Regularized Isothermal Phase-Field Type Model of a Two-Phase Compressible Fluid and Its One-Dimensional Spatial Discretization

Abstract

We consider a quasi-hydrodynamic regularized model of the phase field type that describes the dynamics of an isothermal compressible two-phase viscous mixture with allowance for interphase effects. The dissipativity of the model, i.e., the lack of growth in the complete energy of the closed system as it tends to an equilibrium state, is discussed. A spatially discrete approximation to the one-dimensional model that possesses the dissipativity property with respect to total energy is constructed for the plane-parallel case. The working capacity of the discretization constructed is demonstrated using the spinodal decomposition of the two-phase mixture.

INTRODUCTION

Multiphase multicomponent microflows, in which capillary effects play a significant role, often occur in nature and technology. They are encountered in motors, reactors, pipelines, pores of container rocks, living organisms, and many other natural and technical systems. Therefore, creating new and perfecting the existing mathematical models to describe such systems and developing counterpart numerical methods is a topical problem.

Depending on the typical spatial scale of the problem, various approaches are used to construct the mathematical model for describing the multiphase system. In the present paper, we consider models in which the interface and its dynamics are resolved. All such models can be divided into two groups. The first group includes models that represent the interface as a mathematical surface (of “zero” thickness) with prescribed contact conditions. These conditions determine the nature of interaction between the phases and are an indispensable part of the model. The interface position can be tracked both explicitly (on Lagrangian meshes) and implicitly (on Eulerian meshes). In the latter case, one can use, for example, the Volume of Fluid (VoF) method [1] or the level set method [2]. The second group incorporates models with a diffuse boundary [3,4,5,6,7], in which the interface is described by a thin layer of finite thickness within which the properties of the medium undergo “rapid” but smooth changes.

These models are based on using a specially defined (in space) function, referred to as the order parameter, that plays the role of a phase indicator. The order parameter can be one of the medium characteristics (for example, density or concentration) or, alternatively, some artificially introduced variable. Note that there can be several order parameters and they can be scalars or quantities of a higher tensor dimension. One can consider phase-field models [8, 9] as a particular case of models with diffuse boundary. The nature of interaction between the phases is determined in these models by a special form of the Helmholtz free energy (or another thermodynamic potential of the system): it depends on both the order parameter and its spatial derivatives (this is why such models are sometimes called “weakly nonlocal” or “gradient”); in this case, the dependence on the order parameter is nonconvex. The indicated special form of free energy determines the thickness of the interface and the interphase energy (the surface tension coefficient in problems of hydrodynamics). Phase field models quite naturally, from the viewpoint of the original mathematical model, describe such phenomena as coagulation and fragmentation of droplets (which are topological changes in the spatial distribution of phases).

Note that no explicit concept of a phase is introduced in the models under consideration. However, in view of the special form of free energy, subdomains with virtually uniform (in space) composition form the flow range. It is these subdomains that are interpreted as separate phases. Phase-field type models are exemplified by the Navier–Stokes–Cahn–Hilliard (NSCH) equations [3, 5], which describe the dynamics of a two-component two-phase mixture. The mass or volume concentration of one of the components serves as the order parameter in the NSCH equations. Another example of frequently used phase field models is Navier–Stokes–Korteweg (NSK) equations [3, 4, 10], which describe the dynamics of a one-component two-phase system (a liquid and its vapor). The mass density is considered in this model as the order parameter.

Liu et al. [11] (see also [12]) considered a system that describes the dynamics of a compressible two-phase two-component mixture. In this case, the medium is treated as one-velocity. The order parameters in this system are the mass densities of components (rather than concentrations as in the NSCH model). This choice of phase fields in some sense combines the NSCH and NSK systems. Similar systems in which molar densities are taken for the phase fields were considered in [12,13,14,15,16]. The model in [13,14,15] was constructed with the use of the density functional method.

In the present paper, for the isothermal case, we suggest a QHD-regularized version of the system proposed in [11] (see Eqs. (1) and (2) below). The dissipativity (in the sense of the lack of growth of total energy) of the system under study is discussed. The problem of constructing a spatial discretization is considered for the one-dimensional plane-parallel case. In this case, the discretization is constructed with allowance for the conservation of the system total energy for a discrete analog of the property of dissipativity. It is important that this property be satisfied at the discrete level. In particular, as was noted in [4, 17, 18], this allows one to get rid of the so-called “parasitic currents,” a numerical artifact that is an eddy-like velocity field in a neighborhood of the interface that does not fade as the system approaches the equilibrium state. To this end, it was proposed in [4, 17, 18] to use a representation of capillary stresses in a potential form. In its turn, such a representation leads to a momentum-nonconservative difference scheme, but this is not critical in view of no shock waves or other strong discontinuities being present. Such discretizations were constructed for QHD-regularized NSCH equations in the papers [19, 20].

QHD-regularization of various models in the continuum mechanics presumes that, in the general case, the mass density of the flow of a mixture differs from the average momentum of unit volume, with the case of the two being equal not excluded. Thus, QHD-regularized models include original models as a special case. In view of the above assumption, additional terms of dissipative nature arise in the equations of the original model. On the one hand, these terms allow one to solve relatively simple-to-implement explicit stable central difference schemes. On the other hand, in some cases they ensure more precise coincidence with the results of experiments [21,22,23]. Note that similar regularization techniques were also considered in the papers [24, 25].

QHD-regularized (and also related quasi-gasdynamic) models of the dynamics of viscous fluid, magnetic hydrodynamics, shallow water etc. have been successfully applied before [21,22,23, 26]. For multiphase multicomponent models with surface effects, a QHD-regularization was constructed in [27]. Its particular case corresponding to the isothermal Navier–Stokes–Cahn–Hilliard model was considered in the papers [19, 20, 28, 29].

This paper is organized as follows. Section 1 introduces the notation used in the paper. In Sec. 2, we present the isothermal two-component phase-field model in [11] and its QHD-regularization. In Sec. 3, we discuss the potential form of capillary stresses. Section Sec. 4 proves the dissipativity of the regularized model. In Sec. 5, we propose a space-discrete difference scheme for the one-dimensional plane-parallel case and prove its dissipativity. In Sec. 6, following the paper [11], the equations of the model are rendered dimensionless. In the concluding section 7, we carry out numerical modeling of the spinodal decomposition and analyze the dynamics of the system total energy.

1. NOTATION

For convenience, let us list the notation used in the paper. Consider a bounded domain $\Omega \subset \mathbb {R}^n$ ($n=1$, $2$, or $3$) with piecewise smooth boundary $\partial \Omega$, $\overline {\Omega }=\Omega \cup \partial \Omega$. In $\mathbb {R}^n$, define a Cartesian coordinate system $Ox_1\ldots x_n$ with basis unit vectors $\boldsymbol {e}_1$, . . . , $\boldsymbol {e}_n$. For a vector (point) $\boldsymbol {x}\in \mathbb {R}^n$, we denote its $i$th component in this basis by $x_i$, i.e., $\boldsymbol {x}:=(x_1,\ldots ,x_n)^{\textrm {T}}$, and the time variable, by $t$. Hereinafter the symbol “$:=$” designates equality by definition.

We will index the spatial coordinates by Latin letters $i$ and $j$ and use Greek letters $\alpha$, $\beta$, $\kappa$, and $\upsilon$ for the numbers of mixture components. In this case, if not stated otherwise, repeated indices $i$ and $j$ assume summation from $1$ to $n$ and repeated indices $\alpha$ and $\beta$ indicate summation from $1$ to $2$ (according to the number of mixture components); no summation is performed over repeated indices $\kappa$ and $\upsilon$.

Let $\boldsymbol {u}=u_i\boldsymbol {e}_i$ and $\boldsymbol {v}=v_i\boldsymbol {e}_i$ be arbitrary smooth vector fields, let $\mathbf {A}=A_{ij}\boldsymbol {e}_i\otimes \boldsymbol {e}_j$ and $\mathbf {B}=B_{ij}\boldsymbol {e}_i\otimes \boldsymbol {e}_j$ be arbitrary smooth tensor fields of the second rank, and let $\boldsymbol {e}_i\otimes \boldsymbol {e}_j$ be the basis dyad. In the sequel, we will use the following notation and operations: $\partial _i:=\partial /\partial x_i$, $\partial _t:=\partial /\partial t$, $\boldsymbol {u}\cdot \boldsymbol {v}:=u_iv_i$ is the inner product, $|\boldsymbol {u}|^2:=\boldsymbol {u}\cdot \boldsymbol {u}$, $\mathbf {A}\boldsymbol {u}\equiv \mathbf { A}\cdot \boldsymbol {u}:=A_{ij}u_j\boldsymbol {e}_i$, $\boldsymbol {u}\cdot \mathbf {A}:=A_{ij}u_i\boldsymbol {e}_j$, $\nabla :=(\partial _i)\boldsymbol {e}_i$ stands for the gradient, $\mathop {\mathrm {div}}\thinspace \mathbf {A}\equiv \nabla \cdot \mathbf {A}:=(\partial _i{A}_{ij})\boldsymbol {e}_j$ is the divergence of a tensor, $\mathbf {A}\boldsymbol {:}\mathbf {B}:={A}_{ij}{B}_{ij}$ is the double inner product, $|\mathbf {A}|^2:=\mathbf {A}\boldsymbol {:}\mathbf {A}$, $\boldsymbol {u}\otimes \boldsymbol {v}:=u_iv_j\boldsymbol {e}_i\otimes \boldsymbol {e}_j$ stands for the tensor product of vectors, $\delta _{ij}$ is Kronecker’s delta, and $\mathbf {I}=\delta _{ij}\boldsymbol {e}_i\otimes \boldsymbol {e}_j$ is the identity tensor.

