) = + 0 is constant. Moreover, we note that, if χ = χ E BV (), then H χ, ϕ = (I n n) : ϕ dh d 1 E ( = I χ χ χ ) : ϕ d χ (1.1.1) χ is a natural definition of the mean curvature functional, which coincides with H Γ(t) defined above if E = + (t) and + (t) is a sufficiently smooth surface. Here χ(t)

11 4 CHAPTER 1. INTRODUCTION is the total variation measure of χ(t) and χ the Radon-Nikodym derivative of χ χ(t) with respect to χ. Moreover, n = χ is the exterior normal at the χ reduced boundary E(t) in a measure theoretic sense, cf. [30, Section 5.8]. In the case without surface tension (κ = 0) we only obtain that χ L (Q) is a priori bounded by one. This motivates to look for weak solutions (v, χ) lying in the function spaces above, satisfying (1.1.9) with a suitable substitute of (1.1.8), such that (v, χ) solve (1.1.7) as well as the transport equation t χ + v χ = 0 in Q, (1.1.13) χ t=0 = χ 0 in (1.1.14) for χ 0 = χ + in a suitable weak sense. Note that (1.1.13) is a weak formulation of 0 (1.1.4), cf. [54, Lemma 1.]. In the case κ = 0, Nouri and Poupaud [54] proved existence of weak solutions in the sense above. Their result is based on the results of DiPerna and Lions [6] on renormalized solutions of the transport equation (1.1.13)-(1.1.14) for a velocity field v with bounded divergence. In the paper even a multi-phase flow and general densities are treated. But unfortunately little can be said about the interface Γ(t) since + (t) = {x : χ(x, t) = 1} is merely a measurable set. Some improvements are due to Desjardins [4] in the case of a two-dimensional flow if ν + ν is sufficiently small. In the case of a Stokes flow, i.e., neglecting the convective term v v, there is a result by Giga and Takahashi [33], who solved (1.1.13)-(1.1.14) in the sense of viscosity solutions, where the characteristic functions (χ(t), χ 0 ) are replaced by continuous level-set functions (ψ(t), ψ 0 ) such that ± 0 = {x : ψ 0(x) 0}. But in their work it cannot be excluded that there is a fattening of the boundary in the sense that Γ(t) has positive Lebesgue measure. This comes from the fact that they can only show existence of sub- and super-solutions of the transport equation in the viscosity sense, which might not coincide. We refer to [4] for further discussions. Although the inclusion of surface tension effects, i.e., κ > 0, leads to a better a priori estimate, it also leads to an additional non-linearity in the system namely the surface tension functional H Γ(t), which quadratic in the normal of the interface. In order to construct weak solutions by a suitable approximation scheme, one needs to exclude certain oscillation and concentration effects related to the interface Γ(t). But unfortunately the a priori estimates together with the known techniques from geometric measure theory seem not to be sufficient for the construction of weak solutions with a satisfactory interpretation of the mean curvature term. So far, only some varifold solutions have been constructed, which was first done by Plotnikov [55] in the case of non-newtonian shear thickening fluids in two dimensions and later by the author [3] for a more general class of Newtonian and non-newtonian fluids in two

12 1.. DIFFUSE INTERFACE MODEL FOR MATCHED DENSITIES 5 and three dimensions. In the latter works the mean curvature functional is given by the first variation of a so-called general varifold, which encodes possible oscillation and concentration effects. It was shown in [3] that any varifold solution is a weak solution in the sense above if there is no loss of energy, i.e., the energy estimate (1.1.9) holds for all T > 0 with equality. But of course this condition might not be satisfied in general. We refer to the overview article [4] for a more detailed discussion and further references. Summing up, we have seen that it is possible to define weak solutions of the two-phase flow described by the classical sharp interface model in a quite natural way. But in the case of surface tension existence of weak solutions is open and good regularity properties of the interface are difficult to get in both cases (with and without surface tension). Moreover, it is questionable how physically meaningful weak solution of the sharp interface model are since the Young-Laplace equation (1.1.3) is derived in situations which are sufficiently close to equilibrium. Moreover, during droplet formation or droplet coalescence there might be physical effects which are important for the dynamics and which are not taken into account in the classical sharp interface model. One attempt to describe two-phase flows with droplet formation and coalescence are the diffuse interface models presented in the next sections. 1. Diffuse Interface Model for Matched Densities An old idea, which goes even back to e.g. van der Waals [59], is that the sharp interface between two (macroscopically) immiscible fluids is an idealization and there is a partial mixing in a small interfacial region. Although this region is quite thin, it might play an important role during coalescence of droplets or other topological singularities. Therefore one introduces an order parameter which will be the concentration difference of both fluids c, which is close to 1 in regions mainly filled by one fluid, close to 1 in regions mainly filled by the other fluid and varies continuously between +1 and 1 in the interfacial region. In order to ensure this behavior of the concentration difference c, one introduces a free energy E free (c) = 1 ε c(x) dx + ε 1 Φ(c(x)) dx, (1..1) where ε > 0 is a small parameter related to the interfacial thickness and Φ: R R is a suitable double well potential, meaning that Φ has exactly two local minima at c = ±1 (or c ±1) in our case, where the global minimum Φ(c) = 0 is attained. A typical example is Φ(s) = (1 s ). We will discuss the choice of Φ, which is called homogeneous free energy density, below in more detail. Moreover, we refer to Anderson et al. [9] for a review article on diffuse interface models. Before we introduce the complete diffuse interface model, let us give some comments on the free energy E free (c). This kind of energy is well-known and is fundamental in the so-called Van der Waals-Cahn-Hilliard theory of phase separations. We

