Overview
This research investigates a time-dependent Navier-Stokes-Cahn-Hilliard (NSCH) system. The system models a binary incompressible mixture situated within a periodically perforated three-dimensional ($\mathbb{R}^3$) domain, denoted as $\Omega_p^\varepsilon$. The study focuses on establishing the existence of weak solutions for fixed $\varepsilon$ and subsequently deriving a homogenized system on the entire domain under specific limiting conditions.
Research Context
The system under consideration is the Navier-Stokes-Cahn-Hilliard type, which describes the dynamics of an incompressible binary mixture. The domain's geometry is characterized by periodic perforations, where obstacles have a diameter of order $\varepsilon^\alpha$ and a mutual distance of order $\varepsilon$. A specific condition is imposed on the obstacle size relative to the domain structure: $\alpha > 3$. This leads to the relationship $\varepsilon^\alpha/\varepsilon^3 \to 0$, which is identified as the subcritical dilute regime. The model incorporates a periodically oscillating viscosity tensor and a nonconservative source term within the Cahn-Hilliard equation. Boundary conditions include no-slip conditions on the outer boundary of the domain and free-slip conditions on the surfaces of the internal obstacles. The capillary coefficient, $\lambda^\varepsilon$, is specified to depend on $\varepsilon$, with $\lambda^\varepsilon > 0$ for every fixed $\varepsilon > 0$. The study then considers the behavior as $\lambda^\varepsilon \to \lambda \in [0, \infty)$.
Approach
For every fixed $\varepsilon > 0$, the study first proves the existence of a weak solution for the described Navier-Stokes-Cahn-Hilliard system. This proof includes the derivation of estimates that are uniform with respect to $\varepsilon$, and these estimates explicitly account for the $\lambda^\varepsilon$-scaling. Following the establishment of existence for fixed $\varepsilon$, the research then proceeds to derive the homogenized system on the whole domain. This homogenization process is contingent on the assumption that $\lambda^\varepsilon \to \lambda \in [0, \infty)$. A further aspect of the approach involves proving the convergence of the time-integrated normalized microscopic energy to its corresponding macroscopic energy counterpart.
Findings
- For every fixed $\varepsilon > 0$, the existence of a weak solution for the time-dependent Navier-Stokes-Cahn-Hilliard system is proven.
- Estimates uniform in $\varepsilon$ are derived, with an explicit dependence on the $\lambda^\varepsilon$-scaling.
- Assuming $\lambda^\varepsilon \to \lambda \in [0, \infty)$, the homogenized system on the whole domain is derived.
- The subcritical obstacles, defined by $\alpha > 3$ and mutual distance of order $\varepsilon$, do not introduce an additional resistance term in the homogenized system.
- The cell problems, necessary for homogenization, are posed on the full periodic cell.
- The scalar correctors are found to vanish, resulting in the scalar diffusion operators remaining unchanged in the homogenized system.
- The periodically oscillating viscosity gives rise to a time-dependent effective viscosity tensor in the homogenized system.
- If the limit of the capillary coefficient, $\lambda$, is $0$, the homogenized system decouples into an effective unsteady Stokes system and a Cahn-Hilliard system with a source term.
- If $\lambda \neq 0$, the homogenized system retains the Navier-Stokes-Cahn-Hilliard coupling, specifically involving convection, phase transport, and capillary forcing weighted by $\sqrt{\lambda}$.
- The time-integrated normalized microscopic energy is proven to converge to the corresponding macroscopic energy.
Why This Matters
This study provides foundational mathematical results concerning the behavior of binary incompressible mixtures in complex, perforated environments. Understanding the existence of solutions and the homogenized macroscopic behavior is crucial for modeling such systems under various conditions, particularly those involving microstructural complexities and varying material properties.