2. REGULARIZED MODEL OF A TWO-COMPONENT TWO-PHASE FLUID

The system of equations describing the dynamics of a two-component two-phase isothermal compressible viscous fluid in the domain $\Omega$ allowing for interphase effects and neglecting gravity has the form [11]:

$$ \partial _t\rho _\alpha +\mathop {\mathrm {div}}\thinspace (\rho _\alpha \boldsymbol {u}) =(-1)^{\alpha +1}\mathop {\mathrm {div}}\thinspace (M\nabla \hat {\mu }_{12}),\quad \alpha =1,2, $$

(1)

$$ \partial _t(\rho \boldsymbol {u})+\mathop {\mathrm {div}}\thinspace (\rho \boldsymbol {u}\otimes \boldsymbol {u})+\nabla p =\mathop {\mathrm {div}}\thinspace \boldsymbol {\Pi }^{\mathrm {c}} +\mathop {\mathrm {div}}\thinspace \boldsymbol {\Pi }^{\mathrm {NS}}, $$

(2)

where $\boldsymbol {u}(\boldsymbol {x}, t)$ is the mass-averaged fluid velocity, $\rho _\alpha (\boldsymbol {x}, t)>0$ is the density of component $\alpha$, and $\rho :=\rho _1+\rho _2$ is the total density of the mixture. To determine the remaining variables in this system, we preliminarily introduce the volume density of the Helmholtz free energy $\psi$ [11]

$$ \psi (\rho _1,\rho _2,\nabla \rho _1,\nabla \rho _2):= \psi _0(\rho _1,\rho _2)+\frac 12\lambda _{\alpha \beta }\nabla \rho _\alpha \cdot \nabla \rho _\beta ,$$

(3)

$$ \psi _0(\rho _1,\rho _2) :=k_{\mathrm {B}}T\frac {\rho _\alpha }{m_\alpha }\ln \bigg [\frac {\rho _\alpha {\Lambda }^3_\alpha }{m_\alpha (1-\phi )}\bigg ] - k_{\mathrm {B}}T\frac {\rho _\alpha }{m_\alpha } -a_{\alpha \beta }\frac {\rho _\alpha }{m_\alpha } \frac {\rho _\beta }{m_\beta }, $$

(4)

where $m_\alpha$ is the molecular mass of component $\alpha$, ${\Lambda }_\alpha =(2\pi m_\alpha k_{\mathrm {B}} T)^{-1/2}\hbar$, $\hbar$ is the Planck constant, $\phi =b_\alpha \rho _\alpha /m_\alpha$ is the volume fraction occupied by the molecules, $b_\alpha$ is the molecular volume of component $\alpha$, and $k_{\mathrm {B}}$ is the Boltzmann constant. The function $\psi _0(\rho _1,\rho _2)$ is the volume density of the homogeneous part of Helmholtz’ free energy of the mixture. The capillary coefficients $\lambda _{\kappa \upsilon }>0$ are constants calculated by the formula $\lambda _{\kappa \upsilon }:=k_{\mathrm {B}}TD_{\kappa \upsilon }/(m_\kappa m_\upsilon )$, $\kappa ,\upsilon =1,2$, where $D_{\kappa \upsilon }=D_{\upsilon \kappa }>0$ are constant coefficients. Recall that no summation is performed over the indices $\kappa$ and $\upsilon$.

The pressure $p$ is linked to the densities $\rho _1$ and $\rho _2$ by the relation

$$ p(\rho _1,\rho _2)=\rho _1\mu _1 +\rho _2\mu _2-\psi _0(\rho _1,\rho _2),$$

where $\mu _\alpha (\rho _1,\rho _2)=\partial _{\rho _\alpha }\psi _0$ is the (classical) chemical potential of component $\alpha$.

The Navier–Stokes viscous stress tensor is prescribed in the form

$$ \boldsymbol {\mathrm \Pi }^{\mathrm {NS}}:=2\eta \mathbf {D} +\biggl (\zeta -\frac 23\eta \biggr )(\mathop {\mathrm {div}}\thinspace \boldsymbol {u})\mathbf {I},\quad \mathbf { D}(\boldsymbol {u}):=\frac 12(\nabla \otimes \boldsymbol {u}+ \left (\nabla \otimes \boldsymbol {u})^{\textrm {T}}\right ),$$

where $\eta (\rho _1,\rho _2)>0$ and $\zeta (\rho _1,\rho _2)\geq 0$ are the dynamic and volume viscosity coefficients, respectively. Further, we set $\eta (\rho _1,\rho _2)=\rho _\alpha \nu _\alpha$, where the $\nu _\alpha >0$ are the constant kinematic viscosity coefficients of component $\alpha =1,2$.

The capillary stress tensor has the form

$$ \boldsymbol {\Pi }^{\mathrm {c}}:= \biggl (\lambda _{\alpha \beta } \rho _\alpha \Delta \rho _\beta +\frac 12\lambda _{\alpha \beta }\nabla \rho _\alpha \cdot \nabla \rho _\beta \biggr )\mathbf {I} -\lambda _{\alpha \beta }\nabla \rho _\alpha \otimes \nabla \rho _\beta .$$

Note that $\boldsymbol {\Pi }^{\mathrm {c}}\approx 0$ in domains where $\rho _\alpha \approx \mathrm{const}\,$. Thus, the tensor $\boldsymbol {\Pi }^{\mathrm {c}}$ significantly differs from zero only within the interphase boundary. The tensor $\boldsymbol {\Pi }^{\mathrm {c}}$, together with the nonconvex dependence of the function $\psi _0$ on $\rho _1$ and $\rho _2$, permits one to take the interphase tension into account.

On the right-hand side of Eqs. (1), the coefficient $M(\rho _1,\rho _2):=M_0\rho _1\rho _2/\rho ^2$ is the mobility of components, where $M_0>0$ is a constant; $\hat {\mu }_{12}:=T^{-1}(\hat {\mu }_1-\hat {\mu }_2)$, where $T>0$ is the temperature, which is a constant parameter by virtue of the assumption about the problem being isothermal; $\hat {\mu }_\alpha$ is the generalized chemical potential of component $\alpha$, and

$$ \hat {\mu }_\alpha (\rho _1,\rho _2,\Delta \rho _1,\Delta \rho _2)=\mu _\alpha (\rho _1,\rho _2) -\lambda _{\alpha \beta }\Delta \rho _\beta .$$

(5)

System (1), (2) is equipped with the boundary conditions

$$ \boldsymbol {u}=\boldsymbol {0},\quad \boldsymbol {n}\cdot \nabla \rho _\alpha =0,\quad \boldsymbol {n}\cdot \nabla \hat \mu _\alpha =0 \quad \text {on}\quad \partial \Omega $$

(6)

and the initial conditions

$$ \rho _\alpha (\boldsymbol {x}, 0)=\rho _{\alpha ,0}(\boldsymbol {x}),\quad \boldsymbol {u}(\boldsymbol {x},0)=\boldsymbol {u}_0(\boldsymbol {x})\quad \text {in}\quad \Omega .$$

Here $\boldsymbol {n}$ is the unit outward normal to the domain boundary $\partial \Omega$. The first condition in (6) corresponds to the standard no-slip condition, the second condition prescribes a neutral wetting angle (the interface is perpendicular to the domain boundary at the points where they intersect), and the third condition ensures the lack of mass flux of the component $\alpha$ through the domain boundary.

