Abstract
In two-phase flows typically a change of topology arises. This happens when two drops merge or when one bubble splits into two. In such a case the concept of classical solutions to two-phase flow problems, which describe the interface as a smooth hypersurface, breaks down.
This contribution discusses two possible approaches to deal with this problem. First of all weak formulations are discussed which allow for topology changes during the evolution. Such weak formulations involve either varifold solutions, so-called renormalized solutions or viscosity solutions.
A second approach replaces the sharp interface by a diffuse interfacial layer which leads to a phase field-type representation of the interface. This approach leads typically to quite smooth solutions even when the topology changes.
This contribution introduces the solution concepts, discusses modeling aspects, gives an account of the analytical results known, and states how one can recover the sharp interface problem as an asymptotic limit of the diffuse interface problem.
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1 Introduction
In the flow of immiscible fluids with interfaces in general, topological transitions like droplet breakup and coalescence occur. In such situations classical formulations based on an explicit parameterization break down as singularities will appear at points where the topology changes. This contribution discusses two approaches to deal with this issue. First of all, weak formulations of the two-phase flow problem for incompressible fluids are introduced which all allow for singularities in the geometry. The known results for the different approaches are stated, and the advantages and disadvantages of the different formulations are discussed. Secondly, diffuse interface methods provide an alternative way to allow for topological transitions. In these models quantities which in traditional sharp interface models are localized to the interfacial surface are now distributed over a diffuse interfacial region. For example, quantities like the density and the viscosity are suitably averaged in the diffuse interface, and the surface tension, which is supported on the interface in a sharp interface model, is now a distributed stress within the diffuse interfacial layer, cf. Fig. 1.
Sharp versus diffuse interface models
In the classical sharp interface approach for incompressible viscous flows, the Navier-Stokes equations have to hold in the two phases, described by disjoint open sets \(\Omega _{\pm }(t)\), which are separated by a hypersurface \(\Gamma (t)\), which evolves in time. In this contribution slip is not allowed at the interface which leads to the fact that the tangential part of the fluid velocity does not jump at the interface and also it is assumed that no phase transitions occur which implies that also the normal part of the velocity does not jump at the interface and that the interface is transported with the fluid velocity. One hence obtains
where v is the fluid velocity, [.] +− denotes the jump across the interface \(\Gamma (t)\), \(\boldsymbol{\nu }\) is a unit normal at the interface \(\Gamma (t)\), chosen as interior normal with respect to \(\Omega _{+}(t)\), and \(\mathcal{V}\) is the normal velocity.
In addition a tangential stress balance has to hold at the interface. In cases where the interface itself does not produce stresses at the interface, the normal stresses have to balance, i.e., on the interface it has to hold
where T+ and T− are the values of the stress tensor on both sides of the interface. For a viscous incompressible fluid, the simplest choice for T is \(\mathbf{T} = 2\eta D\mathbf{v} - p\mathop{\mathrm{Id}}\nolimits\), where \(D\mathbf{v} = \frac{1} {2}(\nabla \mathbf{v} + \nabla \mathbf{v}^{T})\). In the following mainly the case with surface tension will be discussed, i.e., surface energy effects are taken into account and in this case the stress balance at the interface is given as the Young-Laplace law
where σ is the surface tension and H is the mean curvature, which is chosen to be the sum of the principal curvatures with respect to \(\boldsymbol{\nu }\).
For weak formulations it is often convenient to reformulate the fact that the interface is transported with the fluid velocity. Defining χ as the characteristic function of one of the phases, one can formally rewrite the equation \(\mathcal{V} = \mathbf{v}\cdot \boldsymbol{\nu }\) as
which for a velocity field that has zero divergence is formally equivalent to
Of course the last two equations need to be interpreted in a suitable weak sense, and different formulations will be discussed in Sect. 2. All these formulations will allow for singularities of the interface and in particular allow for topological transitions. The weak formulations mentioned above are:
-
approaches based on the theory of viscosity solutions, see [42, 69, 71],
-
methods which use the concept of renormalized solutions of transport equations, see [56, 57].
These approaches work for the case without surface tension. If surface tension effects are present, also the mean curvature has to be interpreted in a weak sense, and this can be done in the context of varifolds; see [2, 14, 16, 21, 58, 72] and Sect. 2.2.
In diffuse interface models (which are also called phase field models), the sharp interface is replaced by an interfacial layer of finite width, and a smooth order parameter is used to distinguish between the two bulk fluids and the diffuse interface. The order parameter takes distinct constant values in each of the bulk fluids and varies smoothly across the thin interfacial layer. In the sharp interface case with surface tension , the total energy is given as
where \(\Omega\) is the domain occupied by the fluid, ρ is the mass density, \(\Gamma\) is the interface, and \(\mathcal{H}^{d-1}\) is the (d − 1)-dimensional surface measure. The first term is the kinetic energy, and the second term accounts for interfacial energy. It is well known based on the work of van der Waals, Korteweg, Cahn, and Hilliard that interfacial energy and also related capillary forces can be modeled with the help of density variables which vary continuously across the interface. In these approaches the term \(\sigma \mathcal{H}^{d-1}(\Gamma )\) is replaced by a multiple of
where ɛ > 0 is a small parameter, φ is an order parameter taking the values ± 1 in the two phases, and ψ is a double well potential which simplest form is \(\psi (\varphi ) = \frac{1} {4}(1 -\varphi ^{2})^{2}\). One can now try to model the physics at the interface with the help of φ and would obtain a new problem which should approximate the above sharp interface problem.
As new energy one obtains
and the transport equation becomes
with
where \(\frac{\delta \mathcal{F}} {\delta \varphi }\) is the first variation of \(\mathcal{F}\).
One obtains the law \(\mathcal{V} = \mathbf{v}\cdot \boldsymbol{\nu }\) from the free boundary problem in the limit ɛ → 0 by choosing m ɛ ∼ ɛ; see Sect. 4. It will turn out, see Sect. 3, that the term
contributing to the stress balance at the interface will become a multiple of
which is a term which is distributed over the diffuse interfacial layer. In the simplest case, the momentum balance in the Navier-Stokes equation can be written as
where the term \(\mathop{\mathrm{Id}}\nolimits -\frac{\nabla \varphi } {\vert \nabla \varphi \vert } \otimes \frac{\nabla \varphi } {\vert \nabla \varphi \vert }\) corresponds to \(\mathop{\mathrm{Id}}\nolimits -\boldsymbol{\nu }\otimes \boldsymbol{\nu }\) which is a multiple of the classical interfacial stress tensor which is just the projection onto the tangent space, cf., weak formulation (17) below. Diffuse interface models for incompressible two-phase flows were introduced and studied in [13, 23, 28, 37, 46, 51, 52].
2 Weak Formulations
In this section different notions of weak/generalized solutions of the two-phase flow of two incompressible, immiscible Newtonian fluids inside a bounded domain \(\Omega \subseteq \mathbb{R}^{d}\), d = 2, 3, are discussed. The fluids fill disjoint domains \(\Omega _{+}(t)\) and \(\Omega _{-}(t)\), t > 0, and the interface between both fluids is denoted by \(\Gamma (t) = \partial \Omega _{+}(t)\). It is assumed that \(\Gamma (t)\) is compactly contained in \(\Omega\), which means that one excludes flows, where a contact angle problem occurs. Hence \(\Omega = \Omega _{+}(t) \cup \Omega _{-}(t) \cup \Gamma (t)\). The flow is described using the velocity \(\mathbf{v}: \Omega \times (0,\infty ) \rightarrow \mathbb{R}^{d}\) and the pressure \(p: \Omega \times (0,\infty ) \rightarrow \mathbb{R}\) in both fluids in Eulerian coordinates. Cases with and without surface tension at the interface are considered. Precise assumptions are made below. Under suitable smoothness assumptions, the flow is obtained as solution of the system
together with \(\Omega _{\pm }(0) = \Omega _{0}^{\pm }\). Here \(\mathcal{V}\) and H denote the normal velocity and mean curvature of \(\Gamma (t)\), resp., taken with respect to the interior normal \(\boldsymbol{\nu }\) of \(\partial \Omega _{+}(t) = \Gamma (t)\), σ ≥ 0 is the surface tension constant (σ = 0 means that no surface tension is present), and ρ± > 0 and η± > 0 are the (constant) densities and viscosities of the fluids, respectively. The equations (2) and (3) describe the conservation of linear momentum and mass for both fluids. Furthermore, (4) is a no-slip boundary condition at \(\Gamma (t)\), implying continuity of v across \(\Gamma\), (5) is the balance of forces at the boundary, (6) is the kinematic condition that the interface is transported with the flow of the mass particles, and (7) is the no-slip condition at the boundary of \(\Omega\). Here exterior forces are neglected for simplicity.
There are many results on well-posedness locally in time or global existence close to equilibrium states for quite regular solutions of this two-phase flow and similar free boundary value problems for viscous incompressible fluids, cf. Solonnikov [66, 68], Beale [24, 25], Tani and Tanaka [70], Shibata and Shimizu [64], Shibata and Shimizu [65] or Prüss and Simonett [59], and the references given there. 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.
In the following different notions of generalized solutions, which allow for singularities of the interface and which exist globally in time for general initial data, are discussed. A similar and more detailed discussion can be found in [2]. To this end, first a suitable weak formulation of the system above is needed. By multiplication of (2) with a divergence-free vector field \(\boldsymbol{\varphi }\) and integration by parts using in particular the jump relation (6), one obtains
for all \(\boldsymbol{\varphi }\in C_{(0)}^{\infty }(\Omega \times [0,\infty ))^{d}\) with \(\mathop{\mathrm{div}}\nolimits \boldsymbol{\varphi } = 0\), where \(D\mathbf{v} = \frac{1} {2}(\nabla \mathbf{v} + \nabla \mathbf{v}^{T})\), \(\chi (x,t) =\chi _{\Omega _{+}(t)}(x)\) for all \(x \in \Omega\), t > 0, \(\chi _{0} =\chi _{\Omega _{0}^{+}}\), χ A denotes the characteristic function of a set A, ρ(1) = ρ+, ρ(0) = ρ−, η(1) = η+, η(0) = η−, and
Here \(\mathcal{H}^{d-1}\) denotes the (d − 1)-dimensional Hausdorff measure. Now, if v and \(\Gamma\) are sufficiently smooth, one obtains by choosing \(\boldsymbol{\varphi }= \mathbf{v}\) the energy inequality
for all T > 0 (even with equality), where \(\Gamma _{0} = \partial \Omega _{0}^{+}\). Here one uses
which is due to (6), cf. [43, Equation 10.12]. More details for a more general model can be found in [13, Section 5].
Since η±, ρ± > 0, (11) yields the a priori estimate
for any sufficiently smooth solution of (2), (3), (4), (5), (6), (7), and (8). Here Lp(M), 1 ≤ p ≤ ∞, denotes the usual Lebesgue space, L p loc (M) its local and Lp(M; X) its vector-valued analog for a given Banach space X. Moreover, if \(A \subset \mathbb{R}\), then Lp(M; A) consists of all f ∈ Lp(M) with f(x) ∈ A for a.e. x ∈ M. Finally, \(L_{\sigma }^{p}(\Omega ) = \overline{\{\boldsymbol{\varphi } \in C_{0}^{\infty }(\Omega )^{d}:\mathop{ \mathrm{div}}\nolimits \boldsymbol{\varphi } = 0\}}^{L^{2}(\Omega ) }\) is the set of all weakly divergence free vector fields \(f \in L^{p}(\Omega )^{d}\).
As will be shown below, if σ > 0, then (11) yields an a priori bound of
where \(BV (\Omega ) =\{ f \in L^{1}(\Omega ): \nabla f \in \mathcal{M}(\Omega )\}\) denotes the space of functions with bounded variation, cf., e.g., [22, 39] and \(\mathcal{M}(\Omega ) = C_{0}(\Omega )\prime\) is the space of finite Radon measures. In the case without surface tension, i.e., σ = 0, one only obtains that χ ∈ L∞(Q) is a priori bounded by one, where \(Q:= \Omega \times (0,\infty )\). This motivates to look for weak solutions (v, χ) lying in the function spaces above, satisfying (11) with a suitable substitute of (10), such that (v, χ) solve (9) as well as the transport equation
for \(\chi _{0} =\chi _{\Omega _{0}^{+}}\) in a suitable weak sense. Note that (14) is a weak formulation of (6), cf. [56, Lemma 1.2].
2.1 Two-Phase Flow Without Surface Tension
Throughout this subsection, it is assumed that σ = 0, i.e., no surface tension is present. Then the two-phase flow consists of a coupled system of the Navier-Stokes equation with variable viscosities and a transport equation for the characteristic function \(\chi (t) =\chi _{\Omega _{+}(t)}\). Then this is a special case of the so-called density-dependent Navier-Stokes equation, cf., e.g., Desjardins [36] and references given there. For given χ it is not difficult to construct a weak solution of the Navier-Stokes equation (9) with the aid of a suitable approximation scheme (e.g., Galerkin approximation). New difficulties arise due to the mean curvature term \(\langle H_{\Gamma (t)},.\rangle\), which depends nonlinearly on the normal of \(\Gamma (t)\).