13 6 CHAPTER 1. INTRODUCTION note that, if ε > 0 is small, then qualitatively E free (c) R implies that c(x) has to be sufficiently close to the minima of Φ(s) since ε 1 Φ(s) will be large outside a small neighborhood of the minima. On the other hand, since ε c(x) dx < and the mean value of c will be a prescribed number in ( 1, 1), c(x) cannot coincide with the minima ±1 of Φ(s) in all of. Therefore there has to be a region where c(x) varies between 1 and 1, which means that the set Γ δ = {x : c(x) ( 1 + δ, 1 + δ)}, which is the diffuse interface, will be non-empty if ε > 0 is sufficiently small in dependence of R, δ > 0, and m := 1 c(x) dx ( 1, 1). In these diffuse interface models E free (c) replaces the interfacial energy κh d 1 (Γ(t)) in the sharp interface model above. We note that the famous results due to Modica and Mortola [5, 51] shows that E free (c) together with the constraint 1 c(x) dx = m converges to the functional E(χ) = { σ χ M() 1 if χ BV (), χ(x) {+1, 1} a.e., + else in the sense of Γ-convergence in L 1, where σ = 1 1 Φ(s)1 ds. (See in particular [51, Section 3].) A diffuse interface model based on the latter choice of the free energy for two fluids with the same density first appeared in Hohenberg and Halperin [40] under the name Model H. The model was derived in an alternative way based on the concept of micro-forces and in the framework of rational continuum mechanics by Gurtin et al. [37]. The model leads to a Navier-Stokes/Cahn-Hilliard type system t v + v v div(ν(c)dv) + p = ε div( c c) in (0, ), (1..) div v = 0 in (0, ), (1..3) t c + v c = m µ in (0, ), (1..4) µ = ε 1 φ(c) ε c in (0, ). (1..5) Here v is the mean velocity, Dv = 1 ( v + vt ), p is the pressure, c is related to the concentration of the fluids (e.g. the concentration difference as before or the concentration of one component), µ is the so-called chemical potential, and is a suitable bounded domain. Moreover, ν(c) > 0 is the viscosity of the mixture, ε > 0 is the same parameter as above, which will be related to the thickness of the interfacial region, and φ = Φ for some suitable free energy density Φ. It is assumed that the densities of both components as well as the density of the mixture are the same and for simplicity equal to one. We note that capillary forces due to surface tension are modeled by an extra contribution ε c c in the stress tensor leading to the term on the right-hand side of (1..). Let us note the following analogy with

14 1.. DIFFUSE INTERFACE MODEL FOR MATCHED DENSITIES 7 the mean curvature functional for the sharp interface model in the weak formulation (1.1.1): If ϕ C0 () d with div ϕ = 0, then ε div( c c) : ϕ dx = ε c c : ϕ dx ( = I c c c ) : ϕ ε c dx. c Hence c c corresponds to the (interior) normal n = χ χ and ε c dx corresponds to the surface measure κ χ. But, besides the fact that the sharp interface is replaced by an interfacial region, there is a second fundamental difference with the sharp interface model, namely: The transport equation (1.1.4), (1.1.13) resp., is replaced by a diffusion equation with convection (1..4)-(1..5), where m > 0 is a mobility coefficient, which is assumed to be constant for simplicity. This means that diffusion of the fluid components relative to each other is taken into account. The case m = 0 corresponds to a pure transport of the components without diffusion. But in this case existence of weak solutions to (1..)-(1..5) is open as in the case of the sharp interface model. The reasons are comparable since the a priori estimates in this case are not enough to ensure enough compactness of approximating sequence to be able to pass to the limit in the non-linear term c c, which corresponds to the difficulties one has to pass to the limit in the mean curvature functional (1.1.1). In the case with diffusion m > 0 there is an additional a priori estimate of µ as will be seen below, which provides additional regularity of c due to (1..5). Moreover, in this case the system (1..)-(1..5) is purely parabolic, in contrast to the sharp interface model which is parabolic-hyperbolic. We close the system by adding the boundary and initial conditions v = 0 on (0, ), (1..6) n c = n µ = 0 on (0, ), (1..7) (v, c) t=0 = (v 0, c 0 ) in (1..8) Here (1..6) is the usual no-slip boundary condition for viscous fluids, n is the exterior normal on, n µ = 0 means that there is no mass flux through the boundary, and n c = 0 describes a contact angle of π/ of the diffused interface and the boundary of the domain. Besides the no-slip boundary condition (1..6), we will also consider the case of Navier boundary conditions n v = (n ν(c)dv) τ + γ(c)v τ = 0, (1..9) where the second equation describes the balance of the tangential part of the inner forces acting on the boundary with a friction force γ(c)v τ due to the tangential motion of the fluids along the boundary, where γ(c) > 0 is a suitable friction constant depending on c. We note that one motivation for considering Navier boundary conditions besides the no-slip boundary condition is the fact that the no-slip boundary