Consider the QHD-regularization of system (1), (2):

$$ \partial _t\rho _\alpha +\mathop {\mathrm {div}}\thinspace (\rho _\alpha \boldsymbol {u}_m) =(-1)^{\alpha +1}\mathop {\mathrm {div}}\thinspace (M\nabla \hat {\mu }_{12}),\quad \alpha =1,2, $$

(7)

$$ \partial _t(\rho \boldsymbol {u})+\mathop {\mathrm {div}}\thinspace (\rho \boldsymbol {u}_m\otimes \boldsymbol {u})+\nabla p =\mathop {\mathrm {div}}\thinspace \boldsymbol {\Pi }^{\mathrm {c}} +\mathop {\mathrm {div}}\thinspace \boldsymbol {\Pi }^{\mathrm {NS}} +\mathop {\mathrm {div}}\thinspace \boldsymbol {\Pi }^\tau . $$

(8)

In both equations (7) and (8), the expression $\rho \boldsymbol {u}$ has been replaced by $\rho \boldsymbol {u}_m$, where $\boldsymbol {u}_m=\boldsymbol {u}-\boldsymbol {w}$ is the regularized velocity. Also, on the right-hand side of the momentum balance equation (8) we added the term $\mathop {\mathrm {div}}\thinspace \boldsymbol {\Pi }^\tau$, where $\boldsymbol {\Pi }^\tau :=\rho \boldsymbol {u}\otimes \boldsymbol {w}$ is the regularizing stress tensor. A thermodynamically consistent derivation of Eqs. (7) and (8) can be conducted by analogy with the paper [27].

The presence of the auxiliary term $\boldsymbol {w}$ in $\boldsymbol {u}_m$ provides a basis for the QHD-regularization of various models of continuum mechanics and corresponds to the mass flux density $\rho \boldsymbol {u}_m$ being, in general, distinct from the average momentum of the volume unit $\rho \boldsymbol {u}$, with them being equal not ruled out. In other words, we do not presume the standard hypothesis about the average momentum of a unit volume being equal to the mass flux density to be satisfied [23]. Similar to other QHD-regularized models (see, e.g., [23, 27, 20]), the expression for $\boldsymbol {w}$ (see the proof of Theorem 1 below) is constructed taking into account the necessity of the condition of dissipativity of total energy (entropy in the nonisothermal case) to be satisfied and has the form

$$ \boldsymbol {w}=\rho ^{-1}\tau [\rho (\boldsymbol {u}\cdot \nabla )\boldsymbol {u}+\rho _\alpha \nabla \hat \mu _\alpha ]. $$

(9)

Here the relaxation parameter $\tau (\rho _1,\rho _2)>0$ has the dimension of time. Terms having the order $\mathcal O(\tau )$ can be considered as physically motivated regularizers ensuring the stability of explicit central difference approximations to Eqs. (7), (8). Obviously, $\boldsymbol {w}=\boldsymbol {0}$ for $\tau =0$, and system (7), (8) becomes system (1), (2).

Performing summation of Eq. (7) over $\alpha$, we obtain the total mass balance equation

$$ \partial _t\rho +\mathop {\mathrm {div}}\thinspace (\rho \boldsymbol {u}_m)=0. $$

(10)

Throughout the rest of the paper, the parameter $\tau$ in (9) will be calculated using the formula $\tau (\rho _1,\rho _1)=\alpha ^\ast {\eta (\rho _1,\rho _2)}/p(\rho _1,\rho _2)$, where $\alpha ^\ast >0$ is a dimensionless parameter whose value is chosen from the considerations of stability of numerical calculations.

Remark$.$ The boundary conditions (6) being satisfied implies the equality $\boldsymbol {n}\cdot \boldsymbol {w}=0$ on the domain boundary; this implies that $\boldsymbol {n}\cdot \boldsymbol {u}_m=0$. Therefore, for system (7), (8), conditions (6) ensure the lack of flux through the boundary in the form of the total mass of the mixture as well as the mass of its separate components. Taking this into account, we have the relations

$$ \int _\Omega \rho _\alpha (\boldsymbol {x},t)\thinspace d\boldsymbol {x}= \int _\Omega \rho _{\alpha ,0}(\boldsymbol {x})\thinspace d\boldsymbol {x},\quad \alpha =1,2;\quad \int _\Omega \rho (\boldsymbol {x},t)\thinspace d\boldsymbol {x} =\int _\Omega (\rho _{1,0}(\boldsymbol {x})+\rho _{2,0}(\boldsymbol {x}))\thinspace d\boldsymbol {x}, $$

which express the law of conservation of the total mass of the mixture and the total mass of individual components.

3. POTENTIAL FORM OF CAPILLARY STRESSES

Below we will use a representation for capillary stresses in the so-called potential form. This form is based on the following assertion, which we provide here with its proof for the presentation to be complete.

Assertion$.$ Under the condition that the capillary coefficients be symmetrical, $\lambda _{\alpha \beta }=\lambda _{\beta \alpha }$, one has the relation

$$ \nabla p-\mathop {\mathrm {div}}\thinspace \boldsymbol {\Pi }^{\mathrm {c}}=\rho _\alpha \nabla \hat \mu _\alpha . $$

(11)

Proof. By virtue of $\lambda _{\alpha \beta }=\lambda _{\beta \alpha }$ and $\partial _i\partial _j=\partial _j\partial _i$, we obtain

$$ \begin {aligned} \lambda _{\alpha \beta }\nabla \rho _\alpha \cdot (\nabla \otimes \nabla \rho _\beta ) &=\lambda _{\alpha \beta }(\partial _i\rho _\alpha )(\partial _i\partial _j\rho _\beta )\boldsymbol {e}_j\cr &=\lambda _{\alpha \beta }(\partial _i\rho _\alpha )(\partial _j\partial _i\rho _\beta )\boldsymbol {e}_j\cr &=\lambda _{\alpha \beta }\partial _j(\partial _i\rho _\alpha \partial _i\rho _\beta )\boldsymbol {e}_j -\lambda _{\alpha \beta }(\partial _i\rho _\beta )(\partial _j\partial _i\rho _\alpha )\boldsymbol {e}_j\cr &=\lambda _{\alpha \beta }\partial _j(\partial _i\rho _\alpha \partial _i\rho _\beta )\boldsymbol {e}_j -\lambda _{\alpha \beta }(\partial _i\rho _\alpha )(\partial _i\partial _j\rho _\beta )\boldsymbol {e}_j\cr &=\nabla (\lambda _{\alpha \beta }\nabla \rho _\alpha \cdot \nabla \rho _\beta ) -\lambda _{\alpha \beta }\nabla \rho _\alpha \cdot (\nabla \otimes \nabla \rho _\beta ). \end {aligned} $$

(12)

It readily follows from (12) that

$$ \lambda _{\alpha \beta }\nabla \rho _\alpha \cdot (\nabla \otimes \nabla \rho _\beta ) =\frac 12\nabla (\lambda _{\alpha \beta } \nabla \rho _\alpha \cdot \nabla \rho _\beta );$$

the last relation and the Leibniz rule imply that

$$ \mathop {\mathrm {div}}\thinspace (\lambda _{\alpha \beta }\nabla \rho _\alpha \otimes \nabla \rho _\beta )= \lambda _{\alpha \beta }(\nabla \rho _\beta )\Delta \rho _\alpha +\nabla \biggl (\frac 12\lambda _{\alpha \beta }\nabla \rho _\alpha \cdot \nabla \rho _\beta \biggr ). $$

(13)

Using the representation $\nabla \psi _0=(\partial _{\rho _\alpha }\psi _0) \nabla \rho _\alpha \equiv \mu _\alpha \nabla \rho _\alpha$, we write the pressure gradient in the form

$$ \nabla p=\nabla (\rho _\alpha \mu _\alpha )-\nabla \psi _0 =\mu _\alpha \nabla \rho _\alpha + \rho _\alpha \nabla \mu _\alpha -\nabla \psi _0= \rho _\alpha \nabla \mu _\alpha . $$

(14)

In what follows, considering relation (13), the Leibniz rule, and the fact that the coefficients $\lambda _{\alpha \beta }$ are symmetric, we obtain the relation

$$ \mathop {\mathrm {div}}\thinspace \boldsymbol {\Pi }^{\mathrm {c}} =\nabla \biggl (\lambda _{\alpha \beta }\rho _\alpha \Delta \rho _\beta +\frac 12\lambda _{\alpha \beta }\nabla \rho _\alpha \cdot \nabla \rho _\beta \biggr ) -\mathop {\mathrm {div}}\thinspace (\lambda _{\alpha \beta }\nabla \rho _\alpha \otimes \nabla \rho _\beta ) =\rho _\alpha \nabla (\lambda _{\alpha \beta }\Delta \rho _\beta ). $$

Subtracting this relation from relation (14) and using the expression for the generalized chemical potential in (5), we arrive at relation (11). The proof of the assertion is complete.

By applying formula (11), we write the momentum balance equation (8) in the form

$$ \partial _t(\rho \boldsymbol {u}) +\mathop {\mathrm {div}}\thinspace (\rho \boldsymbol {u}_m\otimes \boldsymbol {u}) +\rho _\alpha \nabla \hat \mu _\alpha =\mathop {\mathrm {div}}\thinspace \boldsymbol {\Pi }^{\mathrm {NS}} +\mathop {\mathrm {div}}\thinspace \boldsymbol {\Pi }^\tau . $$

(15)

Thus, in the representation (15), the capillary forces have a nondivergence (potential) representation. It was noted in the papers [4, 17, 18] that writing the momentum balance equation in such a form permits one to construct energy-dissipative approximations even if they lead to a momentum-nonconservative difference scheme, with the latter not being critical in view of the absence of shock waves and other strong discontinuities. The representation (15) (for $\tau =0$) was used in the papers [17, 12].