For the coupled system (9) together with (14) and (15), there are two different approaches. The essential difference is in which sense the transport equation is solved. One approach is due to Giga and Takahashi [42], who solved (14) and (15) in the sense of viscosity solutions, where the characteristic functions (χ(t), χ0) are replaced by continuous level-set functions (ψ(t), ψ0) such that
For simplicity they consider periodic boundary conditions, i.e., \(\Omega = \mathbb{T}^{d}\). Since v is in general not Lipschitz continuous, the existence of a viscosity solution of (14) and (15) with (χ, χ0) replaced by continuous level-set functions (ψ, ψ0) is not known. There are only a least super-solution ψ+(t) and a largest sub-solution ψ−(t) of the transport equation. Then one defines
With this definition \(\Omega _{\pm }(t)\) are disjoint open sets, but the “boundary” \(\Gamma (t) = \mathbb{T}^{d}\setminus (\Omega _{+}(t) \cup \Omega _{-}(t))\) might have interior points and might have positive Lebesgue’s measure. Giga and Takahashi call this possible effect “boundary fattening.” With this definition they construct weak solutions of a two-phase Stokes flow, i.e., the convective term v ⋅ ∇v is neglected in (9), assuming that the viscosity difference | η+ −η− | is sufficiently small and ρ+ = ρ−; cf. [42] for details. This approach was adapted to the case of a Navier-Stokes two-phase flow by Takahashi [69] under similar assumptions and to a one-phase flow for an ideal, irrotational, and incompressible fluid by Wagner [71].
The other approach was established by Nouri and Poupaud [56] and Nouri et. al. [57] and is based on the results of DiPerna and Lions [38] on renormalized solutions of the transport equation (14) and (15) for a velocity field v with bounded divergence. Here χ ∈ L∞(Q) is called a renormalized solution of (14) and (15) if for all \(\beta \in C^{1}(\mathbb{R})\) which vanish near 0 the function β(χ) solves (14) and (15) with initial values β(χ0), cf. [38] for details. In particular, this implies that \(\chi (t,x) \in \overline{\{\chi _{0}(x): x \in \Omega \}}\) for almost all \(t> 0,x \in \Omega\). Due to [38, Theorem II.3], for every \(\chi _{0} \in L^{\infty }(\mathbb{R}^{d})\), there is a unique renormalized solution of (14) and (15) under general conditions on v, which are weaker than the condition (13). Based on this notion the following result for the two-phase flow without surface tension holds true:
Theorem 1 (Existence of Weak Solutions, [56, Theorem 1.1]).
For every \(\mathbf{v}_{0} \in L_{\sigma }^{2}(\Omega )\), \(\chi _{0} \in L^{\infty }(\Omega;\{0,1\})\) there are \(\mathbf{v} \in L^{\infty }(0,\infty;L_{\sigma }^{2}(\Omega )) \cap L^{2}(0,\infty;H_{0}^{1}(\Omega )^{d})\) and χ ∈ L∞(Q; {0, 1}) that are a weak solution of the two-phase flow (2), (3), (4) (5), (6), (7), and (8) without surface tension (σ = 0) in the sense that (9) holds true for all \(\boldsymbol{\varphi }\in C_{(0)}^{\infty }(\Omega \times [0,\infty ))^{d}\) with \(\mathop{\mathrm{div}}\nolimits \boldsymbol{\varphi } = 0,\chi\) is the unique renormalized solution of the transport equation of (14) and (15), and (11) holds for almost all t > 0 with σ = 0.
Here \(C_{(0)}^{\infty }(\Omega \times [0,T))\), T ∈ (0, ∞], is the space of all smooth \(\varphi: \Omega \times [0,T) \rightarrow \mathbb{R}\) with compact \(\mathop{\mathrm{supp}}\nolimits \varphi \subseteq \Omega \times [0,T)\). Moreover, \(H_{0}^{k}(\Omega )\), \(k \in \mathbb{N}\) is the closure of compactly supported, smooth functions \(\varphi: \Omega \rightarrow \mathbb{R}\) in \(H^{k}(\Omega )\).
The result was proved by Nouri and Poupaud [56] for the case of a bounded domain \(\Omega\) with Lipschitz boundary. The authors even considered the case of a multiphase flow with more than two components. The result was extended to generalized Newtonian fluids of power-law type for a power-law exponent \(q \geq \frac{2d} {d+2} + 1\) in [3].
In order to prove the latter theorem, a key step is to show strong compactness of the sequence χ k in Lp(Q T ), 1 ≤ p < ∞, where \(Q_{T} = \Omega \times (0,T),T> 0\) and (v k , χ k ) is a suitably constructed approximation sequence. This is done by using the fact that
if χ k are solutions of (14) and (15) with v replaced by v k and \(\mathop{\mathrm{div}}\nolimits \mathbf{v}_{k} = 0\). Using that
for a suitable subsequence, one shows that χ solves (14) and (15), cf. [3, Lemma 5.1]. Here ⇀∗ denotes the weak-∗ convergence. Therefore
This implies strong convergence χ k →k → ∞χ in Lp(Q T ), 1 ≤ p < ∞, for every T > 0. Based on this, one can pass to the limit in all terms in (9).
Remark 1.
Using the solution of Theorem 1, one can define the sets \(\Omega _{+}(t) =\{ x \in \Omega:\chi (t) = 1\}\) and \(\Omega _{-}(t) =\{ x \in \Omega:\chi (t) = 0\}\). Then one knows that | Ω+(t) | = | Ω +0 | and \(\Omega \setminus (\Omega _{+}(t) \cup \Omega _{-}(t))\) has Lebesgue measure zero. But, since only χ ∈ L∞(Q) is known, it is not clear whether \(\Omega _{\pm }(t)\) have interior points. In particular, it is not excluded that \(\overline{\Omega _{+}(t)} = \Omega\) and \(\mathop{\mathrm{int}}\nolimits \Omega _{+}(t) =\emptyset\). Therefore it is not immediately clear what the “interface” between both fluids should be. If one naively defines the interface as \(\Gamma (t) = \partial \Omega _{+}(t)\), then \(\Gamma (t)\) can have positive Lebesgue measure as in the result by Giga and Takahasi.
It seems that by neglecting surface tension in the two-phase flow, one looses a “good control” of the interface between both fluids. At least the precise regularity of the interface seems to be unknown in general. Some results in this direction can be found in the contribution by Danchin and Mucha [34], where existence and uniqueness of more regular solutions for the inhomogeneous Navier-Stokes equation with discontinuous initial density are shown under several smallness assumptions.
2.2 Case with Surface Tension: Varifold Solutions
As discussed in the previous section, a deficit of the two-phase flow without surface tension is that there is no good information on the properties of the interface. As mentioned in the introduction, if σ > 0, the energy equality (11) for sufficiently smooth solutions provides an a priori estimate of the interface:
This implies an a priori bound of χ in the space \(BV (\Omega )\) as follows: Note that, if \(\Gamma (t) = \partial \Omega _{+}(t)\) is sufficiently smooth, Gauss’ theorem yields
for all \(\boldsymbol{\varphi }\in C_{0}^{\infty }(\Omega )^{d}\). Hence the distributional gradient ∇χ(t) is a finite Radon measure and
Thus, if σ > 0, then \(\chi (t) \in BV (\Omega )\) for all t > 0, and (16) gives an a priori estimate of
Conversely, if \(\chi (t) =\chi _{E} \in BV (\Omega )\) for some set E = E(t), then E is said to be of finite perimeter, and the following characterization holds, cf. [39, Section 5.7, Theorem 2]:
where ∂∗E is the reduced boundary of E, cf. [39, Definition 5.7], \(\boldsymbol{\nu }_{E} = \frac{\nabla \chi _{E}} {\vert \nabla \chi _{E}\vert }\), and ∂∗E is countably (d − 1)-rectifiable in the sense that
where K k are compact subsets of C1-hypersurfaces S k , \(k \in \mathbb{N}\), \(\mathcal{H}^{d-1}(N) = 0\), and \(\nu _{E}\vert _{S_{k}}\) is normal to S k . Moreover, by [39, Section 5.8, Lemma 1] ∂∗E ⊆ ∂∗E and \(\mathcal{H}^{d-1}(\partial ^{{\ast}}E\setminus \partial _{{\ast}}E)\), where ∂∗E is the measure theoretic boundary of E consisting of all \(x \in \Omega\) such that
where \(\mathcal{L}^{d}\) is the Lebesgue measure on \(\mathbb{R}^{d}\).
Based on these properties, one can define the mean curvature functional of a set of finite perimeter E as
where P τ = I −ν E (x) ⊗ν E (x) and \(C_{0}^{1}(\Omega )\) is the closure of smooth functions \(\varphi: \Omega \rightarrow \mathbb{R}\) with compact support in \(\Omega\). Note that \(\mathop{\mathrm{tr}}\nolimits (P_{\tau }\nabla \varphi )\) corresponds to the divergence of φ along the “surface” ∂∗E and that by integration by parts (17) coincides with the usual definition if ∂∗E is a C2-surface, cf., e.g., Giusti [43, Chapter 10].
In the following it is assumed that ρ+ = ρ− = 1 and \(\Omega = \mathbb{R}^{d}\). Motivated by the considerations above, one defines weak solutions of the two-phase flow in the case of surface tension as follows:
Definition 1 (Weak Solutions).
Let σ > 0. Then
are called a weak solution of the two-phase flow for initial data \(\mathbf{v}_{0} \in L_{\sigma }^{2}(\mathbb{R}^{d})\), \(\chi _{0} =\chi _{\Omega _{0}^{+}}\) for a bounded domain \(\Omega _{0}^{+} \subset \subset \mathbb{R}^{d}\) of finite perimeter if the following conditions are satisfied:
-
(i)
(9) holds for all \(\boldsymbol{\varphi }\in C_{(0)}^{\infty }(\mathbb{R}^{d} \times [0,\infty ))^{d}\) with \(\mathop{\mathrm{div}}\nolimits \boldsymbol{\varphi } = 0\), where \(H_{\Gamma (t)}\) is replaced by Hχ(t) defined as in (17).
- (ii)
-
(iii)
The energy inequality
$$\displaystyle\begin{array}{rcl} & & \frac{1} {2}\|\mathbf{v}(t)\|_{2}^{2} +\sigma \| \nabla \chi (t)\|_{ \mathcal{M}(\Omega )} \\ & & \ \ +\int _{0}^{t}\int _{ \Omega }2\eta (\chi )\vert D\mathbf{v}\vert ^{2}\,dx\,d\tau \leq \frac{1} {2}\|v_{0}\|_{2}^{2} +\sigma \| \nabla \chi _{ 0}\|_{\mathcal{M}(\Omega )} {}\end{array}$$(18)holds for almost all t ∈ (0, ∞).
Unfortunately, the existence of weak solutions as defined above is open. The reasons are possible oscillation and concentration effects related to the interface, which cannot be excluded so far. One can hence not pass to the limit in the mean curvature functional (17) during an approximation procedure used to construct weak solutions.
In order to demonstrate these effects, let E k be a sequence of sets of finite perimeter such that \(\chi _{k} \equiv \chi _{E_{k}}\) is bounded in \(BV (\Omega )\) and let \(\Omega = \mathbb{R}^{d}\). Then after passing to a suitable subsequence, one assumes that
But then the question arises how | ∇χ | and μ are related and whether
holds. The continuity result due Reshetnyak, cf. [22, Theorem 2.39], gives a sufficient condition for (19): If
then (19) holds. But in general (20) will not hold, for example, because of the following oscillation/concentration effects at the reduced boundary of E:
-
(i)
Several parts of the boundary ∂∗E k might meet.
-
(ii)
Oscillations of the boundary might reduce the area in the limit.
-
(iii)
There might be an “infinitesimal emulsion.”
These effects are sketched in Fig. 2.
Some possible oscillation/concentration effect
It is an open problem how to exclude such kind of oscillation/concentration effects. This might even not be possible in general since the model might not describe the behavior of both fluids appropriately when, e.g., a lot of small-scale drops are forming. One way out of this problem is to define so-called varifold solution of a two-phase flow, which was first done by Plotnikov [58] in the case of d = 2 for shear-thickening non-Newtonian fluids. Here a general (oriented) varifold V on a domain \(\Omega\) is simply a nonnegative measure in \(\mathcal{M}(\Omega \times \mathbb{S}^{d-1})\), where \(\mathbb{S}^{d-1}\) denotes the unit sphere in \(\mathbb{R}^{d}\). Here \(\mathcal{M}(\Omega \times \mathbb{S}^{d-1}) = C_{0}(\Omega \times \mathbb{S}^{d-1})\prime\) is the space of finite Radon measures on \(\Omega \times \mathbb{S}^{d-1}\), where \(\Omega \times \mathbb{S}^{d-1}\) is equipped with the product of the Lebesgue and (d − 1)-dimensional Hausdorff measure. Moreover, \(C_{0}(\Omega \times \mathbb{S}^{d-1})\) is the closure of smooth, compactly supported functions \(\varphi: \Omega \times \mathbb{S}^{d-1} \rightarrow \mathbb{R}\). By disintegration, cf. [22, Theorem 2.28], a varifold V can be decomposed in a nonnegative measure \(\vert V \vert \in \mathcal{M}(\Omega )\), and a family of probability measures \(V _{x} \in \mathcal{M}(\mathbb{S}^{d-1})\), \(x \in \Omega\), such that
Moreover, | V | corresponds to the measure of the “area of the interface,” and V x defines a probability for the “normal at the interface” for | V | -a.e. x.