15 8 CHAPTER 1. INTRODUCTION condition does not permit the motion of a contact line along the boundary in the sharp interface model with finite stress, cf. e.g. Hocking [38, 39] and the discussion and references in Schweizer [60]. Moreover, assuming Navier boundary conditions leads to several better mathematical properties, which will be essential for the analysis in the case on non-matched densities. The total energy of the system above is given by E(c, v) = E free (c) + E kin (v), where E kin (v) = 1 v(x) dx (1..10) is the kinetic energy of the fluid. For every sufficiently smooth solution of (1..)- (1..8) d E(c(t), v(t)) = ν(c(t)) Dv(t) dx m µ(t) dx dt holds and therefore t t E(c(t), v(t)) + ν(c(τ)) Dv(τ) dxdτ + m µ(t) dxdτ E(c 0, v 0 ) 0 for all t > 0, cf. Section.3 below. Hence, if m > 0, there is an additional L -estimate of µ, which has no corresponding term in the energy estimate for the sharp interface model (1.1.9) and is related to the additional diffusion taken into account. For the system (1..)-(1..8) it is possible to obtain analytical results on existence of weak solutions, regularity of weak solutions, uniqueness and existence of strong solutions, which are comparable with the results of a single viscous, incompressible Newtonian fluid. In particular, if the initial data (v 0, c 0 ) are sufficiently regular, there is a unique strong solution globally in time in the case d =. Moreover, if d = 3, there are weak solutions globally in time (not necessarily unique), which are regular for small and large times. Furthermore, using the regularity for large times, it is possible to show convergence to stationary solutions as t. The results are presented in Section 6.1 in more detail. Finally, we give some comments on the choice of the homogeneous free density, which is also essential for our analysis. In the original derivation of the Cahn-Hilliard equation, i.e., (1..4)-(1..5) with v 0, Cahn and Hilliard [19] suggested the following choice of the free energy: 0 Φ(c) = θ ((1 + c) ln(1 + c) + (1 c) ln(1 c)) θ c c, (1..11) where the logarithmic terms are related to the entropy of the system, θ > 0 is related to the absolute temperature and the quadratic term represents the energy needed to mix the fluids, where θ c > 0. We note that f is convex if and only if θ θ c. In this case the mixed phase is stable. On the other hand, if 0 < θ < θ c, the mixed phase is unstable and phase separation occurs.

16 1.. DIFFUSE INTERFACE MODEL FOR MATCHED DENSITIES 9 Because of the logarithmic terms, the free energy density (1..11) is often approximated by a suitable double well potential like Φ(c) = (1 c ). This simplifies the mathematical analysis a lot. But, since (1..4)-(1..5) is a parabolic equation of fourth order, in general c(t, x) will not be in the physically reasonable interval [ 1, 1] for all (x, t) due to a lack of a maximum principle. Interestingly, using the logarithmic free energy density (1..11), it is possible to construct unique weak solutions of the Cahn-Hilliard system (1..4)-(1..5) (with v = 0) such that c(t, x) ( 1, 1) for almost all t, x. This was first proven by Elliott and Luckhaus [9]. Alternative proofs and further discussions of the Cahn-Hilliard system with the latter free energy density can be found in Debussche and Dettori [0], Dupaix [8] and Miranville and Zelik [50], Kenmochi and Pawlow [4], and Abels and Wilke [6]. Motivated by the properties of the free energy density (1..11), we will assume that the free energy density Φ satisfies the following assumptions: Assumption 1..1 Let Φ: [a, b] R be a continuous functions, which is continuously differentiable in (a, b) such that φ = Φ satisfies for some κ R. lim φ(s) =, lim s a+ φ(s) =, s b φ (s) κ It is easy to check that Φ as in (1..11) satisfies the latter assumption with [a, b] = [ 1, 1]. We note that every Φ satisfying the latter assumption has a decomposition Φ(s) = Φ 0 (s) κ c, φ(s) = φ 0 (s) κc (1..1) for some κ R, where Φ 0 C([a, b]) C ((a, b)) is convex. This will be the key point in the following analysis of the Cahn-Hilliard equation (1..4)-(1..5). The condition lim c a+ φ 0 (c) =, lim c b φ 0 (c) = for φ 0 = Φ 0 will keep the concentration difference c in the (physical reasonable) interval [a, b] and ensures that the subgradient of the associated functional is a single valued function with a suitable domain, cf. Theorem 3.1. below. In Figure 1.1 the free energy density is plotted for some choices of θ, θ c. Note that, if θ > 0, then the minima are never ±1. (Although Figure 1.1 for θ = 0. suggests that the minima are ±1.) But the minima are close to ±1 if θ is small in comparison with θ c. Since for two macroscopically immiscible fluids mixing costs a lot of energy, we think that a small θ in comparison with θ c is physically the most meaningful choice. Since qualitatively the free energy density for θ = 0. in Figure 1.1 looks already the same as for θ = 0, it is very reasonable to choose θ = 0 directly, i.e., to choose the free energy density { θc Φ(c) = c if c [ 1, 1] + else.