4. DISSIPATIVITY OF QHD-REGULARIZED EQUATIONS

One fundamental property described by systems (1), (2) and (7), (8) is their dissipativity. For isothermal processes, this implies the lack of growth in the total energy $\mathcal {E}_{\mathrm {tot}}(t)$ of the closed system as the system moves to the equilibrium state. In this section, we prove the dissipativity of the system of QHD-regularized equations (7), (8) with the boundary conditions (6). The following assertion holds.

Theorem 1$.$ For the QHD-regularized system of equations (7, (8) (or (15)), one has the total-energy local balance equation

$$ \partial _t e_{\mathrm {tot}} +\mathop {\mathrm {div}}\thinspace \boldsymbol {a} +TM|\nabla \hat {\mu }_{12}|^2+2\eta |\mathbf {D}|^2 +\biggl (\zeta -\frac 23\eta \biggr )(\mathop {\mathrm {div}}\thinspace \boldsymbol {u})^2 +\rho \tau ^{-1}|\boldsymbol {w}|^2 =0. $$

(16)

Here $e_{\mathrm {tot}}:=\psi _0 +(\lambda _{\alpha \beta }/2)\nabla \rho _\beta \cdot \nabla \rho _\alpha +(\rho |\boldsymbol {u}|^2)/2$ is the volume density of total energy of the system and

$$ \eqalign { \boldsymbol {a}&=(\partial _{\rho _\alpha }\psi _0)\rho _\alpha \boldsymbol {u}_m - TM\hat {\mu }_{12}\nabla \hat {\mu }_{12} -\partial _t(\lambda _{\alpha \beta }\rho _\alpha )\nabla \rho _\beta \cr &\qquad {}- (\lambda _{\alpha \beta }\Delta \rho _\beta )\rho _\alpha \boldsymbol {u}_m+ \biggl (\frac 12|\boldsymbol {u}|^2\rho \boldsymbol {u}_m -\boldsymbol {\Pi }^\tau \boldsymbol {u} -\boldsymbol {\Pi }^{\mathrm {NS}}\boldsymbol {u}\biggr )} $$

(17)

is the total-energy flux vector.

Proof. 1. Multiply Eq. (7) by $\mu _\alpha \equiv \partial _{\rho _\alpha }\psi _0$ and perform summation over $\alpha$. Then, taking into account the relation $\partial _t\psi _0=(\partial _{\rho _\alpha }\psi _0) \partial _t\rho _\alpha$, we have

$$ \partial _t\psi _0+\mu _\alpha \mathop {\mathrm {div}}\thinspace (\rho _\alpha \boldsymbol {u}_m) =\mu _\alpha (-1)^{\alpha +1}\mathop {\mathrm {div}}\thinspace (M\nabla \hat {\mu }_{12}). $$

(18)

Let us transform the second term on the left-hand side in relation (18) using the Leibniz rule:

$$ \mu _\alpha \mathop {\mathrm {div}}\thinspace (\rho _\alpha \boldsymbol {u}_m) =\mathop {\mathrm {div}}\thinspace (\mu _\alpha \rho _\alpha \boldsymbol {u}_m) -\rho _\alpha \boldsymbol {u}_m\cdot \nabla \mu _\alpha , $$

(19)

and represent the right-hand side of relation (18) in the form

$$ \eqalign { \mu _\alpha (-1)^{\alpha +1}\mathop {\mathrm {div}}\thinspace (M\nabla \hat {\mu }_{12}) &=(\mu _1-\mu _2)\mathop {\mathrm {div}}\thinspace (M\nabla \hat {\mu }_{12})\cr &=(T\hat {\mu }_{12}+\lambda _{1\beta }\Delta \rho _\beta -\lambda _{2\beta }\Delta \rho _\beta )\mathop {\mathrm {div}}\thinspace (M\nabla \hat {\mu }_{12})\cr &=\mathop {\mathrm {div}}\thinspace (TM\hat {\mu }_{12}\nabla \hat {\mu }_{12})-TM|\nabla \hat {\mu }_{12}|^2+(\lambda _{1\beta }-\lambda _{2\beta })\Delta \rho _\beta \mathop {\mathrm {div}}\thinspace (M\nabla \hat {\mu }_{12}),} $$

(20)

where we have also used the Leibniz rule when deriving the last expression.

Let us substitute the expressions (19) and (20) into (18) to arrive at the relation

$$ \eqalign { &\partial _t\psi _0 +\mathop {\mathrm {div}}\thinspace (\mu _\alpha \rho _\alpha \boldsymbol {u}_m-TM\hat {\mu }_{12}\nabla \hat {\mu }_{12}) -\rho _\alpha \boldsymbol {u}_m\cdot \nabla \mu _\alpha \cr &\qquad \qquad {}=-TM|\nabla \hat {\mu }_{12}|^2 +(\lambda _{1\beta }-\lambda _{2\beta })\Delta \rho _\beta \mathop {\mathrm {div}}\thinspace (M\nabla \hat {\mu }_{12}).} $$

(21)

2. Multiply Eq. (7) by $-\lambda _{\alpha \beta }\Delta \rho _\beta$ and perform summation over $\alpha$ to obtain

$$ -(\lambda _{\alpha \beta }\Delta \rho _\beta )\partial _t\rho _\alpha -(\lambda _{\alpha \beta }\Delta \rho _\beta )\mathop {\mathrm {div}}\thinspace (\rho _\alpha \boldsymbol {u}_m) =(-1)^{\alpha +2}(\lambda _{\alpha \beta }\Delta \rho _\beta )\mathop {\mathrm {div}}\thinspace (M\nabla \hat {\mu }_{12}). $$

(22)

Let us transform the first term on the left-hand side in relation (22) as

$$ -(\lambda _{\alpha \beta }\Delta \rho _\beta )\partial _t\rho _\alpha =-\mathop {\mathrm {div}}\thinspace (\nabla \rho _\beta )\lambda _{\alpha \beta }\partial _t\rho _\alpha =-\mathop {\mathrm {div}}\thinspace (\lambda _{\alpha \beta }\partial _t\rho _\alpha \nabla \rho _\beta ) +\partial _t\biggl (\frac 12\lambda _{\alpha \beta }\nabla \rho _\beta \cdot \nabla \rho _\alpha \biggr ), $$

(23)

where we have used the Leibniz rule as well as the identity $\partial _t\nabla =\nabla \partial _t$ and the relation

$$ \lambda _{\alpha \beta }\nabla \rho _\beta \cdot \partial _t\nabla \rho _\alpha = \partial _t(\lambda _{\alpha \beta }\nabla \rho _\beta \cdot \nabla \rho _\alpha /2), $$

(24)

which holds by virtue of the condition $\lambda _{\alpha \beta }=\lambda _{\beta \alpha }$. Recall that that repeated indices $\alpha$ and $\beta$ assume summation from 1 to 2.

We bring the second term on the left-hand side in relation (22) to the form

$$ -(\lambda _{\alpha \beta }\Delta \rho _\beta )\mathop {\mathrm {div}}\thinspace (\rho _\alpha \boldsymbol {u}_m) =-\mathop {\mathrm {div}}\thinspace [(\lambda _{\alpha \beta }\Delta \rho _\beta )\rho _\alpha \boldsymbol {u}_m] +\rho _\alpha \boldsymbol {u}_m\cdot \nabla (\lambda _{\alpha \beta }\Delta \rho _\beta ). $$

(25)

Substituting the expressions (23) and (25) into (22), we arrive at the relation

$$ \eqalign { \partial _t\biggl (\frac 12\lambda _{\alpha \beta }\nabla \rho _\beta \cdot \nabla \rho _\alpha \biggr ) +\rho _\alpha \boldsymbol {u}_m\cdot \nabla (\lambda _{\alpha \beta }\Delta \rho _\beta ) &{}-\mathop {\mathrm {div}}\thinspace [\partial _t(\lambda _{\alpha \beta }\rho _\alpha )\nabla \rho _\beta +(\lambda _{\alpha \beta }\Delta \rho _\beta )\rho _\alpha \boldsymbol {u}_m]\cr &\qquad \qquad {}=- (\lambda _{1\beta }-\lambda _{2\beta })\Delta \rho _\beta \mathop {\mathrm {div}}\thinspace (M\nabla \hat {\mu }_{12}),} $$

(26)

where we have also used the identity $(-1)^{\alpha +2}(\lambda _{\alpha \beta }\Delta \rho _\beta ) \equiv -(\lambda _{1\beta }-\lambda _{2\beta })\Delta \rho _\beta$.