The reduced boundary ∂∗E of a set of finite perimeter induces naturally a varifold by setting | V | = | ∇χ E | and \(V _{x} =\delta _{\nu _{E}(x)}\) for x ∈ ∂∗E, where δ ν denotes the Dirac measure at \(\nu \in \mathbb{S}^{d-1}\). Hence the associated varifold V E is
Now let E k be a sequence of sets of finite perimeter as above. Then by the weak-∗ compactness of \(\mathcal{M}(\Omega \times \mathbb{S}^{d-1})\), there is a limit varifold \(V \in \mathcal{M}(\Omega \times \mathbb{S}^{d-1})\) such that
for a suitable subsequence. Hence using \(\psi (s,x) =\mathop{ \mathrm{tr}}\nolimits ((I - s \otimes s)\nabla \boldsymbol{\varphi }(x))\) for \(\boldsymbol{\varphi }\in C_{0}^{1}(\Omega )^{d}\), it follows
for all \(\boldsymbol{\varphi }\in C_{0}^{1}(\Omega )^{d}\). Here \(\delta V \in C_{0}^{1}(\Omega; \mathbb{R}^{d})\prime\) defined as above is called the first variation of the generalized varifold V. Moreover,
Hence V can be used to describe the limit of \(H_{\chi _{E_{ k}}}\) as well as the limit of \(\nabla \chi _{E_{k}}\).
Now a varifold solution of the two-phase flow is defined as follows:
Definition 2 (Varifold solutions).
Let σ > 0. Then
are called a varifold solution of the two-phase flow for initial data \(\mathbf{v}_{0} \in L_{\sigma }^{2}(\mathbb{R}^{d})\) and \(\chi _{0} =\chi _{\Omega _{0}^{+}}\) for a bounded domain \(\Omega _{0}^{+} \subset \subset \mathbb{R}^{d}\) of finite perimeter if the following conditions are satisfied:
-
(i)
(9) holds for all \(\boldsymbol{\varphi }\in C_{(0)}^{\infty }(\mathbb{R}^{d} \times [0,\infty ))^{d}\) with \(\mathop{\mathrm{div}}\nolimits \boldsymbol{\varphi } = 0\), where \(\langle H_{\Gamma (t)},\boldsymbol{\varphi }\rangle\) is replaced by
$$\displaystyle{ \langle \delta V (t),\boldsymbol{\varphi }\rangle =\int _{\mathbb{R}^{d}\times \mathbb{S}^{d-1}}\mathop{ \mathrm{tr}}\nolimits ((I - s \otimes s)\nabla \boldsymbol{\varphi }(x))\,dV (s,x),\quad \boldsymbol{\varphi } \in C_{0}^{1}(\Omega )^{d}. }$$ -
(ii)
The modified energy inequality
$$\displaystyle\begin{array}{rcl} & & \frac{1} {2}\|\mathbf{v}(t)\|_{2}^{2} +\sigma \| V (t)\|_{ \mathcal{M}(\Omega \times \mathbb{S}^{d-1})} \\ & & +\int _{0}^{t}\int _{ \mathbb{R}^{d}}2\eta (\chi )\vert D\mathbf{v}\vert ^{2}\,dx\,d\tau \leq \frac{1} {2}\|\mathbf{v}_{0}\|_{2}^{2} +\sigma \| \nabla \chi _{ 0}\|_{\mathcal{M}(\Omega )} {}\end{array}$$(22)holds for almost all t ∈ (0, ∞).
-
(iii)
The compatibility condition
$$\displaystyle{ -\langle \nabla \chi (t),\varphi \rangle =\int _{\Omega \times \mathbb{S}^{d-1}}s \cdot \boldsymbol{\varphi } (x)\,dV (x,s),\quad \boldsymbol{\varphi } \in C_{0}(\Omega )^{d} , }$$(23)holds for almost all t > 0.
Here L ∞ω∗ (0, T; X′) denotes the space of weakly-∗ measurable essentially bounded functions f: (0, T) → X′.
Remark 2.
-
(i)
Let (V x (t), | V (t) | ), \(x \in \mathbb{R}^{d}\), denote the disintegration of \(V (t) \in \mathcal{M}(\mathbb{R}^{d} \times \mathbb{S}^{d-1})\) as described above. Then (23) implies that | ∇χ(t) | (A) ≤ | V (t) | (A) for all open sets A and almost all t ∈ (0, ∞). Hence | ∇χ(t) | is absolutely continuous with respect to | V (t) | and
$$\displaystyle{ \int _{\mathbb{R}^{d}}f(x)\,d\vert \nabla \chi (t)\vert =\int _{\mathbb{R}^{d}}f(x)\alpha _{t}(x)\,d\vert V (t)\vert,\quad f \in C_{0}(\mathbb{R}^{d}), }$$for a | V (t) | -measurable function \(\alpha _{t}: \mathbb{R}^{d} \rightarrow [0,\infty )\) with | α t (x) | ≤ 1 almost everywhere. In particular, this implies \(\mathop{\mathrm{supp}}\nolimits \nabla \chi _{t} \subseteq \mathop{\mathrm{supp}}\nolimits V (t)\) and \(\|\nabla \chi (t)\|_{\mathcal{M}}\leq \| V (t)\|_{\mathcal{M}}\) for almost all t ∈ (0, ∞). Hence every varifold solution satisfies the energy inequality (18) for almost every t > 0.
Moreover, if \(E(t) =\{ x \in \mathbb{R}^{d}:\chi (x,t) = 1\}\), t > 0, then (23) yields the relation
$$\displaystyle{ \int _{\mathbb{S}^{d-1}}s\,dV _{x}(t)(s) = \left \{\begin{array}{@{}l@{\quad }l@{}} \alpha _{t}(x)\boldsymbol{\nu }_{E(t)}(x)\quad &\text{if}\ x \in \partial ^{{\ast}}E_{t} \\ 0 \quad &\text{else}\end{array} \right. }$$for | V (t) | -almost every \(x \in \mathbb{R}^{d}\) and almost every t > 0. In other words, the expectation of V x (t) is proportional to the normal \(\boldsymbol{\nu }\) on the interface described by ∇χ and zero away from it.
-
(ii)
In general, it is an open problem whether V (t) is a so-called countably (d − 1)-rectifiable varifold, which implies that up to orientation V x (t) is a Dirac measure for | V (t) | -almost every x. Then V (t) can naturally be identified with a countably (d − 1)-rectifiable set – a “surface” – equipped with a density θ t ≥ 0. So far only a sufficient condition for the rectifiability of V (t) in terms of the first variation δV (t) is known, cf. [2, Section 4].
-
(iii)
As noted above, the existence of weak solutions to the two-phase flow with surface tension is open. But a general property of varifold solutions is that a varifold solution is a weak solution if the energy equality holds, i.e., (18) holds with equality for almost every t > 0. See [3, Proposition 1.5] for details.
Theorem 2 (Existence of Varifold Solutions, [3, Theorem 1.6]).
Let σ > 0, d = 2, 3. Then for every \(\mathbf{v}_{0} \in L_{\sigma }^{2}(\mathbb{R}^{d})\) and \(\chi _{0} =\chi _{\Omega _{0}^{+}}\) where \(\Omega _{0}^{+} \subset \subset \mathbb{R}^{d}\) is a bounded C1-domain, there is a varifold solution of the two-phase flow with surface tension σ > 0 in the sense of Definition 2.
In [3, Theorem 1.6] further properties are stated, which can be shown for the constructed varifold.
Further and related results: The result was extend by Yeressian [73], where the existence of axisymmetric varifold solutions in the case of axisymmetric initial values in \(\mathbb{R}^{3}\) was shown. In the case d = 2 and ρ± = η± = 1, existence of varifold solutions was also obtained by Ambrose et al. [21]. Their definitions and statements are slightly different; but the result is essentially the same. Moreover, they discuss possible defects in the surface tension functional. Earlier generalized solutions for the two-phase flow with surface tension were also constructed by Salvi [61]. But in the latter work, the meaning of the mean curvature functional is not specified and can be chosen arbitrarily within in a certain function space. Moreover, a Bernoulli free boundary problem with surface tension was discussed by Wagner [72]. Existence of varifold solutions was also obtained in [14] by a sharp interface limit of a diffuse interface model, which will be discussed in the next section. But the definition of varifold solution and their properties are slightly different. The limit system obtained in this sharp interface limit depends on the scaling of a mobility coefficient in the diffuse interface model. In one case the classical model (2), (3), (4), (5), (6), (7), and (8) is obtained; in another case the system studied in the next section is obtained.
2.3 Existence of Weak Solutions for a Navier-Stokes/Mullins-Sekerka System
In this subsection we consider only the case ρ+ = ρ− = 1 for simplicity. In this case an alternative model to the classical two-phase flow model (2), (3), (4), (5), (6), (7), and (8) is the following:
for t > 0 together with \(\Omega _{\pm }(0) = \Omega _{0}^{\pm }\). The system arises naturally as a sharp interface limit of the diffuse interface models discussed in Sect. 3.1 if the mobility coefficient m does not vanish in the limit. If m = 0 in the system above, then the equations for μ decouple from the rest of the system and can be deleted from the system. Then the system coincides with the classical model (2), (3), (4), (5), (6), (7), and (8). Here \(\mu: \Omega \times (0,\infty ) \rightarrow \mathbb{R}\) is a new quantity in the system and plays the role of a chemical potential associated to a free energy, which is \(\sigma \mathcal{H}^{d-1}\) restricted to the interface \(\Gamma (t)\). Moreover, m > 0 is a mobility coefficient, which influences the strength of a (nonlocal) diffusion in the system. The system (26), (27), (30), and (32) for v = 0 is the so-called Mullins-Sekerka system (or two-phase Hele-Shaw system), which arises as sharp interface limit of the Cahn-Hilliard equation, which models phase separation in a two-component mixture. It is well known that solutions of this system show the so-called Ostwald ripening effect in the long-time dynamics, which is the diffusion of mass from smaller droplets to larger droplets until finally one large droplet remains. This effect is also present in the full system (24), (25), (26), (27), (28), (29), (30), (31), (32), and (33).
In the following a result on existence of weak solutions for the Navier-Stokes/Mullins-Sekerka system above is discussed. It is noted that sufficiently smooth solutions of (24), (25), (26), (27), (28), (29), (30), (31), (32), and (33) satisfy the following energy dissipation identity,
where η(0) = η− and η(1) = η+ as before. This identity can be verified by multiplying (24) and (26) with v, μ, resp., integrating and using the boundary and interface conditions (26), (27), (28), (29), (30), (31), and (32). This energy equality motivates the choice of solution spaces in the weak formulation and shows that the regularization introduced for m > 0 yields an additional dissipation term. In particular, one expects \(\mu (\cdot,t) \in H^{1}(\Omega )\) for almost all \(t \in \mathbb{R}_{+}\) and, formally, using Sobolev inequality and (7), that \(H(\cdot,t) \in L^{4}(\Gamma (t))\) for d ≤ 3. This gives some indication of extra regularity properties of the interface in the model with m > 0 and is in big contrast to the classical model (the case m = 0), where no control of the mean curvature of \(\Gamma (t)\) can be derived from the energy identity in a straightforward manner.
The following result on existence of weak solutions of (24), (25), (26), (27), (28), (29), (30), (31), (32), and (33) was proved in [16].
Theorem 3 (Existence of Weak Solutions, [16, Theorem 1.1]).
Let d = 2, 3, T > 0, let \(\Omega \subseteq \mathbb{R}^{d}\) be a bounded domain with smooth boundary or let \(\Omega = \mathbb{T}^{d}\), let η(0): = η−, η(1): = η+ and σ, m > 0. Then for any \(\mathbf{v}_{0} \in L_{\sigma }^{2}(\Omega )\), \(\chi _{0} \in BV (\Omega;\{0,1\})\) there are
that satisfy (24), (25), (26), (27), (28), (29), (30), (31), (32), and (33) in the following sense: For almost all t ∈ (0, T), the phase interface ∂∗{χ(⋅ , t) = 1} has a generalized mean curvature vector \(\mathbf{H}(t) \in L^{s}(\Omega;\vert \nabla \chi (t)\vert )^{d}\), cf. [60], with s = 4 if d = 3 and 1 ≤ s < ∞ arbitrary if d = 2, such that
holds for all \(\boldsymbol{\varphi }\in C^{\infty }([0,T];C_{0,\sigma }^{\infty }(\Omega ))\) with \(\boldsymbol{\varphi }\vert _{t=T} = 0\),
holds for all \(\psi \in C^{\infty }([0,T] \times \overline{\Omega })\) with ψ |t = T = 0 and
holds for almost all 0 < t < T.
Here \(L^{s}(\Omega;\vert \nabla \chi (t)\vert )\) is the standard Lebesgue space with respect to the measure | ∇χ(t) | on \(\Omega\), and the concept of generalized mean curvature for non-smooth phase interfaces is taken from [60] and can also be found in [16, Definition 4.4].
Remark 3.
Equation (35) is the weak formulation of (24), (29), and (33). It is obtained from testing (24) with \(\boldsymbol{\varphi }\) in \(\Omega _{\pm }(t)\), integrating over \(\Omega _{+}(t) \cup \Omega _{-}(t)\) and using Gauss’ theorem, (29), and (33). Moreover, (36) is a weak formulation of (26), (30), (32), and \(\Omega _{+}(0) = \Omega _{0}^{+}\). The conditions (25), (28), and (31) are included in the choice of the function spaces, namely, \(\mathbf{v}(t) \in H_{0}^{1}(\Omega )\) for almost every t ∈ (0, T), and (27) is formulated in (37).