17 10 CHAPTER 1. INTRODUCTION Figure 1.1: Logarithmic free energy density for θ = 0.8, 0.7, 0.6, 0.5, 0.4, 0., 0.0 (from top to bottom) and θ c = 1 The free energy density is called to be of double obstacle type because of the constraint c [ 1, 1] for all c with Φ(c) <. The Cahn-Hilliard equation with the latter free energy was first studied by Blowey and Elliott [13]. In [9] it is also shown that as θ 0 the solutions of the Cahn-Hilliard equation with the logarithmic free energy density converge to solutions of the latter free energy of double obstacle type. For our diffuse interface model we can show a similar convergence result as θ 0 and obtain existence of weak solutions also in the limit case θ = 0 in that manner, cf. Section 6.5 for details. 1.3 Diffuse Interface Model for General Densities The derivation of the system (1..)-(1..8) is essentially based on the assumption that both fluids have the same densities. For the general case Lowengrub and Truskinovsky [48] derived the following extension of the Model H above: ρ t v + ρv v div S(c, Dv) + p = div(ρ c c) (1.3.1) t ρ + div(ρv) = 0 (1.3.) ρ t c + ρv c = div(m(c) µ) (1.3.3) µ = ρ ρ c p + Φ(c) + ρφ(c) 1 div(ρ c), (1.3.4) ρ

18 1.3. DIFFUSE INTERFACE MODEL FOR GENERAL DENSITIES 11 where ρ = ˆρ(c) for some given function ˆρ: [ 1, 1] (0, ) the so-called viscous part of the stress tensor is given by S(c, Dv) = ν(c)dv + η(c) div v with an additional viscosity coefficient η(c) related to the bulk viscosity of the mixture. We close the system with the initial and boundary condition (1..7)-(1..9). In particular, we will only deal with the case of Navier boundary conditions, which will be essential for getting good estimates of the pressure in our analysis for this model. The total energy of the system is given by E(c, v) = E free (c) + E kin (c, v) with E free (c) = E kin (c, v) = ρ c ρ v dx dx + and, similarly as in the case of matched densities, d E(c(t), v(t)) = dt ( ν(c(t)) Dv(t) + η(c(t)) divv(t) ) dx ρφ(c) dx γ(c) v τ dσ m µ(t) dx, which leads to the same kind of energy estimate as before, cf. Section.3 below. Here the extra boundary integral comes from the fact that we use Navier boundary conditions instead of no-slip boundary conditions. Obviously, if ˆρ 1, then (1.3.1)-(1.3.4) reduces to (1..)-(1..5) (with ε = 1). But, if ˆρ(c) const., then div v 0 and the pressure p enters the equation for the chemical potential (1.3.4). This has severe consequences for the mathematical analysis since it is no longer possible to work in function spaces of divergence free vector fields and to use the Helmholtz projection to get rid of the pressure. Moreover, (1.3.4) no longer implies good a priori estimates for the regularity of the concentration c since the pressure p has low regularity. Because of the presence of p in (1.3.4), we are not able to use the logarithmic free energy (1..11) or a general singular free energy density as in Assumption In particular, we cannot use these kind of free energies to ensure that the concentration difference c stays in the interval [ 1, 1] for almost all (t, x). This causes severe mathematical problems since in the case of a so-called simple mixture the density ρ = ˆρ(c) is given by 1 ˆρ(c) = c c. ρ 1 ρ Hence ˆρ(c) is a rational functions, which becomes negative and singular outside of [ 1, 1]. As will be seen in Section.4, this is not only a physical natural assumption. It also leads to some mathematical simplifications which will be essential in the

19 1 CHAPTER 1. INTRODUCTION mathematical analysis for this model done in Chapter 7. Hence one needs a different mechanism to ensure that the concentration stays in [ 1, 1] or at least in a suitable neighborhood [ 1 ε, 1 + ε]. In order to get such a mechanism, we modify the gradient term in the free energy E free (c) and consider instead E free (c) = a(c) c q q dx + Φ(c) dx for some q > d =, 3 and a suitable 0 < a a(c) a. Since the gradient term c in the Cahn-Hilliard type free energy is based on phenomenological considerations and plays the role of a regularizing term to ensure a sufficiently smooth variant of the concentration, we believe that there is no fundamental reason not to replace it by c q. In Section we will derive a variant of Lowengrub and Truskinovsky s model based on the modified free energy above. 1 We expect qualitatively the same behavior for the solutions of the modified model. Mathematically, the modified free energy implies that c L (0, ; Wq 1 ()) with q > d. Therefore c(t) will be a Hölder continuous function for almost all t > 0. This can be used to ensure that c(t, x) [ 1 ε, 1 + ε] for all x and almost all t > 0 provided one chooses the free energy density Φ(s) sufficiently steep outside of [ 1, 1] (in dependence on the size of the initial data). This is explained in detail in Section 7.1. Finally, we note that the main mathematical difficulties, which prevent from using the free energy density (1..11), are due to a part of the pressure p related to the term ρ t v in (1.3.1). Therefore, if one considers a creeping flow and neglects the terms ρ t v + ρv v in (1.3.1), then it is possible to construct weak solutions with singular free energies as in Assumption 1..1, cf. Section 7.5. We note that this quasi-stationary approximation corresponds to a limit where the Reynolds number tends to zero. 1.4 Overview of the Analytical Results There are only few results on the mathematical analysis of diffuse interface models in fluid mechanics and the diffuse interface model with matched densities presented above. First results on existence of strong solutions, if = R and Φ is a suitably smooth double well potential were obtained by Starovoitov [70]. More complete results were presented by Boyer [15] in the case that R d is a periodical channel and f is a suitably smooth double well potential. The author showed the existence of global weak solutions, which are strong and unique if either d = or d = 3 and t (0, T 0 ) for a sufficiently small T 0 > 0. Moreover, the case of the physical relevant logarithmic potential (1..11) presented below is also considered in connection with a degenerate mobility m = m(c) 0 suitably as c ±1. In this case existence of weak solutions with c(t, x) [ 1, 1] is shown. The system (6.1.1)-(6.1.4) was also briefly 1 Actually, part of the derivation in [48] is done for a general free energy density E free (c) = f(c, c)dx. But we will give an alternative derivation based on considerations similarly to [37].