3. Let us obtain the kinetic-energy balance equation. First, note that by virtue of the Leibniz rule, one has the formulas

$$ \boldsymbol {u}\cdot \partial _t(\rho \boldsymbol {u}) =\frac 12|\boldsymbol {u}|^2\partial _t\rho +\partial _t\biggl (\frac 12|\boldsymbol {u}|^2\rho \biggr ) =-\frac 12|\boldsymbol {u}|^2 \mathop {\mathrm {div}}\thinspace (\rho \boldsymbol {u}_m)+\partial _t\biggl (\frac 12|\boldsymbol {u}|^2\rho \biggr ),$$

(27)

$$ \boldsymbol {u}\cdot \mathop {\mathrm {div}}\thinspace (\rho \boldsymbol {u}_m\otimes \boldsymbol {u}) =\frac 12|\boldsymbol {u}|^2\mathop {\mathrm {div}}\thinspace (\rho \boldsymbol {u}_m) +\mathop {\mathrm {div}}\thinspace \biggl (\frac 12|\boldsymbol {u}|^2\rho \boldsymbol {u}_m\biggr ),$$

(28)

$$ \boldsymbol {u}\cdot \mathop {\mathrm {div}}\thinspace \boldsymbol {\Pi }^{\mathrm {NS}}+ \boldsymbol {u}\cdot \mathop {\mathrm {div}}\thinspace \boldsymbol {\Pi }^{\tau } = \mathop {\mathrm {div}}\thinspace \left (\boldsymbol {\Pi }^{\tau }\boldsymbol {u}+\boldsymbol {\Pi }^{\mathrm {NS}}\boldsymbol {u}\right ) -\boldsymbol {\Pi }^{\mathrm {NS}}\boldsymbol {:}(\nabla \boldsymbol \otimes \boldsymbol {u}) -\boldsymbol {\Pi }^{\tau }\boldsymbol {:}(\nabla \boldsymbol {\otimes }\boldsymbol {u}), $$

(29)

where, in relation (27), we have also used the total-mass balance equation (10). Let us transform the last two terms on the right-hand side in relation (29) as

$$ \boldsymbol {\Pi }^{\mathrm {NS}}\boldsymbol {:}(\nabla \boldsymbol {\otimes }\boldsymbol {u})= 2\eta |\mathbf {D}|^2 +\biggl (\zeta -\frac 23\eta \biggr )(\mathop {\mathrm {div}}\thinspace \boldsymbol {u})^2, \quad \boldsymbol {\Pi }^{\tau }\boldsymbol {:}(\nabla \boldsymbol {\otimes }\boldsymbol {u})= \rho \boldsymbol {w}\cdot [(\boldsymbol {u}\cdot \nabla )\boldsymbol {u}], $$

(30)

where the first relation is derived taking into account the identity $\mathbf {D}\boldsymbol {:}(\nabla \boldsymbol {\otimes }\boldsymbol {u})\equiv |\mathbf { D}|^2$, which holds in view of the tensor $\mathbf {D}$ being symmetric.

Taking the inner product of the momentum balance equation (15) by $\boldsymbol {u}$ and using the relations in (27)–(30), we obtain

$$ \eqalign { \partial _t\biggl (\frac 12\rho |\boldsymbol {u}|^2\biggr )+\mathop {\mathrm {div}}\thinspace \biggl (\frac 12|\boldsymbol {u}|^2\rho \boldsymbol {u}_m&{}- \boldsymbol {\Pi }^{\mathrm {NS}}\boldsymbol {u}-\boldsymbol {\Pi }^\tau \boldsymbol {u}\biggr )+\rho _\alpha \boldsymbol {u}\cdot \nabla \hat \mu _{\alpha }\cr &=-2\eta |\mathbf {D}|^2 - \biggl (\zeta -\frac 23\eta \biggr )(\mathop {\mathrm {div}}\thinspace \boldsymbol {u})^2 -\rho \boldsymbol {w}\cdot [(\boldsymbol {u}\cdot \nabla )\boldsymbol {u}].} $$

(31)

Adding relations (21), (26), and (31), we arrive at Eq. (16), and this completes the proof of the theorem. Note that it is at this stage that it becomes clear that the vector $\boldsymbol {w}$ has the form (9) and the vector $\boldsymbol {a}$ has the form (17).

Corollary 1$.$ For system (7), (8) (or (15)) with the boundary conditions (6), one has the law of lack of growth in the system total energy ,

$$ \frac {d\mathcal {E}_{\mathrm {tot}}}{dt} =-\int _\Omega \biggl (2\eta |\mathbf {D}|^2 +\biggl (\zeta -\frac 23\eta \biggr )(\mathop {\mathrm {div}}\thinspace \boldsymbol {u})^2 +TM|\nabla \hat {\mu }_{12}|^2 +{\rho }{\tau }^{-1}|\boldsymbol {w}|^2\biggr )\thinspace d\boldsymbol {x} \leq 0, $$

(32)

where $\mathcal {E}_{\mathrm {tot}}(t):=\int _\Omega e_{\mathrm {tot}}(\boldsymbol {x}, t)\thinspace d\boldsymbol {x}$ is the total energy of the system.

Proof. The proof is conducted by a straightforward integration of Eq. (16) over the domain $\Omega$ and application of the Gauss divergence theorem with allowance for conditions (6) and the relation

$$ |\mathbf {D}|^2 - \frac 13(\mathop {\mathrm {div}}\thinspace \boldsymbol {u})^2 \equiv \bigg |\mathbf {D}-\frac 13 (\mathrm {tr}\thinspace \mathbf {D})\mathbf {I}\bigg |^2\geq 0.$$

Corollary 2$.$ System (1), (2) with the boundary conditions (6) is dissipative.

Proof. The proof follows from the fact that for $\tau =0$ system (7), (8) transforms into system (1), (2), while inequality (32) remains true.

5. SPATIAL APPROXIMATION

The dissipativity is a fundamental property of system (7), (8) (or (15)) that determines its thermodynamic correctness. Therefore, among all feasible numerical algorithms for solving this system, one should prefer the ones that ensure this property (in one or another form) at the discrete level. This section is devoted to constructing such a spatial discretization of Eqs. (7), (8) that is dissipative in the above-indicated sense for the one-dimensional plane-parallel case.

Consider the spatial domain ${\Omega }=[0,L]\subset \mathbb {R}$. We will use uniform difference meshes with step $h=L/N$. The mesh nodes with integer indices will be denoted by $x_i=hi$ and those with half-integer ones, by $x_{i-1/2}=h(i-1/2)$, with $x_{0}=0$ and $x_{N}=L$. Here and below, the subscripts $i$, $j$, and $k$ are reserved for node numbers. In $\Omega$, introduce the main mesh $\bar {\omega }_h:=\{x_i\}_{i=0}^{N}$ and internal meshes ${\omega }_h:=\{x_i\}_{i=1}^{N-1}$ and ${\omega }^\ast _h:=\{x_{i-1/2}\}_{i=1}^{N}$. We will need the auxiliary extended meshes $\bar {\omega }_h^\ast :=\{x_{i-1/2}\}_{i=0}^{N+1}$, $\bar {\bar {\omega }}_h^\ast :=\{x_{i-1/2}\}_{i=-1}^{N+2}$, and $\bar {\bar {\omega }}_h:=\{x_i\}_{i=-1}^{N+1}$. Note that not all the nodes of the extended meshes lie in the domain $\Omega$.

Let $H(\omega )$ be the set of functions defined on some mesh $\omega$. We introduce the mesh averaging operators $s: H(\bar {\bar {\omega }}_h)\to H(\bar {\omega }_h^\ast )$, $s^\ast : H(\bar {\bar {\omega }}_h^\ast )\to H(\bar {\bar {\omega }}_h)$ and difference relations $\delta : H(\bar {\bar {\omega }}_h)\to H(\bar {\omega }_h^\ast )$, $\delta ^\ast : H(\bar {\bar {\omega }}_h^\ast )\to H(\bar {\bar {\omega }}_h)$ by the formulas

$$ \begin {aligned} (s{u})_{i+1/2}&:=\frac 12(u_{i+1}+u_i),&\quad (s^\ast {v})_{i}&:=\frac 12(v_{i+1/2}+v_{i-1/2}),\cr (\delta {u})_{i+1/2}&:=\frac 1h(u_{i+1}-u_i),&\quad (\delta ^\ast {v})_{i}&:=\frac 1h(v_{i+1/2}-v_{i-1/2}). \end {aligned} $$

Let $u$ and $v$ be some mesh functions in $H(\bar {\bar {\omega }}_h)$ and let $\tilde {u}$ and $\tilde {v}$ be mesh functions in $H(\bar {\bar {\omega }}_h^\ast )$. Define the inner products $(u,v)$ on $H(\omega _h)$, $(u,v)_{\bar x}$ on $H(\bar {\omega }_h)$, and $(\tilde {u},\tilde {v})_{\ast }$ on $H(\omega ^\ast _h)$ by the formulas

$$ (u,v):=h\sum _{i=1}^{N-1}u_i v_i,\quad (u,v)_{\bar {x}}:=\frac 12hv_0u_0+(u,v)+\frac 12hv_Nu_N,\quad (\tilde {u},\tilde {v})_\ast :=h\sum _{i=1}^{N}\tilde {u}_{i-1/2}\tilde {v}_{i-1/2}. $$

