Multiphase flows are important to various engineering processes. In many applications such as contaminant transport in karst aquifers, oil recovery, the development of sinkholes, the biogeochemical processes in hyporheic zone of river ⇤ beds, the proton exchange membrane fuel cell technology and cardiovascular modeling, multiphase flows in conduits/channels and in porous media interact with each other, and therefore need to be considered together. See Fig. 2.1 for an illustration of the coupled domain. In this article, we aim to study the well-posedeness of a di↵use interface model for multiphase flows in conduits and porous media where the Navier-Stokes-Cahn-Hilliard equations (NSCH) are coupled with the Darcy-Cahn-Hilliard equations (DCH) through a set of domain interface boundary conditions.
The well-posedness of either the NSCH system or the DCH system in single domains has been intensively investigated in recent years. Boyer in [1] studies existence and uniqueness as well as asymptotic stability of solutions of the NSCH system with both regular and degenerate mobility. Global (weak solutions, strong solutions in 2D) and local well-posedness (strong solutions in 3D), and regularity of solutions are further examined by Abels [2] and more recently by Giorgini et al in [3] for the NSCH system of singular free energy densities and matched densities, see [4,5,6,7] for results regarding the NSCH type equations with general densities. Long time behavior of solutions to the NSCH system can be found in [8,9,10]. As for the DCH system (also referred to as Cahn-Hilliard-Hele-Shaw), the global existence of weak solutions is first established by Feng and Wise in [11]. Wang and Zhang [12] establish the existence and uniqueness of regular solutions (global in 2D and local in 3D) for the DCH system of variable viscosities, cf. [13] for the study on long-time behavior. Global well-posedness (resp., local) is also established by Zhao et. al. [14] in 2D (resp., 3D) for the DCH system modeling tumour growth, see also [15,16,17,18,19,20]. The CHD system with the singular potential has been extensively analysed by Giorgini et al in [16,21].
The di↵use interface model for two-phase flows in the coupled conduit and porous media setting is first derived by Han et al. in [22] via Onsager's extremum principle. The derivation only takes into account the irreversible part of the dynamics resulting in the coupling of the Stokes-Cahn-Hiliard equations in conduit and the DCH system in porous media. The existence and unique-ness of global weak finite energy solutions is shown in [23], see [24] for numerical methods solving the coupled system. A numerical model consisting of the NSCH system and Richards equation in a coupled free flow and porous media system is proposed by Chen et al. [25] in which the well-posedness is not analysed.
In this article, we propose a di↵use interface model for two-phase flows in the superposed free flow and porous media where the free flow is necessarily governed by the Navier-Stokes equations. The model comprises the NSCH system in free flow (hence incorporating the reversible dynamics) and the DCH system in porous media coupled via a set of domain interface boundary conditions. We establish the global existence of weak solutions in three dimension. Moreover, provided that there exists a strong solution (not established in this article), we show that the strong solution agrees with weak solutions (weak-strong uniqueness). These results are in parallel to those in [23] for the Cahn-Hilliard-Stokes-Darcy model. Central to our analysis is the utilization of the Lions interface boundary condition, cf. (2.10), which states that the stress in the normal direction to the domain interface including the dynamic pressure in free flow is balanced by the flow pressure in porous media. As a consequence one can show that the model obeys an energy law which implies the necessary a priori estimates for compactness argument. Compared to the work [23] for the Cahn-Hilliard-Stokes-Darcy system, the adoption of Navier-Stokes equations and the nonlinear Lions interface boundary conditions introduces extra nonlinearity and strong coupling among the equations. For establishing the existence of weak solutions we develop a divide-and-conquer strategy by taking advantage of the coupling of the equations via the chemical potential (an idea from [26]), and by application of the Leray-Schauder principle. We should also emphasize that the coupling between Cahn-Hilliard-Navier-Stokes system and Cahn-Hilliard-Darcy system poses new challenge for analysis. For instance, the uniqueness of weak solution in two dimensions and (local) existence of strong solutions remain open, even for the case of Cahn-Hilliard-Stokes-Darcy system [23].
There is a vast literature on single phase flows in the context of coupled free flow and porous media. Interested readers can refer to [27,28,29,30,31,32,33,34,35,36,37,38,39]. The rest of the article is organized as follows. In Section 2, we present the Cahn-Hilliard-Navier-Stokes-Darcy model, introduce the weak formulation and state the main theorem of the article. We prove existence of weak solutions in Section 3 based on solutions to a time-discrete elliptic system and compactness arguments. In Section 4 we establish the weak-strong uniqueness result. We give a brief derivation of the model in the Appendix.