20 1.4. OVERVIEW OF THE ANALYTICAL RESULTS 13 discussed by Liu and Shen [46]. As far as the authors knows there are no analytical results on existence of solutions to the diffuse interface model in the case of general densities as proposed by Lowengrub et al. [48]. A different diffuse interface model for fluids with non-matched densities was studied by Boyer [16], where short existence of strong solutions and global in time existence of weak solutions for sufficiently small variations of the density was shown. Finally, we mention that in Abels and Feireisl [5] a system similar to (1.3.1)-(1.3.4) for compressible fluids is studied and existence of weak solutions is proven. For the diffuse interface model in the case of matched densities we present a more complete mathematical theory on existence, uniqueness, regularity of solutions to (1..)-(1..8) and asymptotic behavior as t. Qualitatively, our results are similar to the known result on the un-coupled Navier-Stokes system, cf. e.g. Sohr [66]. Of course the results are limited by the fact that the regularity and uniqueness of weak solution of the non-stationary Navier-Stokes system in three space dimensions is an unsolved problem. Moreover, it is the purpose of this work to present a theory for a class of physically relevant free energy densities Φ, including the logarithmic free energy in (1..11), which keep the concentration in the physically reasonable range, cf. Section 6.1 below. We note that the latter results are extensions of the authors results [1] (unpublished), where only the case of no-slip boundary conditions is considered. Finally, we also study the double obstacle limit θ 0 as mentioned at the end of Section 1.. For the diffuse interface model with non-matched densities due to Lowengrub and Truskinovsky our goals have been the same. But, because of the much stronger coupling of the system, already the existence of weak solutions was much more difficult, cf. Section 7.1 below. We have shown existence of weak solutions for a suitable variant of (1.3.1)-(1.3.4) in the case of a sufficiently smooth free energy density such that the concentration is in [ 1 ε, 1 + ε] with ε > 0 arbitrary but fixed. Moreover, we constructed weak solutions for a quasi-stationary variant of (1.3.1)-(1.3.4) with singular free energy densities as in Assumption We refer to Section 7.1 for a more detailed overview. The structure of the work is as follows: In Chapter we give a brief derivation of the class of diffuse interface models, we are considering in this contribution. In particular we discuss the local dissipation inequality, which substitutes the second law of thermodynamics. Moreover, we present the energy estimates, discuss conserved quantities, and reformulate the system suitably. Finally, we consider the case of a simple mixtures, which leads to an important simplification of the system. Chapter 3 is devoted to a summary of the mathematical background needed throughout the work. In particular, we discuss the used function spaces and recall some result from the theory of monotone operators. Moreover, we derive some important characterizations of the subgradients of the energy functional we are using. The analysis of the diffuse interface model with matched densities is done in Chapter 6. It is based on the results of the Chapter 4, where the Cahn-Hilliard system with convection term, i.e., (1..4)-(1..5) for given v, is studied, and on the results of Chapter 5, where the

21 14 CHAPTER 1. INTRODUCTION associated Stokes and Navier-Stokes system with variable viscosity ν(c) is studied. The existence weak solutions for the diffuse interface models with general densities is done in Chapter 7. The Sections 6.1 and 7.1 give an overview of the main analytical results of this work in the case of matched and general densities, respectively. Acknowledgments: The author expresses his gratitude for the excellent working conditions and nice atmosphere he found during the last three years at the Max Planck Institute for Mathematics in the Sciences. In particular, he wants to thank Prof. Stefan Müller for the support during the last years, for creating such an inspiring scientific environment in the working group, and for many interesting lectures and discussions. Moreover, the author is grateful to Prof. Stephan Luckhaus, Prof. Eduard Feireisl, Prof. Yoshikazu Giga, Prof. Richard D. James, Prof. Pavel Plotnikov, Dr. Matthias Röger, and Dr. Mathias Wilke for many interesting discussions on topics related to the thesis. Thanks are also due to all the other interesting scientific discussions with numerous people through the last years, from which I learned a lot. In particular, the author wants to thank Dr. Moritz Kassmann, Dr. Katrin Schumacher, Dr. Anja Schlömerkemper, Dr. Yutaka Terasawa, and everybody who might be forgotten. Special thanks also belong to the colleagues who helped me to improve the manuscript which are Matthias Röger, Anja Schlömerkemper, Katrin Schumacher, and Mathias Wilke. Furthermore, the author is very grateful for the efficient and helpful administrative staff of the research institute, who help a lot to concentrate on the scientific research in everyday life. Finally, I am deeply indebted to my wife Lydia for all the patience and support through the years and especially during the last months.