For the inner products thus introduced, one has the identities

$$ \eqalign { (\delta v,\tilde {u})_\ast &=-(v,\delta ^\ast \tilde {u})_{\bar x}+(s^\ast \tilde {u})_Nv_N-(s^\ast \tilde {u})_0v_0,\cr (sv,\tilde {u})_{\ast }&=(v,s^\ast \tilde {u})_{\bar {x}} +\frac 14h^2v_0(\delta ^\ast \tilde {u})_0 - \frac 14h^2v_{N}(\delta ^\ast \tilde {u})_{N}.}$$

In particular, if $v_0=v_N=0$, then $(u,v)=(u,v)_{\bar x}$ and

$$ (\delta v,\tilde {u})_\ast =-(v,\delta ^\ast \tilde {u}), $$

(33)

$$ (s v,\tilde {u})_\ast =(v, s^\ast \tilde {u}). $$

(34)

Consider the following space-discrete and time-continuous method for system (7) and (15) in the spatially uniform statement:

$$ \partial _t{\rho }_\alpha +\delta [(s^\ast \rho _\alpha ) u_m] =(-1)^{\alpha +1}\delta [(s^\ast M)\delta ^\ast \hat {\mu }_{12}] \quad \text {on}\quad \omega ^\ast _h,\quad \alpha =1,2, $$

(35)

$$ \partial _t{[(s^\ast \rho ) u]}+\delta ^\ast [(s j_m) su] +(s^\ast \rho _{\alpha })\delta ^\ast \hat \mu _{\alpha } =\delta ^\ast \Pi ^{\mathrm {NS}}+\delta ^\ast \Pi ^{\tau } \quad \text {on}\quad \omega _h, $$

(36)

where

$$ j_m=(s^\ast \rho )u_m,\quad u_m=u-w,\quad \Pi ^{\mathrm {NS}}=\eta \frac 43\delta {u}+\zeta \delta u,$$

(37)

$$ \hat {\mu }_{12}=T^{-1}(\hat {\mu }_1-\hat {\mu }_2),\quad \Pi ^{\tau }=s[(s^\ast \rho ) u w],\quad w=\tau (s^\ast \rho )^{-1}[(s^\ast \rho )u s^\ast \delta u +(s^\ast \rho _\alpha )\delta ^\ast \hat \mu _\alpha ],$$

(38)

$$ \hat \mu _\alpha =\mu _\alpha -\lambda _{\alpha \beta }\delta \delta ^\ast \rho _\beta . $$

(39)

The main unknown functions are $u\!\in \!H(\bar {\omega }_h)$ and $\rho _\alpha \!\in \!H(\omega ^\ast _h)$. In addition, $\{w,\tau (s^\ast \rho _1, s^\ast \rho _2)\}\!\subset \!H(\bar {\omega }_h)$ and $\{\Pi ^{\mathrm {NS}}$, $\mu _\alpha$, $\hat {\mu }_\alpha$, $\eta (\rho _1,\rho _2)$, $\zeta (\rho _1,\rho _2)$, $M(\rho _1,\rho _2)\}$$\subset$$H(\omega ^\ast _h)$.

On the boundaries $x=0$ and $x=L$ of the computational domain, we set

$$ u_0=u_N=0, \ \ (\delta ^\ast \rho _\alpha )_0 =(\delta ^\ast \rho _\alpha )_N=0, \ \ (\delta ^\ast \hat \mu _\alpha )_0 =(\delta ^\ast \hat \mu _\alpha )_N=0,\quad \alpha =1,2. $$

(40)

It follows from relations (37)–(39) that for all necessary discrete functions to be determined correctly, the desired functions $u$ and $\rho _\alpha$, as well as $\mu _\alpha$ and $\hat {\mu }_\alpha$, must be continued to wider domains. To this end, we set $u\in H(\bar {\omega }_h)$, $\rho _\alpha \in H(\bar {\bar {\omega }}^\ast _h)$, and $\{\mu _\alpha ,\hat {\mu }_\alpha ,M\}\subset H(\bar {\omega }^\ast _h)$. In this case, for the second and third boundary conditions in (40) to be satisfied, we continue $\rho _\alpha$ in an even manner with respect to the boundary nodes $x_0$ and $x_N$. Note also that (40) implies the lack of the mass flux through the boundary; i.e.,

$$ w_0=w_N=0,\quad (u_m)_0=(u_m)_N=0. $$

(41)

By analogy with the continuous case, performing summation of Eqs. (35) over $\alpha =1,2$, we obtain the discrete complete mass balance equation for the mixture,

$$ \partial _t\rho +\delta [(s^\ast \rho ) u_m]=0. $$

(42)

Also note that this discretization obeys the conservation laws for the total mass and the masses of components,

$$ (\rho _\alpha ,1)_\ast =(\rho _{\alpha ,0},1)_\ast =\mathrm{const}\, ,\quad \alpha =1,2,\quad (\rho ,1)_\ast =(\rho _{1,0}+\rho _{2,0},1)_\ast =\mathrm{const}\, . $$

(43)

The following difference analog of the corollary of Theorem 1 about the lack of growth in the system total energy holds true.

Theorem 2$.$ For the semidiscrete method (35), (36) one has the relation

$$ \frac {d{\mathcal E}_{\mathrm {tot}}}{dt}= -\frac 43\left ({\eta (\delta {u})^2},1\right )_\ast -\left ({\zeta (\delta {u})^2},1\right )_\ast -\left (T(s^\ast M)(\delta ^\ast \hat {\mu }_{12})^2, 1\right ) -\left (\tau w^2,(s^\ast \rho )^{-1}\right )\leq 0, $$

(44)

where ${\mathcal E}_{\mathrm {tot}}(t):=(e_{\mathrm {tot}}, 1)_\ast$ is the discrete analog of the system total energy and

$$ e_{\mathrm {tot}}:=\frac 12\rho su^2+\psi _0 +\frac 12\lambda _{\alpha \beta } s\left [\left (\delta ^\ast \rho _\alpha \right )\left (\delta ^\ast {\rho }_\beta \right )\right ] $$

is the discrete analog of the volume density of total energy.

Proof. The derivation of relation (44) presented below is based on the proof of Theorem 1 and consists of similar steps.

1. Take the inner product of Eq. (35) by $\partial _{\rho _\alpha }{\psi _0}\equiv \mu _\alpha$, perform summation over $\alpha =1,2$, and take the relation $(\partial _{\rho _\alpha }{\psi _0})\partial _t\rho _\alpha =\partial _t\psi _0$ into account to obtain

$$ \partial _t(\psi _0,1)_\ast +(\mu _\alpha ,\delta [(s^\ast \rho _\alpha )u_m])_\ast =(\mu _\alpha , (-1)^{\alpha +1}\delta [(s^\ast M)\delta ^\ast \hat {\mu }_{12}])_\ast . $$

(45)

Considering relations (41), we use formula (33) to transform the second term on the left-hand side in relation (45). We have

$$ (\mu _\alpha ,\delta [(s^\ast \rho _\alpha )u_m])_\ast = -(\delta ^\ast \mu _\alpha , (s^\ast \rho _\alpha )u_m). $$

(46)

Transforming the right-hand side of relation (45) by analogy with (20), we obtain

$$ \eqalign { (\mu _\alpha , (-1)^{\alpha +1}\delta [(s^\ast M)\delta ^\ast \hat {\mu }_{12}])_\ast &= (T\hat {\mu }_{12}+\lambda _{1\beta }\delta \delta ^\ast \rho _\beta -\lambda _{2\beta }\delta \delta ^\ast \rho _\beta ,\delta [(s^\ast M)\delta ^\ast \hat {\mu }_{12}])_\ast \cr &=-(T(s^\ast M)(\delta ^\ast \hat {\mu }_{12})^2, 1)+([\lambda _{1\beta } -\lambda _{2\beta }]\delta \delta ^\ast \rho _\beta ,\delta [(s^\ast M)\delta ^\ast \hat {\mu }_{12}])_\ast ,} $$

(47)

where we have used the identity $\mu _\alpha (-1)^{\alpha +1}\equiv \mu _1-\mu _2$ and applied formula (33) with allowance for the third condition in (40) when deriving the last expression.