In this section, we present the Cahn-Hilliard-Navier-Stokes-Darcy model (CH-NSD) for two phase flows of matched densities in superposed free flow and porous media; then we introduce the weak formulation of the model; finally we state the main results of this article. We will focus on the three dimensional case with the understanding that similar result holds for the two dimensional domain.
The physical setting of the problem is that there is a mixture of two fluids (say oil and water) occupying the free flow region and porous media region.
Through the domain interface of the two regions fluid in the two systems can exchange. Detailed discussion of the physical background and the derivation of the CH-NSD model are given in the Appendix. We also refer to [22] for a similar model (Cahn-Hilliard-Stokes-Darcy) where the Navier-Stokes equations are replaced by the Stokes equation equipped with some linear interface boundary conditions.
We consider a bounded domain
where ⌦ c is the free-flow region and ⌦ m is the porous media region. Let @⌦ c and @⌦ m , which are assumed to be Lipschitz continuous, denote the boundaries of ⌦ c and ⌦ m , respectively. Let = @⌦ m \ @⌦ c , m = @⌦ m \ , and c = @⌦ c \ .
A two-dimensional geometry is shown in Figure 2.1 for illustration. For analysis purpose, we take the background density ⇢ 0 and the gravitational constant g to be unity throughout the rest of the article. Define
the Navier-Stokes-Cahn-Hilliard (NSCH) equations in free flow
)
subject to the following domain interface boundary conditions on
)
)
and the following initial and boundary conditions
where n c = n m is the unit outer normal vector relative to ⌦ c , cf. the illustration in Fig. 2.1.
In the model for i = c, m, u i are the fluid velocity; p i are the pressure; i are the order parameters; w i are the chemical potentials. In addition, we denote by ⇧ the permeability matrix of the porous media, ⌫ the viscosity, M the scalar mobility function, the mixing energy density coe cient proportional to surface tension, T(u c , p c ) = 2⌫( c )D(u c ) p c I the Cauchy stress tensor with
) the rate of deformation tensor and I the 3 ⇥ 3 identity matrix. In the domain interface boundary conditions (2.9)-(2.13), ↵ B is an empirical friction coe cient, tr(⇧) is the trace of ⇧, ⌧ j (j = 1, 2) denote mutually orthogonal unit tangential vectors to the interface . We may also use P ⌧ to denote the orthogonal projection onto . The domain interface boundary condition (2.10) expresses the balance of force (including the dynamic pressure) in the normal direction of the interface, also known as the Lions interface boundary condition. The Navier slip condition (2.11) is the celebrated Beavers-Joseph-Sa↵man-Jones (BJS) interface condition [40].
One can verify that the CH-NSD system satisfies an energy law.
Then (u m , u c , , w) satisfies the following energy law:
where the total energy E and the dissipation function D are defined as
(2.19)
We now provide the weak formulation of the Cahn-Hilliard-Navier-Stokes-Darcy model (2.1)-(2.13). We use the standard notation for the Sobolev space
, where m is a nonnegative integer and
, which is a Hilbert space with inner product (u, v) H 1 = R ⌦i ru • rv dx due to the classical Poincaré inequality for functions with zero mean. Its dual space is simply denoted by ( Ḣ1 (⌦ i )) 0 . For our coupled system, the spaces that we utilize are
P ⌧ denotes the projection onto the tangent space on , i.e.
For the domain ⌦ i (i = c, m), (•, •) i denotes the L 2 inner product on the domain ⌦ j indicated by the subscript of integrated functions, and h•, •i denotes the L 2 inner product on the interface . For convenience, we define the inner product
where u c = u| ⌦c and u m = u| ⌦m , and denote
We now introduce the weak formulation for the Cahn-Hilliard-Navier-Stokes-Darcy model, similar to the weak form defined in [23] for the Cahn-Hilliard-Stokes-Darcy system.
)
and for almost all t 2 (0, T ) there hold
where
)
) Hence the initial conditions in Definition 2.1 make sense.
