Navier-Stokes-Cahn-Hilliard system in a $3$D perforated domain with free slip and source term: Existence and homogenization

1. Problem Statement

The paper investigates the mathematical analysis and asymptotic behavior of a Navier-Stokes-Cahn-Hilliard (NSCH) system modeling a binary incompressible fluid mixture within a 3D periodically perforated porous medium.

Key Physical and Mathematical Features:

• Domain: A perforated domain $\Omega^\varepsilon_p \subset \mathbb{R}^3$ representing the fluid pores, surrounded by solid inclusions $\Omega^\varepsilon_s$. The geometry is periodic with a small scale parameter $\varepsilon > 0$.
• Governing Equations:
  • Momentum: Time-dependent Navier-Stokes equations with a variable viscosity tensor $A^\varepsilon(t,x)$ and a Korteweg-type capillary force term $\lambda_\varepsilon \phi^\varepsilon \nabla \mu^\varepsilon$.
  • Mass Conservation: Incompressibility constraint $\nabla \cdot u^\varepsilon = 0$.
  • Phase Field: A Cahn-Hilliard equation with a non-conservative source term $G(\phi^\varepsilon)$ (representing bulk reactions or external driving), coupled with the chemical potential $\mu^\varepsilon = -\Delta \phi^\varepsilon + F'(\phi^\varepsilon)$.
• Boundary Conditions:
  • Outer Boundary ($\partial \Omega$): No-slip condition ($u^\varepsilon = 0$).
  • Solid Inclusions ($\Gamma^\varepsilon_s$): Free-slip (Navier slip with zero tangential stress) conditions: $u^\varepsilon \cdot n^\varepsilon = 0$ and $(A^\varepsilon D(u^\varepsilon)n^\varepsilon) \cdot \tau^\varepsilon = 0$.
• Scaling: The capillary strength $\lambda_\varepsilon$ scales with $\varepsilon$ such that $\lim_{\varepsilon \to 0} \lambda_\varepsilon = \lambda \in [0, +\infty)$. The external body force $g^\varepsilon$ scales as $\sqrt{\lambda_\varepsilon}$.

2. Methodology

The authors employ a rigorous two-step analytical approach:

A. Existence and Uniform A Priori Estimates (Microscopic Level)

For a fixed $\varepsilon > 0$, the authors prove the existence of weak solutions and derive estimates uniform in $\varepsilon$.

• Galerkin Approximation: They construct approximate solutions using a four-step Galerkin scheme based on Stokes-type eigenfunctions adapted to the mixed boundary conditions (no-slip on outer boundary, free-slip on inclusions).
• Extension Operators: A critical technical challenge is the $\varepsilon$-dependent domain. The authors construct:
  • Bogovskiĭ Operators: Defined on both the fluid domain $\Omega^\varepsilon_p$ and the disconnected solid domain $\Omega^\varepsilon_s$ to handle the divergence constraint and pressure reconstruction. Crucially, the norms of these operators are independent of $\varepsilon$.
  • Restriction and Extension Operators: Used to extend the order parameter $\phi^\varepsilon$ and velocity $u^\varepsilon$ from the porous domain to the whole domain $\Omega$ while preserving uniform bounds in Sobolev spaces.
• Handling the Source Term: Unlike classical Cahn-Hilliard models, the source term $G(\phi^\varepsilon)$ breaks mass conservation. The authors derive specific estimates for the spatial mean of $\phi^\varepsilon$ and the chemical potential to handle the lack of zero-mean properties, utilizing the specific structure of $G$ (bi-Lipschitz).
• Energy Estimates: They establish a uniform energy dissipation law, proving that the total energy is bounded by $C\sqrt{\lambda_\varepsilon}$.

B. Periodic Homogenization (Macroscopic Limit)

The authors perform the limit $\varepsilon \to 0$ using the Periodic Unfolding Method ($T^\varepsilon_*$).