Substituting the expressions (46) and (47) into (45), we arrive at the relation

$$ \partial _t(\psi _0,1)_\ast -((s^\ast \rho _\alpha )u_m,\delta ^\ast \mu _\alpha ) =-(T(s^\ast M)(\delta ^\ast \hat {\mu }_{12})^2, 1) +([\lambda _{1\beta } -\lambda _{2\beta }]\delta \delta ^\ast \rho _\beta ,\delta [(s^\ast M)\delta ^\ast \hat {\mu }_{12}])_\ast . $$

(48)

2. In what follows, taking the inner product of Eq. (35) by $-\lambda _{\alpha \beta }\delta \delta ^\ast \rho _\beta$, we have

$$ -(\lambda _{\alpha \beta }\delta \delta ^\ast \rho _\beta ,\partial _t{\rho }_\alpha )_\ast - (\lambda _{\alpha \beta }\delta \delta ^\ast \rho _\beta ,\delta [(s^\ast \rho _\alpha ) u_m])_\ast =(\lambda _{\alpha \beta }\delta \delta ^\ast \rho _\beta , (-1)^{\alpha +2}\delta [(s^\ast M)\delta ^\ast \hat {\mu }_{12}])_\ast . $$

(49)

Let us transform the first term on the left-hand side in relation (49) using formula (33), which holds by virtue of the second condition in (40), and taking into account the fact that the coefficients $\lambda _{\alpha \beta }$ are symmetric,

$$ \eqalign { -(\lambda _{\alpha \beta }\delta \delta ^\ast \rho _\beta ,\partial _t{\rho }_\alpha )_\ast &= \lambda _{\alpha \beta }(\delta ^\ast \rho _\beta ,\delta ^\ast \partial _t{\rho }_\alpha )\cr &=\partial _t\biggl (\frac 12\lambda _{\alpha \beta }\delta ^\ast \rho _\beta \delta ^\ast {\rho }_\alpha ,1\biggr )= \partial _t\biggl (\frac 12\lambda _{\alpha \beta }s[\delta ^\ast \rho _\beta \delta ^\ast {\rho }_\alpha ],1\biggr )_{\ast },} $$

(50)

where the penultimate expression was derived using the relation

$$ \lambda _{\alpha \beta }(\delta ^\ast \rho _\beta )\partial _t\delta ^\ast \rho _\alpha =\partial _t(\lambda _{\alpha \beta }\delta ^\ast \rho _\beta \delta ^\ast {\rho }_\alpha /2),$$

similar to (24), and the relation $\delta ^\ast \partial _t=\partial _t\delta ^\ast$; the last expression was derived using formula (34) with allowance for the identities $1\equiv s^\ast 1$ and $\delta ^\ast 1\equiv 0$ and the second condition in (40).

By directly applying formula (33), which holds by virtue of (41), we can reduce the second term on the left-hand side in relation (49) to the form

$$ {}-(\lambda _{\alpha \beta }\delta \delta ^\ast \rho _\beta ,\delta [(s^\ast \rho _\alpha ) u_m])_\ast =((s^\ast \rho _\alpha ) u_m,\delta ^\ast [\lambda _{\alpha \beta }\delta \delta ^\ast \rho _\beta ]). $$

(51)

Substituting the expressions (50) and (51) into (49), we obtain

$$ \eqalign { \partial _t\biggl (\frac 12\lambda _{\alpha \beta }s [\delta ^\ast \rho _\alpha \delta ^\ast {\rho }_\beta ],1\biggr )_{\ast } &{}+\left ((s^\ast \rho _\alpha ) u_m,\delta ^\ast [\lambda _{\alpha \beta }\delta \delta ^\ast \rho _\beta ]\right )\cr &=-\left ([\lambda _{1\beta }-\lambda _{2\beta }]\delta \delta ^\ast \rho _\beta , \delta [(s^\ast M)\delta ^\ast \hat {\mu }_{12}]\right ),} $$

(52)

where we have also used the identity $(-1)^{\alpha +2}\lambda _{\alpha \beta } \delta \delta ^\ast \rho _\beta \equiv -(\lambda _{1\beta }-\lambda _{2\beta })\delta \delta ^\ast \rho _\beta$.

Adding relations (48) and (52), we have

$$ \partial _t\biggl (\psi _0+\frac 12\lambda _{\alpha \beta }s [\delta ^\ast \rho _\alpha \delta ^\ast {\rho }_\beta ],1\biggr )_{\ast } - ((s^\ast \rho _\alpha ) u_m,\delta ^\ast \hat {\mu }_\alpha ) =-\left (T(s^\ast M)(\delta ^\ast \hat {\mu }_{12})^2, 1\right ). $$

(53)

3. Let us obtain the discrete kinetic energy balance equation for the system. To this end, take the inner product of the momentum balance equation (36) by $u$,

$$ (u,\partial _t[(s^\ast \rho ) u])+(u,\delta ^\ast [(sj_m)s{u}]) +(u,(s^\ast \rho _\alpha )\delta ^\ast \hat \mu _\alpha )= (u,\delta ^\ast {\Pi }^{\mathrm {NS}})+(u,\delta ^\ast {\Pi }^{\tau }). $$

(54)

Let us transform the first term on the left-hand side in relation (54). First, note that, by virtue of the relation $\partial _t s^\ast =s^\ast \partial _t$, formulas (33) and (34), the boundary conditions (40), relations (41), and the discrete total-mass balance equation (42), the following chain of relations holds:

$$ \eqalign { (u^2,\partial _t(s^\ast \rho ))&=(su^2,\partial _t\rho )_\ast =-(su^2,\delta j_m)_\ast =(\delta ^\ast su^2, j_m)\cr &=(s^\ast \delta u^2, j_m)=(\delta u^2, s j_m)_\ast =2(\delta u, (s j_m)su)_\ast ,} $$

(55)

where the relations $s^\ast \delta =\delta ^\ast s$ and $\delta u^2=2 (\delta u) su$ have also been taken into account. Using (34) and (55), we can express the first term on the left-hand side in relation (54) in the form

$$ (u,\partial _t[(s^\ast \rho ) u]) =\frac 12\partial _t(s^\ast \rho ,u^2)+\frac 12(u^2,\partial _t(s^\ast \rho )) =\frac 12\partial _t(\rho , s u^2)_\ast +(\delta u, (s j_m)su)_\ast . $$

(56)

Using the first boundary condition in (40) and formula (33), we write the second term on the left-hand side in relation (54) in the form

$$ (u,\delta ^\ast [(sj_m)su])=-(\delta u, (sj_m)su)_\ast . $$

(57)

Transforming the first and second terms on the right-hand side in relation (54) with the help of formula (33), we have

$$ \left (u,\delta ^\ast {\Pi }^{\mathrm {NS}}\right ) =-\left (\delta u,\Pi ^{\mathrm {NS}}\right )_\ast =- \frac 43\left ({\eta (\delta {u})^2},1\right )_\ast -\left ({\zeta (\delta {u})^2},1\right )_\ast , $$

(58)

$$ \left (u,\delta ^\ast {\Pi }^{\tau }\right ) =-\left (\delta u,\Pi ^{\tau }\right )_\ast =- \left (\delta u, s[u w s^\ast \rho ]\right )_\ast =- \left (us^\ast \delta u, (s^\ast \rho \right ) w), $$

(59)

where in (59) we have also used formula (34), which holds by virtue of the no-slip conditions (59).

Thus, substituting the expressions (56)–(59) into (54), we obtain the kinetic energy balance equation for the system,

$$ \partial _t\biggl (\rho , \frac 12su^2\biggr )_{\ast } +\left (u,(s^\ast \rho )\delta ^\ast \hat \mu _\alpha \right )= - \frac 43\left ({\eta (\delta {u})^2},1\right )_\ast -\left ({\zeta (\delta {u})^2},1\right )_\ast -\left (us^\ast \delta u, (s^\ast \rho ) w\right ). $$

(60)

Adding (53) to (60), we obtain relation (44). The proof of the theorem is complete.

6. NONDIMENSIONALIZATION

Let us nondimensionalize system (7), (15) following the paper [11]. For the typical quantities we take

$$ \rho ^\ast =\frac {m_2}{3b_2},\quad p^\ast =\frac {a_{22}}{27b_2^2}, \quad T^\ast =\frac {8a_{22}}{27k_{\mathrm {B}}b_2},\quad L^\ast =2b_2^{1/3}, \quad u^\ast =\sqrt {\frac {p^\ast }{\rho ^\ast }}.$$

Here $\rho ^\ast$, $p^\ast$, and $T^\ast$ correspond to the density, pressure, and temperature at the critical point for the second component, $l^\ast$ is the typical spatial scale, which is on the order of the thickness of the interface, and $u^\ast$ is the typical velocity. The dimensionless variables will be equipped with the symbol “$\sim$” (for example, $\tilde {\rho })$. Thus, we have

$$ \begin {gathered} \rho _\alpha =\rho ^\ast \tilde {\rho }_\alpha ,\quad \boldsymbol {u}=u^\ast \tilde {\boldsymbol {u}},\quad \boldsymbol {w}=u^\ast \tilde {\boldsymbol {w}},\quad \boldsymbol {x}=L^\ast \tilde {\boldsymbol {x}},\quad p=p^\ast \tilde {p},\cr \psi _0=p^\ast \tilde {\psi }_0,\quad T=T^\ast \tilde {T},\quad \mu _\alpha =\frac {p^\ast }{\rho ^\ast }\tilde {\mu }_\alpha ,\quad t=\frac {L^\ast }{u^\ast }\tilde {t},\quad \eta =\nu _2\rho ^\ast \tilde {\eta },\cr M_0=\frac {{\rho ^\ast }^2 u^\ast L^\ast T^\ast }{p^\ast }\tilde {M}_0,\quad \boldsymbol {\Pi }^{\mathrm {NS}}=\frac {\nu _2\rho ^\ast u^\ast }{L^\ast }\tilde {\boldsymbol {\Pi }}^{\mathrm {NS}},\quad \lambda _{\alpha \beta }=\frac {{L^\ast }^2 p^\ast }{{\rho ^\ast }^2}\tilde {\lambda }_{\alpha \beta },\quad D_{\alpha \beta }=\frac {{m^\ast }^2p^\ast {L^\ast }^2}{{\rho ^\ast }^2 k_{\mathrm {B}}T^\ast }\tilde {D}_{\alpha \beta }. \end {gathered}$$