22 Chapter Derivation of the Diffuse Interface Model.1 Fundamental Quantities and Equations In order to derive the diffuse interface model, one assumes a partial mixing of the macroscopically immiscible fluids, labeled by j = 1,, filling a domain R 3. Hence we consider the mixture of two fluids. The total mass density of the mixture is denoted by ρ. Moreover, ρ j denotes the mass density of the fluid j, i.e., M j = ρ j (x) dx V is the mass of the fluid j contained in a set V. Obviously ρ = ρ 1 +ρ. Moreover, we denote by c j = ρ j the mass concentration, where we note that 1 = c ρ 1 + c. As in Gurtin et al. [37], we assume that the inertia and kinetic energy due to the motion of the fluid relative to the gross motion is negligible. Therefore we consider the mixture as a single fluid which satisfies the laws of conservation of mass, linear, and angular momentum of continuum mechanics with density and stress tensor depending on some additional internal variables like c and c. 1 Therefore we denote by v the velocity of the mixture considered as a single fluid. (Note that v could also be defined as mass averaged velocity by ρv = ρ 1 v 1 +ρ v, cf. Lowengrub and Truskinovsky [48], where v j denotes the velocity of the fluid j and the fluids are considered as co-existing continua.) We assume conservation of mass, linear and angular momentum with respect to the mean velocity, i.e., we assume that ρ t v + ρv v = div T (.1.1) ρ t + div(ρv) = 0 (.1.) 1 Note that this approach is different from continuum theories of mixtures as presented in Ragagopal and Tao [56], where first the fluids are considered as co-existing continua and then equations for the mean quantities are derived. 15

23 16 CHAPTER. DERIVATION OF THE DIFFUSE INTERFACE MODEL for a symmetric stress tensor T describing the material behavior of the mixture, which has to be specified by constitutive assumptions. Here exterior forces are neglected for simplicity. Furthermore, we denote by h j the mass flux of the fluid j relative to the mean velocity v, i.e., t ρ j + div(ρ j v) = div h j. Because of (.1.), div(h 1 +h ) = 0 necessarily. Therefore we assume that h 1 +h = 0. Let c = c 1 c = c 1 1 be the concentration difference. Then ρ t c + ρv c = div h, (.1.3) where h = h 1. In order to specify the behavior of the mixture, we will make several constitutive assumptions on the form of the stress tensor T and the relative mass flux h. But before we make the following basic assumption, which is essentially the same as in [48]. Assumption of quasi-incompressibility: Since we want to model the (partial) mixing of two incompressible fluids, it is reasonable to assume that the total density ρ depends only on the concentration difference c. Hence we assume that there is some function ˆρ: [ 1, 1] (0, ) such that ρ = ˆρ(c). A simple example is the relation 1 ρ 1 ˆρ(c) = c 1 ρ 1 + c ρ = 1(c + 1) + ρ 1 1(1 c), ρ where ρ j are the densities of the separate fluids. This describes the case that the excess volume of the mixture is zero, cf. [48] and will be discussed in Section.4 in more detail. Consequences: An immediate consequence of the latter assumption and the conservation of mass (.1.) is that where ċ = t c + v c. Moreover, by (.1.3) div v = ρ 1 ρ (.1.4) cċ, div v = ρ ρ div h. (.1.5) c In particular, in the case that ρ 0 in some region, we obtain the same constraint c div v = 0 as for a usual incompressible fluid, which implies that the stress tensor T is determined by the deformation history only up to pi, where p is an additional scalar quantity called pressure, cf. e.g. Truesdell and Noll [75, Section 30].

24 .1. FUNDAMENTAL QUANTITIES AND EQUATIONS 17 Even if ρ 0, div v is a given quantity, then depending on c and h. Therefore c we make the ansatz T = S pi where S = Ŝ(Dv, c, c) with an additional scalar quantity p. Here we can choose without loss of generality p such that Tr S = 0 if div v = 0 since otherwise one can replace S e.g. by S 1 Tr SI 3 and p by p 1 Tr S(Dv, c, c)i. Finally, we like to note that, if ρ 0, then we 3 c will be able to eliminate p again from the system derived in the following (after our constitutive assumptions) since p can be determined explicitly by c, c, c and µ. But this is not the case if ρ vanishes. Therefore we make the ansatz above. c Diffusion, free energy: The relative motion of the fluids is assumed to be diffusional. To this end, we introduce a Helmholtz free energy in a given volume V of the form ρf(c(x), c(x)) dx, (.1.6) V which describes the internal energy of the mixture. It will play the role of a surface energy for the diffuse interface. Note that we could also take it in the form f(c(x), c(x)) dx, V but, since ρ is a function of c, this does not change anything. The total energy in a volume V is the sum of the kinetic and the free energy, i.e., E V (c, v) = ρ v dx + ρf(c(x), c(x)) dx = e(v, c, c) dx, V V where e(v, c, c) = ˆρ(c) v + ˆρ(c)f(c, c). In order to describe the change of the free energy due to diffusion, we introduce a chemical potential µ such that µh n dσ. V (t) is the working done by the relative mass flux h = h 1. V Surface forces: Moreover, we assume the existence of a generalized (vectorial) surface force ξ such that ċ ξ n dσ. V (t) represents the working due to changes of the concentration. Concepts of surface forces related to the working due to changes in the microscopic structure can be found very often in the literature, see e.g. [7, 37]. In Gurtin et al. [37], ξ is called micro-force,