The proof is essentially based on a compactness result of Schätzle [62] for (d − 1)-dimensional hypersurfaces with mean curvature given as the trace of an ambient Sobolev function in \(W_{p}^{1}(\mathbb{R}^{d})\) for \(p> \frac{d} {2}\). For the application of this result, the bound of \(\nabla \mu \in L^{2}(0,T;L^{2}(\Omega ))^{d}\) obtained from (34) is used. Such a control of the curvature of the interface is missing for the classical model (2) (3) (4) (5) (6) (7), and (8), which is one of the main reasons that existence of weak solutions to the latter system is open in general if σ > 0.
3 Diffuse Interface Models
In diffuse interface models, a partial mixing of the two incompressible fluids in a thin interfacial region is assumed. In the following two fluids with mass densities ρ− and ρ+ are considered. The mass balance equation for the two fluids in local form is given by
where \(\widehat{\mathbf{J}}_{\pm }\) are the mass fluxes of the fluids + and −. Introducing the velocities \(\mathbf{v}_{\pm } =\widehat{ \mathbf{J}}_{\pm }/\rho _{\pm }\), one can rewrite the mass balance as
The further modeling now crucially depends on the way how an averaged velocity v is defined. Precise choices of v will be given below.
The mass flux of the two fluids relative to the velocity v is denoted by
and the mass balances are rewritten as
where J± are diffusive flow rates. Defining the total mass
one obtains
One observes that the classical continuity equation does not hold if \(\mathop{\mathrm{div}}\nolimits (\mathbf{J}_{+} +\,\mathbf{J}_{-})\,\neq \,0\).
Considering a conservation of linear momentum with respect to the above velocity, one obtains
where \(\widetilde{\mathbf{T}}\) is the stress tensor which has to be specified by constitutive assumptions. It turns out that \(\widetilde{\mathbf{T}}\) in general is not an objective tensor, i.e., the tensor is not invariant under a change of observer; see [20] for details. Rewriting (41) with the help of the mass conservation (40), one gets with \(\widetilde{\mathbf{J}} = \mathbf{J}_{-} + \mathbf{J}_{+}\)
The system now allows for an objective tensor, i.e., a tensor which is frame indifferent,
and one obtains
which is the classical formulation if \(\widetilde{\mathbf{J}} = 0\), which is equivalent to J+ = −J−. The work [12] and [20] give more details concerning the objectivity of the mass–momentum system with \(\widetilde{\mathbf{J}}\neq 0\).
It remains to specify the averaged velocity v with the help of the individual velocities v− and v+. Two choices are used in the literature.
The volume averaged velocity
where u− and u+ are the volume fractions of the two fluids and the mass averaged-velocity
In the following the two modeling variants which result from different choices of the averaged velocity v are discussed.
3.1 Models Based on the Volume Averaged Velocity
In the interfacial zone, the total volume occupied by each fluid is no longer conserved. Insisting on a conservation of volume during the mixing process would lead to the necessity that when fluid + flows out of a region, an amount of fluid − of the same volume would have to enter this region. Defining the specific (constant) density of the unmixed fluid by \(\tilde{\rho }_{\pm }\), one introduces the volume fraction
and the above discussion on the volume conservation leads to
which states that the excess volume is zero. Multiplying (38) with \(1/\tilde{\rho }_{\pm }\), using u− + u+ = 1 and the definition of v as the volume averaged velocity gives
From (39) one derives
where one sets \(\widetilde{\mathbf{J}}_{\pm } = \mathbf{J}_{\pm }/\tilde{\rho }_{\pm }\). Because of 
and u− + u+ = 1 we require, see [13],
Taking the difference of these two equations gives for φ = u+ − u−, the equation
where
It is also noted that (44) and (45) together with ρ = ρ+ + ρ− give
i.e., ρ is an affine linear function of φ. Using \(\rho\prime (\varphi ) = (\tilde{\rho }_{+} -\tilde{\rho }_{-})/2\), one obtains from (47)
where the relation \((\tilde{\rho }_{+} -\tilde{\rho }_{-})\mathbf{J}_{\varphi } = 2\,\widetilde{\mathbf{J}}\) holds; compare (46).
Motivated by the discussion in the introduction, one introduces a total energy density
as the sum of a kinetic and a free energy. As an example one can take \(f(\varphi,\nabla \varphi ) =\hat{\sigma } \left ( \frac{\varepsilon }{2}\vert \nabla \varphi \vert ^{2} + \frac{1} {\varepsilon } \psi (\varphi )\right )\), cf. (1). In an isothermal situation, the appropriate formulation of the second law of thermodynamics is given as the following dissipation inequality, see, e.g., [45], as follows
where V (t) is a test volume which is transported with the flow, described by v, and J e is a general energy flux which will be specified later. Using a transport theorem and the fact that the test volume is arbitrary, one obtains the local form, see [13, 50],
One can now use the Lagrange multiplier method of Liu and Müller [50, 55] to derive constitutive relations which guarantee that the second law is fulfilled. Every field (φ, v) which fulfills the dissipation inequality (49), \(\mathop{\mathrm{div}}\nolimits \mathbf{v} = 0\) and (47) also fulfill
where μ is a Lagrange multiplier which will be specified later.
Denoting by f,φ and f,∇φ, the partial derivatives with respect to φ and ∇φ one gets
where
is the material derivative. Using
yields that (50) gives after some computations
Choosing the chemical potential as
and
one ends up with the dissipation inequality
Often it is convenient, see, e.g., [45], to introduce an extra stress \(\widetilde{\mathbf{S}}\) and the pressure p such that
Here, T is the total stress tensor, and \(\widetilde{\mathbf{S}}\) is a stress tensor from which the part stemming from the hydrostatic pressure is subtracted. It will turn out that \(\widetilde{\mathbf{S}}\) contains viscous stresses and stresses that can be related to capillary forces. Due to the incompressible condition, the pressure p is still indeterminate; see also [45]. With the stress \(\widetilde{\mathbf{S}}\) one obtains
since \(\mathop{\mathrm{div}}\nolimits \mathbf{v} = 0\). The term \(\mathbf{S} =\widetilde{ \mathbf{S}} + \nabla \varphi \otimes f_{,\nabla \varphi }\) is the viscous stress tensor since it corresponds to irreversible changes of energy due to friction.
One now considers specific constitutive assumptions. For a classical Newtonian fluid, one chooses
for some φ-dependent viscosity η(φ) ≥ 0. The simplest form of the flux J φ is of Fick’s type
where m(φ) ≥ 0 in order to guarantee that the dissipation inequality is fulfilled. Choosing
gives in conclusion the following diffuse interface model
It is remarked that
which gives that the term involving \(\widetilde{\mathbf{J}}\) in the momentum equation vanishes for equal densities, i.e., if \(\tilde{\rho }_{+} =\tilde{\rho } _{-}\). In the case of equal densities, one hence recovers the famous “model H” discussed in Hohenberg and Halperin [46]. The model (52), (53), (54), and (55) was first derived in [13]. However, other diffuse interface models based on a volume averaged velocity were also studied in [29, 37]. For both models neither global nor local energy inequalities seem to be known. The model of Ding, Spelt, and Shu [37] is given by (52), (53), (54), and (55) with \(\widetilde{\mathbf{J}}\) being zero which hence drops a term which is important for the dissipation inequality.
Using the fact that the pressure can be redefined, there are a few reformulations of (52) which are convenient. Due to the identity
it is possible to redefine the pressure as follows
and one obtains instead of (52)
For the following one assumes that \(f(\varphi,\nabla \varphi ) =\hat{\sigma } \left (\varepsilon \frac{\vert \nabla \varphi \vert ^{2}} {2} + \frac{\psi (\varphi )} {\varepsilon } \right )\), cf. (1). In some situations it is more convenient to consider the formulation
It turns out that
in some sense converges in the sharp interface limit ɛ → 0 to a multiple of
where \(\delta _{\Gamma }\) is a surface Dirac distribution concentrated on the interface and \(\mathop{\mathrm{Id}}\nolimits -\boldsymbol{\nu }\otimes \boldsymbol{\nu }\) is the projection onto the interface which is up to a factor of the relevant surface stress tensor, cf. (17) below.
Moreover, using (48) one obtains that (52) is equivalent to
Furthermore, in the same way as discussed above, the right-hand side of (57) can be replaced by μ∇φ if p is replaced by \(p +\varepsilon \tfrac{\vert \nabla \varphi \vert ^{2}} {2} + \tfrac{\psi (\varphi )} {\varepsilon }\) and for the new pressure, one obtains
3.2 Model Based on the Mass Averaged Velocity
A model based on a mass averaged velocity was derived by Lowengrub and Truskinovsky [52]. They define the averaged velocity v as
In this case the mass balance becomes
which is obtained by adding both mass balances in (38). Defining the mass concentrations
one now introduces the concentration difference
as phase field variable. One now wants to model the mixing of two incompressible fluids and in the following assumes that the total density ρ depends only on the concentration difference c. Hence, it is assumed that there exists a function \(\hat{\rho }: [-1,1] \rightarrow (0,\infty )\) such that \(\rho =\hat{\rho } (c)\). Adapting a model of a simple mixture, see [47], one obtains
This condition is just the assumption of zero excess volume which was discussed earlier. In this case the functional dependence between ρ and c = c+ − c− is given as (one has to use c+ + c− = 1)
However, in what follows one allows for a more general relation \(\rho =\hat{\rho } (c)\). Taking the difference of the mass balances (39) now yields (using \(\rho _{\pm } =\hat{\rho } (c)c_{\pm }\) and c = c+ − c−)
which is, using (59), equivalent to
where j = J+ − J−. The equation (62) has to be supplemented with the momentum equation (41) which, using (59), can be rewritten as
As in Sect. 3.1 one requires a free energy inequality
where
It turns out that for the mass averaged velocity, it is more convenient to work with a free energy density \(\hat{f}\) per unit mass. For solutions of the mass and momentum equations, one obtains, using Lagrange multipliers λ ρ and μ,
which is equivalent to
Using the identity
which follows using (51), one obtains
This is true for all solutions of the mass and momentum balance equations if
and
where
Interpreting λ ρ ρ as the pressure, i.e., setting
and making the specific choices
leads the model of Lowengrub and Truskinovsky [52]
One can now use \(D_{t}\rho =\hat{\rho }\prime (c)D_{t}c\) to rewrite (65) as
Choosing
gives that (64) and (66) become
and
Sometimes also the following ansatz for \(\hat{f}\) is chosen
see, e.g., [1, 4]. In this case (64) and (66) are given as
and
In the case of a simple mixture, see (60), one has
with
One hence obtains
this then implies that (67) has the following simple divergence structure
In addition, (68) becomes
The major difference between the model studied in Sect. 3.1 which was based on a volume averaged velocity and the model studied in this section is that the model which is based on a mass balanced velocity leads to a velocity which in general is not divergence free and to a pressure-dependent chemical potential. Both facts make the analysis of this model much more involved, cf. Sect. 3.5 below. It is pointed out that both models reduce to “model H” in the case that the two mass densities ρ− and ρ+ are the same.
3.3 Analytic Results in the Case of Same Densities
In this subsection the mathematical results concerning existence and uniqueness of weak and strong solutions and partly their qualitative behavior for large times in the case that ρ(c) ≡ 1. are discussed. In this case (52), (53), (54), and (55) as well as (63), (64), (65), and (66) reduce to the system
where one has chosen \(\hat{f}(c,\nabla c) =\varepsilon \vert \nabla c\vert ^{2} + \frac{1} {\varepsilon } \psi (c)\) and \(\hat{\sigma }= 1\). The system is studied in \(\Omega \times (0,T)\), T ∈ (0, ∞], where \(\Omega \subseteq \mathbb{R}^{d}\), d = 2, 3, is a suitable domain, e.g., a bounded sufficiently smooth domain. It has to be closed by suitable initial and boundary conditions. The standard choice, which was done for most mathematical results, consists of
for suitable initial values (v0, c0). For all results mentioned in the following, it is assumed that \(\eta: \mathbb{R} \rightarrow (0,\infty )\) is sufficiently smooth, strictly positive and bounded. For existence of weak solutions, continuity of η is usually sufficient. But more smoothness is needed for higher regularity and uniqueness. In the following one assumes for simplicity that ρ = 1. However, the results will also be true for general positive constant ρ.
Before the analytic results are discussed, it is noted that every sufficiently smooth solution of (69), (70), (71), (72), (73), (74), and (75) on a suitable domain \(\Omega\) (e.g., bounded with Lipschitz boundary) satisfies
for all t ∈ (0, T), where E(c, v) = E free (c) + E kin (v) and
This is a consequence of the energy dissipation inequality (50) integrated with respect to \(x \in \Omega\) together with the boundary conditions (73) and (74). Alternatively, it follows from testing (69) with v, (71) with μ and (72) with ∂ t c as well as integration by parts. Moreover, it is often useful to replace (69) by
with the new pressure \(g = p + \frac{\varepsilon } {2}\vert \nabla c\vert ^{2} +\varepsilon ^{-1}\psi (c)\), cf. (58).