In what follows, to simplify the notation, we omit the symbol “$\sim$” over all the variables. It is convenient to introduce the following dimensionless variables:

$$ \begin {gathered} m_{21}=\frac {m_2}{m_1},\quad r_{21}=\frac {a_{21}}{a_{22}},\quad r_{{1}{1}} = \frac {a_{11}}{a_{22}},\quad b_{12}=\frac {b_1}{b_2},\quad A=\frac 13\biggl (\frac {27b_2^{1/3}\hbar ^2}{16\pi a_{22} m_2}\biggr )^{3/2},\quad \nu _{12}=\frac {\nu _1}{\nu _2}. \end {gathered}$$

The relation (4) for the Helmholtz energy acquires the form

$$ \eqalign { {\psi }_0({\rho }_1,{\rho }_2)&= {T}\frac 83{\rho }_1{m}_{21}\ln \bigg [\frac {{m}_{21}^{5/2}{\rho }_1A}{{T}^{3/2}(1-\phi )}\bigg ] + {T}\frac 83{\rho }_2\ln \bigg [\frac {{\rho }_2A}{{T}^{3/2}(1-\phi )}\bigg ]\cr &\qquad {}-3({r}_{11} {m}_{21}^2{\rho }^2_1 +2{r}_{21} {m}_{21}{\rho }_1{\rho }_2 +{\rho }^2_2) -{T}\frac 83({m}_{21}{\rho }_1+{\rho }_2),} $$

(61)

where $\phi (\rho _1,\rho _2)=b_{12}\dfrac 13 m_{21}\rho _1+\dfrac 13\rho _2$. For system (7), (15), we obtain

$$ \eqalign { \partial _t\rho _\alpha +\mathop {\mathrm {div}}\thinspace (\rho _\alpha \boldsymbol {u}_m) &=(-1)^{\alpha +1}\mathop {\mathrm {div}}\thinspace (M\nabla \hat {\mu }_{12}),\quad \alpha =1,2,\cr \partial _t(\rho \boldsymbol {u}) +\mathop {\mathrm {div}}\thinspace (\rho \boldsymbol {u}_m\otimes \boldsymbol {u}) +\rho _\alpha \nabla \hat \mu _\alpha &=\frac 1{\mathrm {Re}}\mathop {\mathrm {div}}\thinspace \boldsymbol {\Pi }^{\mathrm {NS}} +\mathop {\mathrm {div}}\thinspace \boldsymbol {\Pi }^\tau ,}$$

where $\mathrm {Re}=L^\ast u^\ast /\nu _2$ and

$$ \begin {gathered} \boldsymbol {\mathrm \Pi }^{\mathrm {NS}}=\eta (\nabla \otimes \boldsymbol {u}+(\nabla \otimes \boldsymbol {u})^{\textrm {T}}) +\biggl (\zeta -\frac 23\eta \biggr ) (\mathop {\mathrm {div}}\thinspace \boldsymbol {u})\mathbf { I},\quad \boldsymbol {\Pi }^\tau =\rho \boldsymbol {u}\otimes \boldsymbol {w},\cr \eta (\rho _1,\rho _2)=\rho _1\nu _{12}+\rho _2,\quad \boldsymbol {w}=\frac {\tau }{\rho }[\rho (\boldsymbol {u}\cdot \nabla )\boldsymbol {u} +\rho _\alpha \nabla \hat {\mu }_\alpha ],\quad \tau =0.3\frac {\eta (\rho _1,\rho _2)}{p(\rho _1,\rho _2)}. \end {gathered}$$

7. NUMERICAL EXPERIMENT

Let us apply the explicit Euler method to discretize the constructed semidiscrete difference scheme (35), (36) in time $t$. To this end, we replace all derivatives with respect to time $\partial _t f$ with $\delta _t f:=(f^{n+1}-f^n)/\Delta t$, where $f$ is the corresponding difference function, the superscript is used for the time layer number, and $\Delta t$ designates the time step.

To demonstrate the capabilities of the constructed difference scheme, we consider the problem of modeling stratification into different phases of a virtually homogeneous mixture whose component composition is unstable under small perturbations. Such a stratification is often referred to as the spinodal decomposition of the mixture (see, e.g., [13, p. 117]). Modeling this process using system (7), (15) is possible since (i) the homogeneous part of the free energy (61) (respectively, (4)) is nonconvex, a fact that accounts for “stratification”; and (ii) the free energy (3) contains gradient terms that prevent the interface from becoming infinitely thin in the course of stratification (therewith the gradient terms also have a regularizing meaning [5, 9]).

Consider a mixture of components described by the following collection of dimensionless parameters: $r_{11}=3.426$, $r_{21}=1.75$, $b_{12}=2.028$, $m_{21}=0.8356$, $T=0.957$, $\nu _{12}=10.0$, $\mathrm {Re}=10.0$, $D_{11}=12$, $D_{22}=4$, $D_{12}=D_{21}=6.9282$, $M_0=0.1$, $A=10^{-5}$, and $\zeta =\eta$.

We take the initial conditions in the form

$$ \rho _{1,0}(x_{i+1/2})=0.054(1+\varepsilon (x_{i+1/2})),\quad \rho _{2,0}(x_{i+1/2})=1+\varepsilon (x_i),\quad u_0(x_{i+1/2})=0, $$

where $\varepsilon (x_{i+1/2})$ is a random variable that has a uniform distribution and assumes values in the interval $[-0.05, 0.05]$. Consider the domain of size $L=200$. We select the space and time steps to be $h=L/100$ and $\Delta t=2\cdot 10^{-3}$, respectively.

Figure 1 presents the dependence $\mathcal {E}_{\mathrm {tot}}(t) -\mathcal {E}_{\mathrm {tot}}(t_{\mathrm {end}})$, where $t_{\mathrm {end}}=4.93\cdot 10^4$ is the moment up to which the modeling was carried out. It can be seen that the total energy is nonincreasing in agreement with inequalities (44). Note that a sharp decrease in the total energy is observed at the moment $t\approx 2.67\cdot 10^4$. In this case, the change occurs gradually as clearly demonstrated by the enlarged fragment of the domain $t\in [26370, 27000]$ (see the bottom of Fig. 1). The kinetic energy $\mathcal {E}_{\mathrm {kin}}(t):=(\rho , 0.5 su^2)_\ast$ of the system also undergoes a dramatic increase at the same time moment and then starts to drop (Fig. 2).

Fig. 1.

Evolution of total energy when modeling spinodal decomposition.

Fig. 2.

Evolution of kinetic energy when modeling spinodal decomposition.

Such a behavior is explained by the “merger” of two regions of one component composition (“droplets”): four subdomains with $\rho _1\approx 0.09$ are observed on the graph of the density distribution $\rho _1$ at the moment $t=26500$ (Fig. 3), while at $t=26800$ (Fig. 4) two of them have already virtually “merged.” Figures 5 and 6 show $\rho _1$ and $\rho _2$ at the moment $t=49\cdot 10^3$, when the “merger” has already completed. The number of interphase subdomains (“boundaries”) where $(\partial _x\rho _\alpha )(\partial _x\rho _\beta )\gg 0$ has decreased, and this has led to an abrupt decrease in the total energy.

Fig. 3.

Distribution of $\rho _1$ immediately prior to “merger,” $t=26500$.

Fig. 4.

Distribution of $\rho _1$ in the course of “merger,” $t=26800$.

Note that the distributions of $\rho _1$ and $\rho _2$ shown in Figs. 5 and 6 are qualitatively very similar while being significantly different quantitatively; namely, $\rho _1\in (0.011, 0.092)$ and $\rho _2\in (0.49, 1.41)$. The minimum and maximum values of $\rho _1$ and $\rho _2$ agree well with the results in the paper [11].

Fig. 5.

Distribution of $\rho _1$, $t=4.9\cdot 10^4$.

Fig. 6.

Distribution of $\rho _2$, $t=4.9\cdot 10^4$.

It has been checked that the total-mass conservation laws (43) are observed in the computations.