25 18 CHAPTER. DERIVATION OF THE DIFFUSE INTERFACE MODEL which is related to the microscopic behavior of the fluid, and n ξ describes the forces transmitted across oriented surfaces with unit normal n. We could justify the surface force ξ and the relation above by this concept. Here, however, we assume the relation above directly. In order to describe capillary effects in viscous compressible fluids in a thermodynamically consistent manner, Dunn and Serrin [7] also introduced a general surface force t which is related to an interstitial work flux u, which would be ċξ in our case. Finally, we note that Tn v dσ V describes, as in usual continuum mechanics, the working in a given volume V due to the macroscopic forces in the fluid.. Local Dissipation Inequality and Constitutive Assumptions Second law of thermodynamics/local dissipation inequality: As in [37], we assume the following dissipation inequality, which substitutes the second law of thermodynamics: d e(v, c, c) dx Tn v dσ + ċξ n dσ + µh n dσ dt V (t) V (t) for every volume V (t) transported with flow. This means that the change of total energy in time is bounded by the working due to macroscopic and microscopic stress and due to diffusion. Recall that in general e.g. by Liu [47, Theorem.1.] d dt V (t) f dx = V (t) t f dx + V (t) V (t) n vf dσ = V (t) V (t) ( t f + div(vf)) dx if the exterior normal velocity of V (t) is given by n v(t) on V (t), i.e., V (t) is transported with the flow described by v. Therefore the equivalent local form is t e + div(ve) div(t v) div(ċξ) div(µh) =: D 0 (..1) Using (.1.1)-(.1.3), we will simplify D. First of all, multiplying (.1.1) with v and using (.1.), one easily obtains ( ) ) t ρ v + div (vρ v = div(t v) T : v. Moreover, t (ρf) + div(ρvf) = ρ t f + ρv f = ρ f c ċ + ρ f c ( c)

26 .. LOCAL DISSIPATION INEQUALITY 19 and div(ċξ) + div(µh) = ċdiv ξ + (ċ) ξ + µρċ + µ h because of (.1.3) and where we have used = (ρµ + div ξ)ċ + ( c) ξ + v : ( c ξ) + µ h, xj ċ = t xj c + v xj c + xj v c = ( xj c) + xj v c. Thus we conclude that (..1) is equivalent to ( D = ρ f ) div ξ ρµ ρ 1 ρ c c p ċ ( + ρ f ) c ξ ( c) ( S + c ξ) : v µ h 0, (..) where we have used (.1.4). In order to motivate the constitutive assumptions, we will derive some restrictions for the constitutive relations specifying T, µ, h, ξ, p by an argument typical for rational continuum mechanics: To this end, we assume that T, µ, h, ξ are functions of Dv, c, c, µ, µ only. Moreover, we assume that p is independent of div v, ċ, respectively. Invoking general exterior forces and mass supplies in the equations, one argues that c, ċ, c, ( c), µ, µ, ( v 1 div vi) can attain arbitrary values for a given point 3 in space and time and since f and ξ do not depend on ċ, ( c), we conclude from (..) that ξ ρ f (c, c) = 0 c necessarily. In particular, ξ depends only on c, c. Hence (..) reduces to D = ( ρ f c ) div ξ ρµ ρ 1 ρ c p ( S + ξ c) : Dv ξ c : 1 ( v vt ) µ h 0, where Dv = 1 ( v + vt ). Hence, since the skew part of v can attain arbitrary values independent of Dv (and in particular independent of div v), we conclude that Thus ξ c = ( c ξ) and therefore for some a(c, c). ċ ξ c = c ξ. ξ(c, c) = a(c, c) c = ˆρ(c) f (c, c) c

27 0 CHAPTER. DERIVATION OF THE DIFFUSE INTERFACE MODEL Furthermore, since we have assumed that Tr S = 0 if ċ = div v = 0, ( S ξ c) : div vi tends quadratically to 0 as ċ 0. On the other hand, the first term in (..) is linear in ċ since f, h, µ and p are assumed to be independent of ċ. Therefore the first term in (..) has to vanish to satisfy (..) in general and ρµ = ρ 1 ρ c p + ρ f c div(a(c, c) c). Finally, the local dissipation inequality is satisfied if and only if ( D = S ρ f ) c c : Dv + µ h 0. (..3) Here S ρ f c is also called viscous stress tensor since it corresponds to irreversible changes of the energy due to friction in the fluids. c Constitutive assumptions: Motivated by Newton s rheological law, we assume that S ρ f c = ν(c, c)dv + η(c, c) div vi S(c, Dv) c for some functions ν(c, c), η(c, c) 0. We note that the form above necessarily holds if one assumes that S ρ f c is isotropic and depends linear on Dv, cf. c e.g. [47, Corollary 4..3]. Finally, we choose f(c, c) in the form f(c, c) = ρ 1 Φ(c) + ρ 11 q a(c) c q where q < and 0 < a a(c) a and h(c, c, µ, µ) in the form h(c, c, µ, µ) = m(c) µ, where m(c) m 0 > 0, which corresponds to a generalized Fick s law. Hence the free energy in a volume V is given by (Φ(c) + 1q ) a(c) c q dx and V ρµ = ρ 1 ρ c (p + Φ(c) + a(c) c q q ) + φ(c) + a (c) c q q div(a(c) c q c). We note that the case q = corresponds to a usual choice of the free energy in the derivation of Allen-Cahn and Cahn-Hilliard type equations. But, since the gradient term plays the role of a regularizing term, which permits discontinuous changes of c, there seems to be no principle reason not to choose c q for some q >. As the