The following conditions on the problem parameters will be assumed throughout the article, cf. [23]:
m 2 and m are positive constants. (iii) The permeability ⇧ is isotropic, bounded from above and below, namely, ⇧ = (x)I with I being the d ⇥ d identity matrix and
The main results of this article are summarized in the following two theorems.
Theorem 2.1 (Existence of weak solutions). Suppose that the assumptions (i)-(iii) are satisfied. Then for any u 0 c 2 L 2 (⌦ c ), 0 2 H 1 (⌦), and T > 0, there exists at least one weak solution to the Cahn-Hilliard-Navier-Stokes-Darcy system (2.1)-(2.16) in the sense of Definition 2.1. Moreover, the following energy inequality holds in the sense of distribution
where E and D are defined in Eqs. (2.18) and (2.19).
Theorem 2.2 (Weak-strong uniqueness). The strong solution to the Cahn-Hilliard-Navier-Stokes-Darcy system, if exists such that
is unique in the class of the weak solutions in the sense of Definition 2.1.
Remark 2.2. The energy inequality (2.32) can be interpreted as
Several remarks are in order. First, for the purpose of establishing the weakstrong uniqueness, the regularity assumption (2.33) can be weakened as in the Cahn-Hilliard-Stokes-Darcy model [23]. Second, in the two-dimensional case, uniqueness of weak solutions to the CH-NSD system is beyond immediate reach, in contrast to the single domain case, see [1,2] for the Cahn-Hilliard-Navier-Stokes system and [21] for the Cahn-Hilliard-Darcy system. This is because the low temporal regularity of the Darcy pressure (cf. Eq. (3.61) in [23]) and the coupling of Navier-Stokes equations and Darcy equations via domain interface boundary condition leads to reduced temporal regularity of @uc @t . Finally, we point out that the (finite-time) existence of the strong solution is an outstanding open question for the coupled Cahn-Hilliard-Navier-Stokes-Darcy system. It is also open for the Navier-Stokes-Darcy type system in the case of single phase flow in superposed free flow and porous media. While the spatial regularity can be iteratively improved in individual domains, to gain further temporal regularity one needs to di↵erentiate in time the whole system due to the presence of domain interface boundary conditions. This will be pursued in another work.
In this section, we establish the existence of weak solutions by following the same semi-discretization method as in our earlier work [23] and the classical compactness argument. That is, one constructs an approximate solution which solves an elliptic system resulting from a temporal discretization of the CH-NSD system, obtains a priori estimates of the approximate solution, and finally passes to the limit.
For a large positive integer N , let = T N . The time-discrete scheme reads as follows. Given (
where
We note that F we note that the following lemma is proved in [23].
(3.7)
For the sake of simplicity, we will omit the superscript k + 1 for the unknown variables in the following subsection. We follow the idea in [41] for showing the existence of solutions to the elliptic system (3. Concerning the solvability of the chemical potential equation, i.e.
the following result is essentially proved in [22].
For a given function w 2 H 1 (⌦), there is a unique solution 2 H 3 (⌦) to the problem (3.8). Moreover, the solution operator (w) : H 1 (⌦) 7 ! H 3 (⌦) is bounded and continuous in the strong topology.
The equation (3.3) can be written as
From Lemma 3.2 we know that is the unique solution of the equation (3.8) for a given w 2 H 1 (⌦). So we can define the source terms f c = w c r c and f m = w m r m , where f c and f m are viewed as functions of w c and w m , respectively.
To establish the well-posedness of (3.9), we define an equivalent norm on the
Proof. We employ the Galerkin method for showing existence of solutions. Since the spaces X c,div and Q m are separable Hilbert spaces, there exists a se-
m . Then a Galerkin approximation to the problem (3.9) is to find
defined by
It is clear that F n is continuous. Next, we recall the definition of ã in (3.5), perform integration by parts and calculate
where
Hence there exists a solution (u c,n , p m,n ) to the Eqs. (3.11).
Now we derive some a priori estimates of (u c,n , p m,n ). By performing inte-gration by parts, one notes the identity
Choosing
By the identity (3.14), the nonlinear term in (3.15) vanishes, i.e.
Since X c,div ⇥ Q m is a reflexive Hilbert space, there exists a subsequence still denoted by {(u c,n , p m,n )} n2N and a pair (u c , p m ) 2 X c,div ⇥ Q m such that
p m,n ! p m strongly in L 2 (⌦ m ).