The analytic results often differ by their assumptions on the mobility m and the (homogeneous) free energy density ψ. Therefore, a brief overview of these assumptions is given. It is always assumed that \(m: \mathbb{R} \rightarrow [0,\infty )\) is sufficiently smooth and bounded. Most of the time it is assumed that the mobility coefficient m is nondegenerate, which means that m is strictly positive. In the case of a degenerate mobility, it is assumed that m(c) = 0 if and only if c ∈ {a, b}, where \(a,b \in \mathbb{R}\) represent the pure phases, which are a = −1, b = +1 in the derivation above. Moreover, a suitable behavior of m(c) as c → ±1 is assumed in the degenerate case. A canonical example is m(c) = m0(1 − c2) with m0 > 0. A mathematical advantage of the degenerate case is that it prevents the concentration c from leaving the physical interval [−1, 1]. But in most cases, one even assumes that m is a positive constant (e.g., m ≡ 1). A standard choice for ψ is that \(\psi: \mathbb{R} \rightarrow \mathbb{R}\) is a sufficiently smooth function satisfying suitable growth conditions for c → ±∞. From the physical point of view, it should be of double well type, which in particular means that ψ(c) ≥ 0 with equality if and only if c ∈ {±1}. A canonical example is
But choosing such a smooth free energy density ψ has the mathematical disadvantage that there is no mechanism known, which prevents c from leaving the physical reasonable interval [−1, 1] even if the initial value c0 attains only values in [−1, 1]. One possibility to ensure that c stays in [−1, 1] is to choose ψ as a singular free energy density, e.g., of the form
if c ∈ [−1, 1] and ψ(c) = +∞ else. Here 0 < θ < θ c and 0ln0: = 0 = lims → 0+slns. Essential properties of this choice of ψ are
Using these properties it is possible to prove existence of weak (or strong) solutions with c(x, t) ∈ (−1, 1) for almost every \(x \in \Omega\), t ∈ (0, T), in many situations if the mobility is nondegenerate. Instead of ψ more general free energy densities with the latter properties can be considered. More details will be given below.
Now the analytic results in the case of matched densities (i.e., ρ ≡ const. ) are discussed in more detail. A first result on existence of strong solutions, in the case that \(\Omega = \mathbb{R}^{2}\) and ψ is a suitably smooth double well potential, was obtained by Starovoitov [67].
More complete results were presented by Boyer [27] in the case of a shear flow in a periodic channel. More precisely, it is assumed that
with periodic boundary conditions with respect to \(x\prime \in \mathbb{R}^{d-1}\) and \(\mathbf{v}\vert _{x_{n}=\pm 1} = \pm Ue_{1}\) with U > 0. Moreover, either the mobility m is nondegenerate and ψ is a suitable smooth potential or m is degenerate and ψ = ψ1 + ψ2, where \(\psi _{1}: (-1,1) \rightarrow \mathbb{R}\) is convex such that mψ″1 has a continuous extension on [−1, 1] and ψ2 ∈ C2([−1, 1]). These assumptions are satisfied for ψ as in (79) if \(\psi _{1}(c) = \frac{\theta } {2}\left ((1 + c)\ln (1 + c) + (1 - c)\ln (1 - c)\right )\) and m(c) = 1 − c2. In the case of nondegenerate mobility, the existence of global weak solutions, which are strong and unique if either d = 2 or d = 3 and t ∈ (0, T0) for a sufficiently small T0 > 0, was shown in [27]. Furthermore, in the degenerate case the existence of weak solutions with c(t, x) ∈ [−1, 1] almost everywhere is proved. The system (69), (70), (71), and (72) was also briefly discussed by Liu and Shen [51].
In the case of a singular free energy density and for constant positive mobility, existence of weak solutions, strong well-posedness, and convergence for large times was proven in [5]. These results are now described in more detail.
Assumption 1.
Let \(\Omega \subseteq \mathbb{R}^{d}\) be a bounded domain with C3-boundary and let ψ ∈ C([−1, 1]) ∩ C2((−1, 1)) such that
for some \(\alpha \in \mathbb{R}\). Furthermore, one assumes that η ∈ C2([a, b]) is a positive function. Finally, ψ(s) is extended by + ∞ if s ∉ [−1, 1].
Definition 3 (Weak Solution).
Let 0 < T ≤ ∞. A triple (v, c, μ) such that
is called a weak solution of (69), (70), (71), (72), (73), (74), and (75) on (0, T) if
for all \(\boldsymbol{\psi }\in C_{(0)}^{\infty }([0,T) \times \Omega )^{d}\) with \(\mathop{\mathrm{div}}\nolimits \boldsymbol{\psi } = 0\),
for all \(\varphi \in C_{(0)}^{\infty }([0,T) \times \overline{\Omega })\), and if the (strong) energy inequality
holds for almost all 0 ≤ t0 < T including t0 = 0 and all t ∈ [t0, T).
Note that for the weak formulation (80), we have used (78) instead of (69).
Here \(L_{\sigma }^{2}(\Omega ) = \overline{\{\boldsymbol{\varphi } \in C_{0}^{\infty }(\Omega )^{d}:\mathop{ \mathrm{div}}\nolimits \boldsymbol{\varphi } = 0\}}^{L^{2}(\Omega ) }\), BC w ([0, T]; X) is the space of all weakly continuous and bounded functions f: [0, T] → X and \(L_{\mathop{\mathrm{loc}}\nolimits }^{2}([0,\infty );X)\) the space of all strongly measurable f: [0, ∞) → X such that f |[0,T] ∈ L2(0, T; X) for all T < ∞, where X is a Banach space. Furthermore, in the following BUC(I; X) denotes the space of all bounded and uniformly continuous f: I → X if \(I \subseteq \mathbb{R}\) is an interval.
Due to (76) sufficiently smooth solutions satisfy (83) with equality for all 0 ≤ t0 ≤ t < T. Moreover, this estimate motivates
As usual weak solutions are constructed by solving a suitable system, which approximates (69), (70), (71), (72), (73), (74), and (75) and satisfies the same kind of energy inequality. Then one passes to the limit using the bounds in the spaces above. To this end, one of the crucial points is to obtain a suitable bound on ψ′(c). To this end the assumptions on ψ due to Assumption 1 play an essential role. If one defines \(\psi _{0}(s) =\psi (s) +\alpha \frac{s^{2}} {2}\), then ψ0 ∈ C([−1, 1]) ∩ C2((−1, 1)) is convex and satisfies
If one replaces ψ by ψ0 in E free , one obtains a lower semicontinuous, convex functional on
with \(m:= \frac{1} {\vert \Omega \vert }\int _{\Omega }c_{0}(x)\,dx\). Its subgradient plays an important role in the analysis of (71) and (72) and can be characterized as follows:
Theorem 4 (Subgradient Characterization, [18, Theorem 4.3]).
Let ψ0 be as above. Moreover, one sets ϕ0(x) = +∞ for x ∉ [−1, 1] and let \(E_{0}: L_{(m)}^{2}(\Omega ) \rightarrow (-\infty,+\infty ]\) be defined as
if \(c \in H^{1}(\Omega )\) with c(x) ∈ [−1, 1] almost everywhere and E0(c) = +∞ else. Moreover, let ∂E0 be its subgradient with respect to the L2-inner product. Then
and
where \(P_{0}: L^{2}(\Omega ) \rightarrow L_{(0)}^{2}(\Omega )\) is the orthonormal projection onto \(L_{(0)}^{2}(\Omega )\). Moreover, there is some C > 0 independent of \(c \in \mathcal{D}(\partial E_{0})\) such that
The result was proven by Abels and Wilke in [18, Theorem 4.3]. Formally, one can obtain (85) by multiplying \(\partial E_{0}(c) = -\Delta c + P_{0}\psi _{0}\prime(c)\) by \(-\Delta c\). This yields
Using regularity results for elliptic equations and Young’s inequality, one obtains (85) formally. These formal arguments are justified rigorously in the proof of [18, Theorem 4.3]. Using (85) together with the a priori estimates for c and μ from the energy inequality, one obtains a bound on \(\nabla ^{2}c,\psi\prime (c) \in L_{\mathop{\mathrm{loc}}\nolimits }^{2}([0,\infty );L^{2}(\Omega ))\). Based on this one obtains:
Theorem 5 (Existence of Weak Solutions, [5, Theorem 1]).
Let m > 0 be independent of c. Then for every \(\mathbf{v}_{0} \in L_{\sigma }^{2}(\Omega )\), \(c_{0} \in H^{1}(\Omega )\) with c0(x) ∈ [−1, 1] almost everywhere there is a weak solution (v, c, μ) of (69), (70), (71), (72), (73), (74), and (75) on (0, ∞). Moreover, if d = 2, then (83) holds with equality for all 0 ≤ t0 ≤ t < ∞. Finally, every weak solution on (0, ∞) satisfies
where r = 6 if d = 3 and 1 < r < ∞ is arbitrary if d = 2 and q > 3 is independent of the solution and initial data. If additionally \(c_{0} \in H_{N}^{2}(\Omega ):=\{ u \in H^{2}(\Omega ):\nu _{\partial \Omega } \cdot \nabla u\vert _{\partial \Omega } = 0\}\) and \(-\Delta c_{0} +\psi\prime _{0}(c_{0}) \in H^{1}(\Omega )\), then it holds \(c \in BUC([0,\infty );W_{q}^{1}(\Omega ))\).
The inclusions \(\nabla ^{2}c,\psi\prime (c) \in L_{loc}^{2}([0,\infty );L^{r}(\Omega ))\) in (86) follows from a generalization of (85), Theorem 4, resp., for \(L^{r}(\Omega )\) instead of \(L^{2}(\Omega )\), cf. [5, Lemma 2]. For further regularity studies and uniqueness results, it is important that Theorem 5 provides \(c \in BUC(\delta,\infty;W_{q}^{1}(\Omega ))\) for some q > d and for all δ > 0 and δ = 0 for suitable initial data. This makes it possible to use a result on maximal regularity for an associated Stokes system with variable viscosity, cf. [5, Proposition 4], to conclude higher regularity for the velocity v in the case of small or large times and in the case d = 2, which is enough to obtain a (locally) unique solution. Then one obtains:
Theorem 6 (Uniqueness, [5, Proposition 1]).
Let m > 0 be independent of c, 0 < T ≤ ∞, q = 3 if d = 3 and let q > 2 if d = 2. Moreover, assume that \(\mathbf{v}_{0} \in W_{q,0}^{1}(\Omega ) \cap L_{\sigma }^{2}(\Omega )\) and let \(c_{0} \in H^{1}(\Omega ) \cap C^{0,1}(\overline{\Omega })\) with c0(x) ∈ [−1, 1] for all \(x \in \Omega\). If there is a weak solution (v, c, μ) of (69), (70), (71), (72), (73), (74), and (75) on (0, T) with \(\mathbf{v} \in L^{\infty }(0,T;W_{q}^{1}(\Omega ))\) and \(\nabla c \in L^{\infty }(\Omega \times (0,T))\), then any weak solution (v′, c′, μ′) of (69) (70), (71), (72), (73), (74), and (75) on (0, T) with the same initial values and \(\nabla c\prime \in L^{\infty }(\Omega \times (0,T))^{d}\) coincides with (v, c, μ).
For the following one denotes \(V _{2}^{1+j}(\Omega ) = H^{1+j}(\Omega )^{d} \cap H_{0}^{1}(\Omega )^{d} \cap L_{\sigma }^{2}(\Omega )\), j = 0, 1. Moreover, for s ∈ (0, 1) one defines \(V _{2}^{1+s}(\Omega ) = (V _{2}^{1}(\Omega ),V _{2}^{2}(\Omega ))_{s,2}\), where (. , . )s, q denotes the real interpolation functor.
Theorem 7 (Regularity of Weak Solutions, [5, Theorem 2]).
Let m > 0 be independent of c and let \(c_{0} \in H_{N}^{2}(\Omega )\) such that E free (c0) < ∞ and \(-\Delta c_{0} +\psi\prime (c_{0}) \in H^{1}(\Omega )\).
-
(i)
Let d = 2 and let \(\mathbf{v}_{0} \in V _{2}^{1+s}(\Omega )\) with s ∈ (0, 1]. Then every weak solution (v, c) of (69), (70), (71), (72), (73), (74), and (75) on (0, ∞) satisfies
$$\displaystyle{ \mathbf{v} \in L^{2}(0,\infty;H^{2+s\prime}(\Omega )) \cap H^{1}(0,\infty;H^{s\prime}(\Omega )) \cap BUC([0,\infty );H^{1+s-\varepsilon }(\Omega )) }$$for all \(s\prime \in [0, \frac{1} {2}) \cap [0,s]\) and ɛ > 0 as well as \(\nabla ^{2}c,\psi\prime (c) \in L^{\infty }(0,\infty;L^{r}(\Omega ))\) for every 1 < r < ∞. In particular, the weak solution is unique.
-
(ii)
Let d = 2, 3. Then for every weak solution (v, c, μ) of (69), (70), (71), (72), (73), (74), and (75) on (0, ∞) there is some T > 0 such that
$$\displaystyle{ \mathbf{v} \in L^{2}(T,\infty;H^{2+s}(\Omega )) \cap H^{1}(T,\infty;H^{s}(\Omega )) \cap BUC([T,\infty );H^{2-\varepsilon }(\Omega )) }$$for all \(s \in [0, \frac{1} {2})\) and all ɛ > 0 as well as \(\nabla ^{2}c,\psi\prime (c) \in L^{\infty }(T,\infty;L^{r}(\Omega ))\) with r = 6 if d = 3 and 1 < r < ∞ if d = 2.