• Two-Scale Convergence: They identify macroscopic fields ($u, \phi, \mu$) and microscopic correctors ($\hat{u}, \hat{\phi}, \hat{\mu}$) defined on the reference cell $Y_p$.
• Case Distinction: The homogenized limit depends critically on the limit value $\lambda = \lim_{\varepsilon \to 0} \lambda_\varepsilon$:
  1. Vanishing Capillarity ($\lambda = 0$): The capillary force becomes negligible relative to other terms in the scaled momentum equation.
  2. Balanced Regime ($\lambda \in (0, +\infty)$): The capillary force, viscous dissipation, and nonlinear convection remain in balance.

3. Key Contributions and Results

1. Existence of Weak Solutions

The paper establishes the existence of weak solutions for the NSCH system in a 3D perforated domain with mixed boundary conditions (no-slip + free-slip) and a non-conservative source term. This extends previous results which typically assumed no-slip everywhere or mass conservation.

2. Derivation of Two Distinct Effective Models

Depending on the scaling of the capillary strength, two different macroscopic models are derived:

• Case I: $\lambda = 0$ (Stokes-Cahn-Hilliard Limit)

  • The macroscopic momentum equation loses the nonlinear convective term $(u \cdot \nabla)u$.
  • The phase field equation loses the advective transport term $u \cdot \nabla \phi$.
  • Result: A Stokes-Cahn-Hilliard (SCH) system where the flow is driven solely by the homogenized capillary force and external body forces, with no macroscopic inertia or advection.
  • Equation (1.5): $\partial_t u - \nabla \cdot (A^{hom} D(u)) + \nabla \tilde{p} + \phi C^{hom} \nabla \mu = g$.

• Case II: $\lambda \in (0, +\infty)$ (Navier-Stokes-Cahn-Hilliard Limit)

  • The nonlinear convection and advection terms survive in the limit, scaled by $\sqrt{\lambda}$.
  • Result: A full Navier-Stokes-Cahn-Hilliard (NSCH) system at the macroscopic scale.
  • Equation (1.7): $\partial_t u + \sqrt{\lambda}(u \cdot \nabla)u - \nabla \cdot (A^{hom} D(u)) + \nabla \tilde{p} + \phi C^{hom} \nabla \mu = g$.
  • The phase field equation includes the advective term $\sqrt{\lambda} u \cdot \nabla \phi$.

3. Homogenized Tensors and Cell Problems

The authors explicitly define the effective (homogenized) tensors $A^{hom}$, $B^{hom}$, and $C^{hom}$ via cell problems on the reference pore $Y_p$. These tensors capture the influence of the microscopic geometry and the free-slip boundary conditions on the macroscopic transport properties.

4. Energy Convergence

The paper proves the convergence of the microscopic total energy to a homogenized energy functional. The limit energy satisfies an analogous dissipation law, confirming the thermodynamic consistency of the derived macroscopic models.

4. Significance

• Rigorous Justification of Upscaling: The work provides a mathematically rigorous derivation of macroscopic models for multiphase flow in porous media, moving beyond heuristic asymptotic expansions.
• Novel Boundary Conditions: It addresses the complex case of free-slip (zero tangential stress) on solid inclusions, which is physically relevant for certain microfluidic or superhydrophobic surfaces but mathematically challenging due to the lack of standard Poincaré inequalities.
• Non-Conservative Dynamics: By incorporating a source term $G(\phi)$, the model accounts for phase change or chemical reactions that alter the total mass, a scenario often ignored in standard Cahn-Hilliard homogenization.
• Regime Transition: The identification of two distinct macroscopic regimes (Stokes vs. Navier-Stokes) based on the capillary strength scaling offers new insights into how interfacial tension drives macroscopic flow in porous media.

In summary, this paper bridges the gap between microscopic diffuse-interface dynamics in complex geometries and macroscopic continuum models, providing a robust theoretical framework for simulating binary fluid mixtures in porous media with reaction terms and specific slip conditions.