(3.20)
To pass to the limit in the nonlinear term, one notes that
By the identity (3.14), and the convergence (3.17), (3.18), one concludes that
Then passing to the limit in (3.11) with n ! 1 we find that
where v c is linear combination of a 1 , ..., a n , ..., and q m is linear combination of Given w 2 H 1 (⌦), let 2 H 3 (⌦) be the unique solution to Eq. (3.8)
according to Lemma 3.2. We show that the mapping w 2
W via Eqs. (3.9) is completely continuous. Suppose (u i c , p i m ), i = 1, 2 are two solutions corresponding to f i , i = 1, 2 respectively. Define
One obtains
where one has applied the identity (3.14) in treating the nonlinear terms. Taking (v c , q) = (u e c , p e m ) in Eqs. (3.22), and noting that
one derives for su ciently small and that
Hence the solution depends continuously on f in the strong topology. On the other hand, the solution operator w 2
completely continuous. This completes the proof.
The following lemma is obvious.
Moreover, there holds the discrete energy law
Proof. Here we apply the Leray-Schauder principle. One defines an operator By taking = w in (3.28), we have
3), performing integration by parts and adding the results together, we have
(3.30)
It follows immediately that
2), we also have
Recall that = T N for T > 0 and a positive integer N , and that We define the approximate solutions to Eqs. (2.27)-(2.30) as follows
With these definitions, one deduces the following equations, cf. (3.1)-(3.3):
and the equation
with initial conditions
As in [23], we also interpolate the discrete-in-time energy and dissipation function introducing
The time-discrete energy law translates to
Integrating (3.39) from 0 to T one immediately derives the following estimates
where the constant C depends on E(u 0 c , 0 ). Based on these estimates and Eqs.
) inequality, the interpolation inequality [44,45] and Sobolev inequality, we have
Hence the inequality (3.46) follows from Eq. (3.32), the estimates (3.40), (3.42) and (3.44).
By the identity (3.14) and the interpolation inequality, one has
It then follows from Eq. (3.35), the trace inequality, the inequality (3.48) and Korn's inequality that
Since the right hand side of (3.50) is in L 4 3 (0, T ), the estimate (3.47) is thus proved. This completes the proof of the lemma.
We are now ready to pass to the limit and prove the main Theorem 2.1.
Proof. The estimates in (3.40)-(3.45) imply the existence of
such that the following convergence (of subsequences) holds as ! 0 û c ! u c weakly ⇤ in L 1 (0, T ; L 2 (⌦ c )), (3.51) weakly in L 2 (0, T ; H 1 (⌦ c )), (3.52)
By the definition of û c and u c , ˆ , ˜ and , we also have
Likewise, one has k ˆ ˜ k L 2 (0,T ;H 1 (⌦)) ! 0 as ! 0. Thus the sequences c and , respectively. Since 2 L 1 0, T ; H 1 (⌦) \L 2 0, T ; H 3 (⌦) and @ @t 2 L 8 5 0, T ; (H 1 (⌦)) 0 , the Aubin-Lions-Simon lemma (cf. [46] Corollary 4) yields
(3.60)
Due to the fact that 2 L 1 0, T ; H 1 (⌦) and @ @t 2 L 8 5 0, T ; (H 1 (⌦)) 0 , it follows that (cf. [42] pp. 178)
Similarly, one has that
and that
by the strong convergence (3.60), one readily derives that
Likewise, the weak strong convergence implies that
in the sense of distributions. For the nonlinear interface term, one has
In addition by assumptions ( 1)-( 2), one also has
These convergence results allow us pass to the limit in Eqs.
Finally, we show that weak solutions satisfy the energy inequality (2.32). The argument is entirely the same as in [23]. We reproduce it here for completeness.
Multiplying the inequality (3.39) by h(t) for h 2 C 1 (0, T ) with h 0, h(T ) = 0 and integrating, one derives
By the strong convergence (3.60), (3.62), the weak convergence (
, one passes to the limit to obtain
where the last inequality follows from the lower semi-continuity of norms and the almost everywhere convergence of ⌫ and M . The energy inequality (2.32) is thus established. This completes the proof of Theorem 2.1.
In this section we prove the weak-strong uniqueness (Theorem 2.2). We largely follow the lines of proof from [23] for the Cahn-Hilliard-Stokes-Darcy system. Special care is paid to the treatment of the nonlinear advection term in the Navier-Stokes equation and the nonlinear Lions interface boundary conditions.