-
(iii)
If d = 3 and \(\mathbf{v}_{0} \in V _{2}^{s+1}(\Omega )\), \(s \in (\frac{1} {2},1]\), then there is some T0 > 0 such that every weak solution (v, c) of (69), (70), (71), (72), (73), (74), and (75) on (0, T0) satisfies
$$\displaystyle{ \mathbf{v} \in L^{2}(0,T_{ 0};H^{2+s\prime}(\Omega )) \cap H^{1}(0,T_{ 0};H^{s\prime}(\Omega )) \cap BUC([0,T_{ 0}];H^{1+s-\varepsilon }(\Omega )) }$$for all \(s\prime \in [0, \frac{1} {2})\) and all ɛ > 0 as well as \(\nabla ^{2}c,\psi\prime (c) \in L^{\infty }(0,T_{0};L^{6}(\Omega ))\). In particular, the weak solution is unique on (0, T0).
The proof of the latter theorem is essentially based on the fact that \(c \in BUC([0,\infty );W_{q}^{1}(\Omega ))\) for some q > d, which implies that \(c: \overline{\Omega } \times [0,\infty ) \rightarrow \mathbb{R}\) is uniformly continuous. This makes it possible to use regularity results for the Stokes system with variable viscosity η(c), which is the linearization of the right-hand side of (69), together with regularity results for the Cahn-Hilliard equation with convection term (71) and (72).
Similar results on existence of weak solutions can also be obtained for the so-called double obstacle potential for ψ, i.e.,
But in this case (72) has to be replaced by the differential inclusion
where I[−1,1] is the indicator function of [−1, 1], i.e., I[−1,1](c) = 0 if c ∈ [−1, 1] and I[−1,1](c) = +∞ else. This double obstacle potential is the pointwise limit of ψ in (79), when θ → 0, cf. Fig. 3. It can also be shown that the corresponding solutions of (69), (70), (71), (72), (73), (74), and (75) converge as θ → 0 to solutions of the system (69), (70), (71), (72), (73), (74), and (75), cf. [1, Section 6.5] or [6].
Logarithmic free energy in (79) for θ → 0
Because of the regularity of any weak solution for large times, it is possible to prove convergence to stationary solutions as t → ∞.
Theorem 8 (Convergence to Stationary Solution, [5, Theorem 3]).
Assume that \(\psi: (-1,1) \rightarrow \mathbb{R}\) is analytic and let (v, c, μ) be a weak solution of (69), (70), (71), (72), (73), (74), and (75). Then (v(t), c(t)) ⇀t → ∞(0, c ∞ ) in \(H^{2-\varepsilon }(\Omega )^{d} \times H^{2}(\Omega )\) for all ɛ > 0 and for some \(c_{\infty }\in H^{2}(\Omega )\) with \(\psi\prime (c_{\infty }) \in L^{2}(\Omega )\) solving the stationary Cahn-Hilliard equation
The proof is based on the so-called Lojasiewicz-Simon inequality, cf. [5] for details. To prove this inequality, it is important that \(\psi: (-1,1) \rightarrow \mathbb{R}\) is analytic, which is the case for the canonical example (79).
The system (69), (70), (71), and (72) was also considered in the case of non-Newtonian fluids of power-law type. In this case 2η(c)Dv in (69) is replaced by general viscous stress tensor S(c, Dv), which satisfies suitable growth conditions with respect to an exponent p > 1. First analytic results in this case were obtained by Kim, Consiglieri, and Rodrigues [48]. They proved existence of weak solutions in the case \(p \geq \frac{3d+2} {d+2}\), d = 2, 3, using monotone operator techniques. In [44] Grasselli and Pražak discussed the longtime behavior of solutions of the system in the case \(p \geq \frac{3d+2} {d+2}\), d = 2, 3, in the case of periodic boundary conditions and a regular free energy density. For the same p existence of weak solutions with a singular free energy density f was proved by Bosia [26] in the case of a bounded domain in \(\mathbb{R}^{3}\). Moreover, the longtime behavior was studied. Finally, existence of weak solutions was shown by Abels, Diening, and Terasawa [11] in the case that \(p> \frac{2d} {d+2}\) using the parabolic Lipschitz truncation method for divergence-free vector fields developed by Breit, Diening, and Schwarzacher [30].
3.4 Analysis for the Model with General Densities Based on the Volume Averaged Velocity
In this subsection analytic results of the system (52), (53), (54), and (55) are discussed, i.e.,
in \(\Omega \times (0,T)\), where \(\Omega \subseteq \mathbb{R}^{d}\), d = 2, 3, is a bounded domain with smooth boundary and
The system is closed with the boundary and initial conditions (73), (74), and (75). Here one sets \(\hat{\sigma }= 1\) for simplicity.
Smooth solutions of (90), (91), (92), and (93) together with (73), (74), and (75) satisfy the same energy dissipation identity as in the case of same densities, i.e., (76), where c is replaced by φ and ρ = ρ(φ) in (77). In particular, this yields a priori bounds for
as in the case of same densities.
So far there are only few results on existence of solutions to the system above. The system was discussed by Abels, Depner, and Garcke in [10] and [9], where existence of weak solutions in the case of singular free energies with nondegenerate and degenerate mobility, respectively, was shown. More precisely, in the nondegenerate case the following result was shown:
Theorem 9 (Existence of Weak Solutions, [10, Theorem 3.4]).
Let \(m \in C^{1}(\mathbb{R})\) be bounded such that \(\inf _{s\in \mathbb{R}}m(s)> 0\), let Assumption 1 hold true and assume that additionally \(\lim _{s\rightarrow \pm 1}\frac{\psi\prime\prime(s)} {\psi\prime (s)} = +\infty\). Then for every \(\mathbf{v}_{0} \in L_{\sigma }^{2}(\Omega )\) and \(\varphi _{0} \in H^{1}(\Omega )\) with | φ0 | ≤ 1 almost everywhere and \(\tfrac{1} {\vert \Omega \vert }\int _{\Omega }\varphi _{0}\,dx \in (-1,1)\) there exists a weak solution (v, φ, μ) of (90), (91), (92), and (93) together with (73), (74), and (75) such that
Here the definition of weak solutions is similar to Definition 3; see [10, Definition 3.3] for the details.
The structure of the proof of Theorem 9 is as follows: System (90), (91), (92), and (93) is first approximated with the aid of a semi-implicit time discretization, which satisfies the same kind of energy identity as the continuous system. Hence one obtains a priori bounds for
where (vN, φN, μN) are suitable interpolations of the time-discretized system with discretization parameter \(h = \frac{1} {N}\). In order to pass the limit N → ∞, it is essential to obtain a bound for
which follows from Theorem 4. The latter theorem is also used to obtain existence of solutions for the discrete-time system with the aid of the Leray-Schauder principle and the theory of monotone operators.
In the case of degenerate mobility, it is assumed that \(\Psi \in C^{1}(\mathbb{R})\),
and η and \(\Omega\) are as in Assumption 1. In this case one does not obtain an a priori bound for ∇μ in \(L^{2}((0,T) \times \Omega )\). Instead one obtains an a priori bound for \(\widehat{\mathbf{J}}:= \sqrt{m(\varphi )}\nabla \mu\) and J: = m(φ)∇μ. There one has to avoid ∇μ in the weak formulation and has to formulate the equations in terms of J. More precisely, weak solutions are defined as follows, cf. [9, Definition 3.3].
Definition 4.
Let T ∈ (0, ∞), \(\mathbf{v}_{0} \in L_{\sigma }^{2}(\Omega )\) and \(\varphi _{0} \in H^{1}(\Omega )\) with | φ0 | ≤ 1 almost everywhere in \(\Omega\). Then the triple (v, φ, J) with the properties
is called a weak solution of (90), (91), (92), and (93) together with (73), (74), and (75) if the following conditions are satisfied:
for all \(\boldsymbol{\psi }\in C_{0}^{\infty }(\Omega \times (0,T))^{d}\) with \(\mathop{\mathrm{div}}\nolimits \boldsymbol{\psi } = 0\),
for all \(\zeta \in C_{0}^{\infty }((0,T);C^{1}(\overline{\Omega }))\) and
for all \(\boldsymbol{\eta }\in L^{2}(0,T;H^{1}(\Omega )^{d}) \cap L^{\infty }(\Omega \times (0,T))^{d}\) which fulfill \(\nu _{\partial \Omega } \cdot \boldsymbol{\eta }\vert _{\partial \Omega } = 0\) on \(\partial \Omega \times (0,T)\).
Here \(C_{0}^{\infty }(\Omega \times (0,T)),C_{0}^{\infty }((0,T);X)\) is the set of all smooth functions \(\varphi: \Omega \times (0,T) \rightarrow \mathbb{R}\), φ: (0, T) → X with compact support.
It is noted that (97) is a weak formulation of
Theorem 10 (Existence of Weak Solutions, [9, Theorem 3.5]).
Let the previous assumptions hold, \(\mathbf{v}_{0} \in L_{\sigma }^{2}(\Omega )\) and \(\varphi _{0} \in H^{1}(\Omega )\) with | φ0 | ≤ 1 almost everywhere in \(\Omega\). Then there exists a weak solution (v, φ, J) of (90), (91), (92), and (93) together with (73), (74), and (75) in the sense of Definition 4 . Moreover for some \(\widehat{\mathbf{J}} \in L^{2}(\Omega \times (0,T))\) it holds that \(\mathbf{J} = \sqrt{m(\varphi )}\,\,\widehat{\mathbf{J}}\) and
for all t ∈ [s, T) and almost all s ∈ [0, T) including s = 0, where E(φ(t), v(t)) is defined as in (76) with c(t) replaced by φ(t). In particular, J = 0 a.e. on the set { | φ | = 1}.
The theorem is proved by approximating m by a sequence of strictly positive mobilities m ɛ and ψ by
where ɛ > 0. Then existence of weak solutions (v ɛ , φ ɛ , μ ɛ ) for ɛ > 0 follows from Theorem 9. In order to pass to the limit, one uses the energy inequality (83). But this does not give a bound for \(\varphi _{\varepsilon } \in L^{2}(0,T;H^{2}(\Omega ))\), which is essential to pass to the limit in the weak formulation of (90). In order to obtain this bound, one tests the weak formulation of (92) with G′ ɛ (φ ɛ ), where \(G\prime\prime(s) = \frac{1} {m_{\varepsilon }(s)}\) for s ∈ (−1, 1) and G′ ɛ (0) = G ɛ (0) = 0; see [9, Proof of Lemma 3.7] for the details.
Existence of weak solutions of (90), (91), (92), and (93) together with (69), (70), (71), and (72) was proven in the case of power-law-type fluids of exponent \(p> \frac{2d+2} {d+2}\), d = 2, 3, in [8]. More precisely, 2η(φ)Dv in (90) is replaced by S(φ, Dv), where \(\mathbf{S}: \mathbb{R} \times \mathbb{R}_{sym}^{d\times d} \rightarrow \mathbb{R}_{sym}^{d\times d}\) satisfies
for all \(\mathbf{M} \in \mathbb{R}_{sym}^{d\times d}\), \(s,s_{1},s_{2} \in \mathbb{R}\) and some C, C1, ω > 0. Furthermore, the case of constant, positive mobility together with a suitable smooth free energy density ψ is considered. Unfortunately, in this case there is no mechanism, which enables to show that ψ ∈ [−1, 1]. Hence one has to modify ρ, defined as in (94) for φ ∈ [−1, 1], outside of [−1, 1] suitably such that it stays positive. But then (48) is no longer valid, and one obtains instead
Here R is an additional source term, which vanishes in the interior of {φ ∈ [−1, 1]}. In order to obtain a local dissipation inequality and global energy estimate, the equation of linear momentum (90) has to be modified to
Under these assumptions existence of weak solutions is shown with the aid of the so-called L∞-truncation method, cf. [8] for the details.
3.5 Analysis for the Model with General Densities Based on the Mass Averaged Velocity
In the following the known results on existence of weak and strong solutions for the model by Lowengrub and Truskinovsky [52] are discussed, i.e., one considers
cf. (63), (64), (65), and (66), in \(\Omega \times (0,T)\), where \(\rho =\hat{\rho } (c)\) with
\(\Omega \subset \mathbb{R}^{d}\), d = 2, 3, is a bounded domain with C3-boundary and T ∈ (0, ∞]. Moreover, one chooses
for some q ≥ 2. Usually one chooses q = 2 for these kinds of diffuse interface models. But for proving existence of weak solutions, it is necessary so far to choose q > d. The reasons will be explained below.
The system is closed by adding the boundary and initial conditions
where 0 < γ < ∞ is a friction coefficient and τ denotes the tangential part of a vector field. For the analysis it is important to choose Navier boundary conditions for v (104) instead of no-slip boundary conditions \(\mathbf{v}\vert _{\partial \Omega } = 0\) as before since this makes it possible to estimate the pressure suitably.
In the following it is assumed that \(\tilde{\rho }_{-}\neq \tilde{\rho }_{+}\), that \(\eta,m,\lambda,\psi: \mathbb{R} \rightarrow \mathbb{R}\) are sufficiently smooth and that η, λ, m are strictly positive and bounded; see [4, 7] for the precise assumptions. Similar as for the other models, smooth solutions of (100), (101), (102), (103), (104), (105), and (106) satisfy the energy dissipation identity
for all t ∈ (0, T), where E(c, v) = E free (c) + E kin (v) and
In order to get a priori estimates for the construction of weak solutions, it is essential that \(\rho =\hat{\rho } (c)\) stays positive. Note that
and
as seen in Sect. 3.2. Hence one needs a mechanism, which guarantees that c stays in [−1, 1] or at least in [−1 −δ, 1 + δ] for some sufficiently small δ > 0. Unfortunately, so far it was not possible to work with a singular free energy because of the pressure appearing in (102). But an alternative is to choose q > d, which yields an a priori bound for \(c \in L^{\infty }(0,T;W_{q}^{1}(\Omega ))\hookrightarrow L^{\infty }(0,T;C^{1-\frac{d} {q} }(\overline{\Omega }))\). In this case c can be trapped in [−1 −δ, 1 + δ] if ψ is chosen “steep enough” outside of the physical interval [−1, 1]. More precisely one has
Lemma 1 (Choice of Free Energy, [4, Lemma 2.3]).