28 .. LOCAL DISSIPATION INEQUALITY 1 previous calculations show, also this choice leads to a thermodynamically consistent model. In order to simplify the expressions, we introduce A(c) such that A (c) = a(c) 1 q and A(0) = 0. Then the free energy can be written as ) (Φ(c) + A(c) q dx. q V and ρµ = ρ 1 ρ c ) (p + Φ(c) + A(c) q + φ(c) a(c) 1 q q A(c) q where q u = div( u q u) denotes the q-laplacian and where we have used that div(a(c) c q c) + a (c) c q q = a(c) 1 q div( A(c) q A(c)) since (a 1 q (c)) a(c) 1 1 q c q c = a (c) c q and a(c) 1 1 q q c q c = A(c) q A(c). Summing up, we derived the following system for our diffuse interface model: ρ t v + ρv v div S(c, Dv) + p = div( A(c) q A(c) A(c)) (..4) ρµ = ρ 1 ρ c t ρ + div(ρv) = 0 (..5) ρ t c + ρv c = div(m(c) µ) (..6) ) (p + Φ(c) + A(c) q + φ(c) a(c) 1 q q A(c) (..7) q which shall hold for all (x, t) (0, ). We close the system by adding initial and boundary conditions as well as n v = 0 on (0, ), (..8) n c = n µ = 0 on (0, ), (..9) (v, c) t=0 = (v 0, c 0 ) in, (..10) n S(c, Dv) τ + γ(c)v τ = 0 if 0 γ(c) < (..11) v τ = 0 if γ, (..1) where τ denotes the tangential part of a vector field at the boundary. Here γ corresponds to the case of no-slip boundary conditions and 0 γ(c) < describes the Navier boundary conditions with a friction coefficient γ(c) 0 depending on c.

29 CHAPTER. DERIVATION OF THE DIFFUSE INTERFACE MODEL.3 Conserved Quantities, Energy Estimates, and Reformulation of the System First of all, since n v = 0, (..5) implies d ρ dx = div(ρv) dx = 0. dt Hence the total mass is constant in time ρ(x, t) dx = ρ 0 (x) dx for all t (0, ), where ρ 0 = ˆρ(c 0 ). Moreover, because of (..5), (..6) is equivalent to t (ρc) + div(ρcv) = div(m(c) µ). Therefore the total mass difference ρc = ρ 1 ρ is a conserved quantity too: ρ(x, t)c(x, t) dx = ρ 0 (x)c 0 (x) dx for all t (0, ). Moreover, integration of (..1) with respect to yields d E(c(t), v(t)) dt = D dx + = D dx + (n T v + ċ n a(c, c) c + µm(c)n µ) dσ (n T) τ v τ dσ because of (..8) and (..9). Due to (..11)-(..1), { (n T) τ v τ dσ = γ(c) v τ dx if 0 γ(c) < 0 if γ(c) where we note that v τ = v on since n v = 0. Hence, using (..3) and the constitutive assumptions above, we obtain d ( E(c(t), v(t)) = ν(c(t)) Dv(t) + η(c(t)) div v(t) ) dx dt m(c) µ dx γ(c) v dσ, (.3.1) where E(c, v) = E free (c) + E kin (c, v) E free (c) = E kin (c, v) = a(c) c q dx + q ˆρ(c) v dx Φ(c) dx,

30 .4. SIMPLE MIXTURES AND CONSEQUENCES 3 and where we use the convention 0 = 0 to explain the meaning in the case γ. Integrating (.3.1) gives the important a priori estimate: T sup E(c(t), v(t)) + 0 t T 0 T ( ν(c(t)) Dv(t) + η(c(t)) divv(t) ) dxdt T + m(c) µ dxdt + γ(c) v dσ dt E(c 0, v 0 ), 0 0 which is essential for the following analysis. In order to see the structure of the system better, we reformulate (..4)-(..7). To this end, we define g = Φ(c) + A(c) q + p ρ ρq ρ µc, where µ = µ 0 + µ and µ = 1 µ dx assuming for simplicity that is a bounded domain. Then ) ρ g = φ(c) c + A(c) q + p ρ (p 1 ρ + Φ(c) + A(c) q c ρ µ c q c q and therefore ρ g ρµ 0 c = p + a(c) 1 q q A(c) c + A(c) q q = p + div( A(c) q A(c) A(c)). Hence we see that (..4)-(..7) is equivalent to the system ρ t v + ρv v div S(c, Dv) + ρ g = ρµ 0 c in (0, ), (.3.) t ρ + div(ρv) = 0 in (0, ), (.3.3) ρ t c + ρv c = div(m(c) µ) in (0, ), (.3.4) and ρµ 0 + ( ρ + ρ ) c c µ = ρ c g a1 q (c) q A(c) + φ(c). (.3.5).4 Simple Mixtures and Consequences In the following mathematical analysis we will only consider the case of a so-called simple mixture, which means that the total volume does not change when mixing the two fluids. More precisely, since the mass density ρ describes the total mass per unit volume, ρ 1 describes the volume filled by a part of the mixture with unit mass. Moreover, since the fluids are assumed to be incompressible, the volume filled by the