Proof. Suppose ( ũc , p m , ũm , ˜ , w) is a strong solution to the Cahn-Hilliard-Navier-Stokes-Darcy system such that
It follows from Eqs. (2.4) and (2.8) that w 2 L 1 0, T ; L 2 (⌦) \L 2 0, T ; H 2 (⌦) , hence w 2 L 4 0, T ; H 1 (⌦) by interpolation. It then follows from the equations that
Owing to the regularity, one can use ( w, ũc , ũm ) as test functions, which gives the energy equality (2.17). That is
For the weak solution (u c , p m , u m , , w) in the sense of Definition 2.1, the energy inequality (2.32) holds (Theorem 2.1), i.e.
Since ũc 2 L 1 (0, T ; X c,div ), for almost all t 2 (0, T ) it permits to use v c = ũc and q m = 0 as test functions in Eq. (2.29). Meanwhile one multiplies by u c the strong form (2.5) for ũc , and performs integration by parts. Adding together the resultants gives
Likewise,
To deal with the Cahn-Hilliard equations, one notes that ˜ 2 L 4 0, T ; H 3 (⌦) , since by the Gagliardo-Nirenberg inequality
Hence in view of Eq. (2.26), for almost every t 2 (0, T ), ( ✏ ˜ ) can be used as a test function in the weak form (2.27). One obtains
Now one adds together Eqs. (4.1) and (4.2), then subtracts Eqs. (4.5) and (4.6)
Since 2 L 1 0, T ; H 1 (⌦) \L 2 0, T ; H 3 (⌦) , it follows that 2 L 8 0, T ; L 1 (⌦) by interpolation. Now
which implies that f ( ) 2 L 4 (0, T ; H 1 (⌦)). Hence f ( ) can be used as a test function in Eq. (2.27). Owing to the monotonicity of 3 , one has for a.e.
Eq. (4.7) can then be written as
where each I j corresponds to the jth line on the right hand side of the inequality (4.8).
The term I 1 is estimated as follows
where is a constant to be determined later, and one has utilized the interpolation, as well as the lower and upper bounds of the viscosity ⌫.
Recall that ˜ is of mean zero. To estimate the rest of the terms, we notice that the Gagliardo-Nirenberg inequality and the Poincaré inequality imply
It follows from the definition of the chemical potential, cf. Eq. (2.28), that
We estimate I j , j = 2 • • • 8 as follows. By the Lipschitz continuity of ⌫, one obtains
Likewise, one has
By the trace theorem and Sobolev imbedding, i.e., H 1 (⌦) ,! H 1 2 ( ) ,! L 4 ( ), one has
Upon performing integration by parts, one derives the estimate for I 6 analogous to the one for I 2
For I 7 , by (4.11), (4.10) and (4.12), one has
where we have used inequality (4.12). Finally,
Collecting the inequalities (4.9), and (4.13)-(4.19), choosing su ciently small , one derives from (4.8) that
Noting that 2 L 8 0, T ; L 1 (⌦) and w 2 L 4 0, T ; H 1 (⌦) , it follows that h 2 L 1 (0, T ). Gronwall's inequality and Poincare's inequality then imply that
This completes the proof of Theorem 2.2.
with S a symmetric tensor, F c the force density, J c the di↵usive flux, to be determined. The total energy of the free flow is
where the first term is the total kinetic energy, and the second term represents the total free energy associated with the free flow. As in our work [22], we identify the dissipation in ⌦ c as
where the first term is due to chemical di↵usion, the second term is due to viscosity, and the last term is because of friction as a result of fluid particles slipping along the domain interface . The friction mechanism along the domain interface is motivated by the study of single phase flow in superposed free flow and porous media, cf [22] and references therein.
Likewise in ⌦ m , we postulate the two-phase flow in porous media satisfies the following conservation of mass
The fluid equations will be derived through the variational procedure. The total energy and dissipation in porous media are as follows
where the second term in m represents the Darcy damping in porous media.
Before we derive the forms of S, F c , J c and F m , J m , we prescribe boundary conditions. On c , the no-slip no penetration boundary condition u c = 0 is imposed for velocity, and no chemical flux condition J c • n c = 0 is imposed.
Similarly, one imposes u m • n m = 0 and J m • n c = 0 on m . On , for conservation of mass one naturally imposes the following continuity interface boundary conditions This completes the derivation of the model.