Let R, δ > 0, q > d, and let ψ ∈ C2([−1, 1]) with ψ(c) > 0, c ∈ [−1, 1], be given. Then there is an extension \(\psi \in C^{2}(\mathbb{R})\), ψ(c) ≥ 0, ψ″(c) ≥ −M > −∞ such that for all \(c \in W_{q}^{1}(\Omega )\)
In order to construct weak or strong solutions, it is essential to reformulate (100), (101), (102), and (103) first. To this end one defines
and \(\mu =\mu _{0}+\bar{\mu }\), \(\bar{\mu }= \frac{1} {\vert \Omega \vert }\int _{\Omega }\mu \,dx\). Moreover, one decomposes \(g = g_{0} +\bar{ g}\), \(\bar{g} = \frac{1} {\vert \Omega \vert }\int _{\Omega }g\,dx\). Then (100), (101), (102), and (103) are equivalent to
together with
cf. [4, Section 3] for the details. Here the specific form of \(\hat{\rho }\) and the corresponding relations above are essentially used.
For the mathematical analysis, it is essential to use a suitable decomposition of g0, namely,
where
and \(\int _{\Omega }G(\mathbf{v})\,dx = 0\). This implies
where \(P_{\sigma }: L^{2}(\Omega )^{d} \rightarrow L_{\sigma }^{2}(\Omega )\) is the Helmholtz projection. Hence (109) is equivalent to
Here the part g1 has relatively good regularity, e.g., \(g_{1} \in L^{2}(0,\infty;L^{p}(\Omega ))\) with \(1 <p <\frac{d} {d-1}\), cf. Theorem 11 below. It is the part ∂ t G(v), which makes the analysis difficult and which does not allow to use a singular free energy as in (79). It is also noted that for the estimates of g1, it is important to consider Navier boundary conditions for v and not no-slip boundary conditions.
Because of (118) one defines:
Definition 5.
Let \(\mathbf{v}_{0} \in L^{2}(\Omega )^{d}\), \(c_{0} \in W_{q}^{1}(\Omega )\), q > d, and let \(\psi: \mathbb{R} \rightarrow [0,\infty )\) be twice continuously differentiable. Then \((\mathbf{v},g_{1},c,\mu _{0},\bar{p})\) with
where \(H_{\nu }^{1}(\Omega ) =\{ \mathbf{v} \in H^{1}(\Omega )^{d}:\nu _{\partial \Omega } \cdot \mathbf{v}\vert _{\partial \Omega } = 0\}\), and such that \(0 <\rho =\hat{\rho } (c) \in L^{\infty }(Q)\) is called a weak solution of (109), (110), (111), (112) and (104), (105), (106) if the following conditions are satisfied:
-
(i)
For every \(\boldsymbol{\varphi }\in C_{0}^{\infty }(0,\infty;H_{\nu }^{1}(\Omega ) \cap L^{\infty }(\Omega )^{d})\)
$$\displaystyle\begin{array}{rcl} & & -\int _{0}^{\infty }\int _{ \Omega }P_{\sigma }\mathbf{v} \cdot \partial _{t}\boldsymbol{\varphi }\,dx\,dt +\int _{ 0}^{\infty }\int _{ \Omega }(\mathbf{v} \cdot \nabla )\mathbf{v} \cdot \boldsymbol{\varphi }\, dx\,dt {}\\ & & \quad +\int _{ 0}^{\infty }\int _{ \Omega }\rho ^{-1}\mathbf{S}(c,D\mathbf{v}): D\boldsymbol{\varphi }\,dx\,dt +\gamma \int _{ 0}^{\infty }\int _{ \partial \Omega }\rho ^{-1}\mathbf{v}_{\tau } \cdot \boldsymbol{\varphi }_{\tau }\,d\sigma \,dt {}\\ & & \quad =\int _{ 0}^{\infty }\int _{ \Omega }g_{1}\mathop{ \mathrm{div}}\nolimits \boldsymbol{\varphi }\,dx\,dt +\int _{ 0}^{\infty }\int _{ \Omega }\left (\mu _{0}\nabla c + \nabla \rho ^{-1} \cdot \mathbf{S}(c,D\mathbf{v})\right ) \cdot \boldsymbol{\varphi }\, dx\,dt.\end{array}$$ -
(ii)
For every \(\phi \in C_{0}^{\infty }(0,\infty;C^{1}(\overline{\Omega })))\)
$$\displaystyle\begin{array}{rcl} \int _{0}^{\infty }\int _{ \Omega }\rho \partial _{t}\phi \,dx\,dt +\int _{ 0}^{\infty }\int _{ \Omega }\rho \mathbf{v} \cdot \nabla \phi \,dx\,dt& =& 0, {}\\ \int _{0}^{\infty } \int _{ \Omega }\rho c\partial _{t}\phi \,dx\,dt +\int _{ 0}^{\infty }\int _{ \Omega }\rho c\mathbf{v} \cdot \nabla \phi \,dx\,dt& =& \int _{0}^{\infty } \int _{ \Omega }m(c)\nabla \mu _{0} \cdot \nabla \phi \,dx\,dt,\end{array}$$and
$$\displaystyle\begin{array}{rcl} & & \int _{0}^{\infty }\int _{ \Omega }\rho ^{-\frac{1} {q} }(\rho \mu _{0} +\rho ^{2}\bar{p} -\psi\prime (c))\phi \,dx\,dt =\beta \int _{ 0}^{\infty }\int _{\Omega }\rho ^{-\frac{1} {q} }\rho ^{2}g_{1}\phi \,dx\,dt {}\\ & & -\beta \int _{0}^{\infty }\int _{ \Omega }G(\mathbf{v})\partial _{t}\left (\rho (c)^{-\frac{1} {q} }\rho ^{2}\phi \right )\,dx\,dt +\int _{ 0}^{\infty }\int _{\Omega }\rho ^{1-\frac{1} {q} }\vert \nabla c\vert ^{q-2}\nabla c \cdot \nabla \phi \,dx\,dt.\end{array}$$ -
(iii)
(v, c) |t = 0 = (v0, c0).
-
(iv)
(v, c, μ0) satisfy the energy inequality
$$\displaystyle\begin{array}{rcl} & & E(c(t),\mathbf{v}(t)) +\int _{ s}^{t}\int _{ \Omega }\left (\mathbf{S}(c,D\mathbf{v}): D\mathbf{v} + m(c)\vert \nabla \mu _{0}\vert ^{2}\right )\,dx\,d\tau {}\\ & & +\gamma \|\mathbf{v}_{\tau }\|_{L^{2}(\partial \Omega \times (s,t))}^{2} \leq E(c(s),\mathbf{v}(s))\end{array}$$for all t ∈ [s, ∞) and almost all 0 ≤ s < ∞ including s = 0.
Theorem 11 (Existence of Weak Solution, [4, Theorem 2.4]).
Let q > d, δ, R > 0. Moreover, let \(\psi \in C^{2}(\mathbb{R})\), ψ(c) ≥ 0, ψ″(c) ≥ −M, be given such that (108) holds. Then for every \(\mathbf{v}_{0} \in L^{2}(\Omega )^{d},c_{0} \in W_{q}^{1}(\Omega )\) with E(c0, v0) ≤ R there exists a weak solution \((\mathbf{v},g_{1},c,\mu _{0},\bar{g})\) of (109), (109), (109), (112) and (104), (104), (106) with the property that
The proof of Theorem 11 is based on a two-level approximation. First (109), (110), (111), and (112) is regularized by adding the terms \(-\delta g_{0}\frac{\mathbf{v}} {2}\) and δg0 to the left-hand sides of (109) and (111), respectively. This gives an extra-term \(-\delta \int _{\Omega }\vert g_{0}\vert ^{2}\,dx\) on the right-hand side of (107). Existence of weak solutions for the regularized system is proved by a semi-implicit time discretization. Afterward, one reformulates (109) as (118) together with the extra-term \(-\delta g_{0}\frac{\mathbf{v}} {2}\), derives suitable a priori estimate for g0, g1 and \(\bar{g}\) and passes to the limit δ → 0.
Finally, a comment on short-time existence of strong solutions in [7] the following result was shown:
Theorem 12 (Existence of Strong Solutions, [7, Theorem 1.2]).
Let \(\mathbf{v}_{0} \in H_{\nu }^{1}(\Omega ),c_{0} \in H^{2}(\Omega )\) with | c0(x) | ≤ 1 for all \(x \in \overline{\Omega }\) and \(\nu _{\partial \Omega } \cdot \nabla c_{0}\vert _{\partial \Omega } = 0\), d = 2, 3, and let the assumption throughout this subsection hold. Then there is some T > 0 such that there is a unique solution \(\mathbf{v} \in H^{1}(0,T;L_{\sigma }^{2}(\Omega ))\cap L^{2}(0,T;H^{2}(\Omega )^{d})),c \in H^{2}(0,T;H_{(0)}^{-1}(\Omega ))\cap L^{2}(0,T;H^{3}(\Omega ))\) solving (109), (110), (111), (112) and (104), (105), (106).
Here \(H_{(0)}^{-1}(\Omega )\) is the dual of \(H_{(0)}^{1}(\Omega ):= H^{1}(\Omega ) \cap L_{(0)}^{2}(\Omega )\).
The prove is based on a fixed-point argument and the unique solvability of the linearized system
where c′ corresponds to ρc. To solve the latter system, one uses the Helmholtz projection P σ to decompose \(\mathbf{v} = P_{\sigma }\mathbf{v} + \nabla G(\mathop{\mathrm{div}}\nolimits \mathbf{v})\), where \((I - P_{\sigma })\mathbf{v} = \nabla G(\mathop{\mathrm{div}}\nolimits \mathbf{v})\). Moreover, P σ and I − P σ are applied to (119). Throughout the analysis one has to solve a kind of damped plate equation of the form
up to lower order terms for some a(c0) > 0. In order to solve this equation, an abstract result by Chen and Triggiani [33] is applied. The same kind of linearized system arises in the analysis of a Korteweg-type model for compressible fluids with capillary stresses, cf. Kotschote [49]. Furthermore the linearized system differs very much from the linearized system of the model with same densities and the model with volume averaged densities.
4 Sharp Interface Limits
In this section it is shown in a formal way that the diffuse interface model of Abels, Garcke, Grün [12] (52), (53), (54), and (55) and the diffuse interface model (63), (64), (65), and (66) of Lowengrub and Truskinovsky both converge to the classical sharp interface model (2), (3), (4), (5), and (6) if the parameter ɛ tends to zero. It was already noted in the introduction that the energy
converges to a multiple of the surface energy
where \(\Gamma\) denotes the sharp interface; see [53, 54]. One would hence expect that all terms involving \(\hat{\sigma }\) will converge to terms involving interfacial energy and curvature, which is the first variation of interfacial energy. This will in fact be the case as one will see in the following analysis.
The method of formally matched asymptotic expansions , which is used in the following, is based on the assumption that for small ɛ the domain \(\Omega\) can at each time be separated into open subdomains \(\Omega ^{\pm }(t,\varepsilon )\) which are separated by a hypersurface \(\Gamma (t,\varepsilon )\). In addition, it is assumed that the solutions have an asymptotic expansion in ɛ in the bulk regions away from \(\Gamma (t,\varepsilon )\) and another suitable scaled expansion close to \(\Gamma (t,\varepsilon )\). The scaling will be needed in the x–variable as the values of the phase field φ will change its value sharply but smoothly in a region of thickness ɛ. That leads to the formation of internal layers.
These expansions then have to be matched in a region where both expansions overlap. A detailed description of the method can be found in [9, 40, 41]. For some phase field models this approach can be justified rigorously, cf. [14, 19, 31, 35].
4.1 Models Based on a Volume Averaged Velocity
In this section the system
with
and
is studied. This is basically the model (52), (53), (54), and (55) with the reformulation (58) and for simplicity m = ɛm0 is taken to be constant; see [9] for a more general case. The solution of (120), (121), (122), and (123) is always denoted by \(\mathbf{v}_{\varepsilon },\widetilde{\mathbf{J}}_{\varepsilon },\varphi _{\varepsilon },\mu _{\varepsilon }\). In addition, it is always assumed that ψ is of double well form with two global minima at ± 1.
4.1.1 Outer Expansions
It is assumed that \(\mathbf{v}_{\varepsilon },\widetilde{\mathbf{J}}_{\varepsilon },\varphi _{\varepsilon },\mu _{\varepsilon }\) have in \(\Omega (t,\varepsilon )\) an expansion of the form
Substituting these expansions into (120), (121), (122), and (123) leads to equations which have to be solved order by order.
The equation (123) gives to leading order ɛ−1
The stable solutions of this equation are ± 1, and \(\Omega ^{\pm }\) are defined to be the sets where φ0 = ±1.
The expansions to order ɛ0 of the fluid equations yield
with the scaling chosen the equation (122) is fulfilled to leading order ɛ0. The paper [9] discusses the case when the mobility is scaled to be of order one.
4.1.2 Inner Expansions and Matching Conditions
It is now assumed that the zero level sets of φ ɛ (⋅ , t) converge for ɛ → 0 to a smooth hypersurface \(\Gamma (t)\) which moves with a normal velocity \(\mathcal{V}\). As \(\Gamma (t)\) is smooth one can define the signed distance function d(x, t) of a point \(x \in \Omega\) to \(\Gamma (t)\) which is defined such that d(x, t) > 0 if \(x \in \Omega _{+}(t)\) and negative if \(x \in \Omega _{-}(t)\). Close to \(\Gamma\) the function d is smooth, and each function u(x, t) close to \(\Gamma\) is expressed in new coordinates U(s, z, t), where s is a tangential spatial coordinate on \(\Gamma\) and z(x, t) = d(x, t)∕ɛ. In the new coordinates, the relevant differential operators transform as follows:
where \(\boldsymbol{\nu }= \nabla _{x}d\) is the unit normal pointing into \(\Omega _{+}(t)\), \(\nabla _{\Gamma }\) is the spatial surface gradient on \(\Gamma\), | S | is the spectral norm of the Weingarten map S, \(\Delta _{\Gamma }\) is the Laplace-Beltrami operator on \(\Gamma (t)\), and h.o.t. denotes terms of higher order in ɛ (see the Appendix of [9] for a proof).
Furthermore, it is assumed that the functions v ɛ , p ɛ , φ ɛ , μ ɛ as functions \((\mathbf{v}_{\varepsilon },p_{\varepsilon },\Phi _{\varepsilon },M_{\varepsilon })\) in the inner variables have an expansion of the form
In an ɛ–dependent overlapping domain, the outer and inner expansions have to coincide in a suitable sense when ɛ tends to zero. This leads to the following matching conditions which are derived in [40] and [41]. At a point \(x \in \Gamma (t)\) with coordinate s, it holds
where u ±0 denotes the limit \(\mathop{\lim }\limits_{\delta \rightarrow 0}u_{0}(x\pm \delta \boldsymbol{\nu })\) at a point \(x \in \Gamma\).
4.1.3 Leading Order Equations
In the interfacial region, the equation (123) gives to leading order \(\frac{1} {\varepsilon }\):
and matching with the outer solutions gives the following boundary condition at ±∞:
The problem (124), (125) has a unique solution with the property
see, e.g., [63, Section 2.6]. This solution is chosen in what follows. The equation \(\mathop{\mathrm{div}}\nolimits \mathbf{v} = 0\) gives to the order \(\frac{1} {\varepsilon }\)
and together with the matching conditions, one obtains that \(\mathbf{V}_{0}\cdot \boldsymbol{\nu }\) needs to be constant. One hence obtains
and this gives
At order \(\frac{1} {\varepsilon }\) the diffusion-type equation (122) leads to
The matching conditions lead to ∂ z M0 → 0 and \(\Phi _{0}(z) \rightarrow \pm 1\) for z → ±∞. Hence (126) implies
and
In addition, one obtains [μ] +− = 0 and hence μ+ = μ− = M0. One now considers the momentum equation to leading order \(\frac{1} {\varepsilon ^{2}}\). Expressing ∇ x v and D x v in the new coordinates gives
With the notation \(\mathcal{E}(\mathbf{A}) = \frac{1} {2}(\mathbf{A} + \mathbf{A}^{T})\), one obtains
where \(\mathop{\mathrm{div}}\nolimits _{\Gamma }\) is the surface divergence. Using \(\partial _{z}\mathbf{V}_{0}\cdot \boldsymbol{\nu } = 0\) one obtains
and hence the momentum equation gives to leading order
The matching conditions imply that V0 is bounded, and hence the above ODE only has constant solutions. The matching property \(\mathop{\lim }\limits_{z \rightarrow \pm \infty }\mathbf{v}_{0}(z) = \mathbf{v}^{\pm }(x)\) for \(x \in \Gamma\) hence implies
4.1.4 Next-Order Equations
The equation (123) which defines the chemical potential gives to the order ɛ0
As \(\partial _{z}\Phi _{0}\) is in the kernel of the differential operator \(u\mapsto \psi\prime\prime(\Phi _{0})u - \partial _{zz}u\) the right-hand side of (127) needs to be L2-orthogonal to \(\partial _{z}\Phi _{0}\); see [9] for details on this Fredholm alternative type of argument. One hence obtains
where \(\sigma =\hat{\sigma } c_{0}\) with
It remains to derive the force balance (5) at the interface. One first observes that the term
in the interfacial region gives no contribution to the order \(\frac{1} {\varepsilon }\). Here one uses the facts that ∂ z M0 = 0 and ∂ z v0 = 0. One hence obtains that (120) to order \(\frac{1} {\varepsilon }\) gives the identity
The matching conditions require \(\mathop{\lim }\limits_{z \rightarrow \pm \infty }\partial _{z}\mathbf{V}_{1}(z) = \nabla \mathbf{v}_{0}^{\pm }(x)\nu\) and hence
Integrating (128) with respect to z now gives
The identity \(\mathcal{V} = \mathbf{v}_{0}\boldsymbol{\nu }\) hence gives
Hence, all equations which appeared in the sharp interface problem (2), (3), (4), (5), and (6) are derived.
4.1.5 The Navier-Stokes/Mullins-Sekerka System as Sharp Interface Limit
It is also possible to obtain the Navier-Stokes/Mullins-Sekerka system (24), (25), (26), (27), (28), (29), and (30) as sharp interface limit. To achieve this, one has to use a different scaling in (122). In fact (122) is replaced by
Expansions in \(\Omega _{\pm }\) immediately give
At order \(\frac{1} {\varepsilon }\) one obtains from (129) that
Matching requires \(\partial _{z}M_{1} \rightarrow \nabla \mu _{0}\cdot \boldsymbol{\nu }\), and integration of the above equation gives
which is precisely equation (30).
4.2 Sharp Interface Expansions for the Lowengrub-Truskinovsky Model
Now the sharp interface limit of the Lowengrub-Truskinovsky model
is considered. Here, \(\mathbf{S}(c,D\mathbf{v}) = 2\eta (c)D(\mathbf{v}) +\lambda (c)\mathop{\mathrm{div}}\nolimits \mathbf{v}\mathop{\mathrm{Id}}\nolimits\), \(m =\varepsilon ^{2}\hat{m}\), and it is assumed that the functional relation between ρ and c is of simple mixture type; see (61).
4.2.1 Outer Expansions
In the phases \(\Omega ^{\pm }\) one obtains as in the preceding section
and hence
This then implies
4.2.2 Inner Expansion to Leading Order
The expansions in the interface are as in the case of the volume averaged velocity with the two exceptions
In the interface the term ɛ∇c ⊗∇c will give a contribution to the order ɛ−1 which has to be balanced by the pressure. This is due to the fact that in contrast to the volume-averaged case, one does not work with μ∇c as a capillarity term. Therefore, the inner expansion of the pressure has P−1 as the leading order term.
For the capillarity-type term \(\varepsilon \mathop{\mathrm{div}}\nolimits (\nabla c \otimes \nabla c)\), one obtains
where an \(\frac{1} {\varepsilon ^{2}}\) contribution of this term in the momentum balance can be observed. Similar as in the previous section, one obtains from (132) that \(\partial _{z}\mathbf{V}_{0}\cdot \boldsymbol{\nu } = 0\) which leads to
The equation (131) gives to leading order
Matching implies
which implies ∂ z C0 ≢ 0 and hence
The momentum balance (130) gives to leading order ɛ−2
Since \(\partial _{z}\mathbf{V}_{0}\cdot \boldsymbol{\nu } = 0\) one obtains from the normal part of the above equation,
Matching requires P−1 → 0 and ∂ z C0 → 0 for z → ±∞ and hence \(\hat{P} \equiv 0\) which gives
Hence (135) boils down to
which implies after matching
The equation (133) gives to leading order ɛ−1
Using β = −ρ′∕ρ2 and \(P_{-1} = -\hat{\sigma }\vert \partial _{z}C_{0}\vert ^{2}\) gives
This ODE has a unique solution fulfilling C0(±∞) = ±1 and C0(0) = 0. In particular C0 is independent of s and t.
4.2.3 Inner Expansions to Next-Leading Order
Using (134) and \(\nabla _{\Gamma }C_{0} \equiv 0\), one obtains from the momentum balance (130) to order ɛ−1
where as above \(\mathcal{V} = \mathbf{v}_{0}\cdot \boldsymbol{\nu }\) is used which yields that the kinetic term gives no contribution. Since C0 is independent of s and t, one obtains that \(\nabla _{\Gamma }P_{-1} \equiv 0\). One computes
Hence, it follows
Integrating and using the matching conditions give similar as in Sect. 4.1.4
where one uses that ∫ ∞−∞ ∂ z (∂ z C0∂ z C1)dz = 0 which follows from matching. One hence obtains that also the Lowengrub-Truskinovsky model yields the sharp interface model (2), (3), (4), (5) and (6) in the asymptotic limit ɛ → 0.
4.3 Known Results on Sharp Interface Limits
First results on the sharp interface limits of Cahn-Hilliard/Navier-Stokes systems are for a simplified situation due to Lowengrub and Truskinovsky [52]. They used the method of formally matched asymptotic expansions. In the general case the sharp interface limit has been analyzed with formally matched asymptotic expansions by Abels, Garcke, and Grün [12], where also different scalings have been analyzed which lead to quite different asymptotic limits.
So far only very few rigorous results for the sharp interface limit exist. Abels and Röger [16] and Abels and Lengeler [14] showed convergence in the sense of varifold solutions, cf. Chen [32], for the case in which m is constant. Abels and Röger [16] studied the case of matched densities and m independent of ɛ. Abels and Lengeler [14] considered the case of a volume averaged velocity and m independent of ɛ as well as m = m(ɛ) →ɛ → 00 sublinearly, i.e., \(\frac{\varepsilon }{m(\varepsilon )} \rightarrow _{\varepsilon \rightarrow 0}0\). Moreover, it is shown that certain radially symmetric solutions of (90), (91), (92), and (93) tend to functions which will not satisfy the Young-Laplace law (6) in the limit ɛ → 0 if the mobility tends to zero faster than ɛ3. A result on a sharp interface limit to solutions which fulfill the limit equations in a stronger sense is still open.
Abels and Schaubeck [17] showed that for mobilities m tending to zero faster than ɛ3 in the convective Cahn-Hilliard equation, i.e., (93), (94) with a given velocity smooth and solenoidal field v, the surface tension term \(-\varepsilon \mathop{\mathrm{div}}\nolimits (\nabla \varphi _{\varepsilon }\otimes \nabla \varphi _{\varepsilon })\) in general does not converge to a multiple of the mean curvature vector as ɛ tends to zero. For a related Allen-Cahn/Stokes system Abels and Liu [15] are able to show converge to solutions which fulfill the sharp interface problem in a strong sense for small times.
5 Conclusions
Because of possible singularities in the interface, the mathematical description of a two-phase of macroscopically immiscible fluids remains a mathematical challenge with many open problems and questions. Weak formulations of the classical sharp interface model for two viscous incompressible, immiscible Newtonian fluids have been discussed. In the absence of surface tension, existence of weak solutions is known, but there is little control of the regularity of the interface known. In particular, it cannot be excluded that it is dense in the domain in general. In the case with surface tension, the energy estimates provide a control of the total surface measure of the interface. But existence of weak solutions is unknown since possible oscillation and concentration effects of the interface prevent from passing to the limit in the weak formulation of the mean curvature vector, which arises due to the Young-Laplace law. Moreover, a nonclassical sharp interface model is discussed, where the classical kinematic condition that the interface is transported by the fluid velocity is replaced by a convective Mullins-Sekerka equation. This model arises as the sharp interface limit of a diffuse interface model if the mobility coefficient in the diffuse interface model does not tend to zero. For this model existence of weak solutions can be shown with similar techniques as for the Mullins-Sekerka system since an additional term in the energy inequality gives rise to a suitable a priori bound of the mean curvature of the interface.
In order to describe two-phase flows beyond the occurrence of topological singularities, diffuse interface models, where the macroscopically immiscible fluids are considered as partly miscible, are an important alternative. In these models the sharp interface and the characteristic function of one phase is replaced by an order parameter, which varies smoothly, but with a steep gradient in a thin interfacial region. In the case of different densities, there are different models in dependence of choice of the mean velocity for the fluid mixture. The choice of a volume averaged velocity leads to a divergence-free velocity field and a system, which is very similar to the case of same densities. Results on existence of weak solutions for different choices of the free energy and mobility are discussed. For a barycentric/mass-averaged velocity, the velocity field is no longer divergence free, and the pressure enters the equation for the chemical potential. This leads to significant new difficulties in the mathematical analysis of this model. Moreover, the linearized system is rather different from the case of same densities. In this case existence of weak solutions is only known in the case of a free energy, which is non-quadratic in the gradient of the concentration.
Finally, the sharp interface limit of the diffuse interface models to the classical sharp interface model has been derived. This convergence can be discussed using the method of formally matched asymptotic expansions. But there are only few mathematical rigorous convergence results. In particular a proof of convergence to strong solutions of the limit equations remains an open problem, even for small times.
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Abels, H., Garcke, H. (2018). Weak Solutions and Diffuse Interface Models for IncompressibleTwo-Phase Flows. In: Giga, Y., Novotný, A. (eds) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Cham. https://doi.org/10.1007/978-3-319-13344-7_29
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