Cahn-Hilliard–Navier–Stokes System

• The CHNS system is a continuum model for two-phase flows that couples Navier–Stokes equations with a Cahn–Hilliard phase-field approach to capture interfacial dynamics.
• It uses a free-energy variational principle to incorporate capillarity and mass conservation, enabling accurate simulation of phase separation and merging phenomena.
• Key advances include rigorous mathematical analysis, energy-stable numerical schemes, and applications in both compressible and incompressible flow regimes.

The Cahn–Hilliard–Navier–Stokes (CHNS) system is a fundamental continuum model for two-phase fluid flow using a diffuse-interface (phase-field) approach. It consists of the incompressible or compressible Navier–Stokes equations for the (possibly variable) mass density and mass-averaged velocity, coupled with a convective Cahn–Hilliard equation for a concentration field representing the local phase composition. The system is derived from a free energy principle and incorporates capillarity and interface effects within a unified PDE framework. Several variants have been developed, including compressible, nonlocal, non-isothermal, and models with mass transfer and active stresses.

1. Mathematical Formulation of the CHNS System

The classical isothermal CHNS system for two-phase flow at constant temperature (isothermal) in domain $\Omega\subset\mathbb R^d$ (for $d=2,3$) consists of the following equations (Freistuhler et al., 2013, Padhan et al., 12 Mar 2025):

• Mass conservation

$\partial_t\rho + \nabla\cdot(\rho u) = 0,$

where $\rho(x,t)$ is the total mass density and $u(x,t)$ is the mass-averaged velocity.

• Momentum equation

$$\partial_t(\rho u) + \nabla\cdot(\rho u\otimes u) = -\nabla p + \nabla\cdot(2\eta D(u)) + \nabla\cdot(\mu \nabla c\otimes\nabla c).$$

Here, $D(u)=\frac12(\nabla u+\nabla u^T)$ is the rate-of-deformation tensor, $\eta>0$ the shear viscosity, $p$ the pressure, $c(x,t)$ the phase concentration, and $\mu$ the chemical potential.

• Cahn–Hilliard concentration equation

$$\partial_t(\rho c) + \nabla\cdot(\rho c u) = \nabla\cdot(M(c)\nabla\mu),$$

with mobility $M(c)>0$ (possibly constant) and the chemical potential

$$\mu = \frac{\partial F}{\partial c}(c, \nabla c) - \kappa\Delta c,$$

where $F(c,\nabla c)$ is the Ginzburg–Landau free energy density (e.g., $F(c,\nabla c)=W(c)+\frac\kappa2|\nabla c|^2$), with $W$ a double-well potential, and $\kappa>0$ the capillarity coefficient.

This formulation encompasses both density variations (compressible models) and sharp or diffuse interfaces, linking hydrodynamics with phase-field evolution.

2. Structural Reductions and the Korteweg Limit

In the regime of two incompressible pure phases with distinct densities, the CHNS model admits a reduction that clarifies its connection to the Navier–Stokes–Korteweg system (Freistuhler et al., 2013):

• The concentration $c$ becomes an algebraic function of $\rho$ via

$$c = \frac{\rho - \hat\rho_2}{\hat\rho_1 - \hat\rho_2},$$

where $\hat\rho_1$ and $\hat\rho_2$ are the (constant) specific densities of the two phases.

• Eliminating $c$ yields a compressible Navier–Stokes–Korteweg system for $(\rho, u)$ with a modified viscous stress that includes a nonlocal operator acting on $\nabla\cdot u$:

$$S_2(u) = 2\eta D(u) + \bigl(\zeta(\rho)+\frac{(\Delta\hat\rho)^2}{4\theta}\bigr)\nabla\cdot u\,\mathbf I + \frac{(\Delta\hat\rho)^2}{4\theta}A^{-1}(\nabla\cdot u)\,\mathbf I,$$

with $A^{-1}$ the inverse Laplacian (Newtonian potential) operator and $K(\rho)$ the classical Korteweg tensor (Freistuhler et al., 2013).

This integro-differential structure is the hallmark of mass-conserving phase dynamics and distinguishes the NSCH reduction from the classical Navier–Stokes–Korteweg and Navier–Stokes–Allen–Cahn models.

3. Analytical Results: Existence, Uniqueness, and Attractors

Extensive advances have been made in the mathematical theory of the CHNS system and its nonlocal and stationary variants.

• Compressible, stationary CHNS: For compressible models in bounded domains, existence of weak stationary solutions is established for broad classes of adiabatic exponent, forces, and free energies (Liang et al., 2020). The weak solution framework involves simultaneous control of total density $\rho$, velocity $u$, concentration difference $c$, and chemical potential $\mu$, under energy dissipation and mass conservation. The construction proceeds via two-level regularization (artificial diffusions and penalizations), energy estimates, compactness arguments, and limit passage in nonlinearities.
• Nonlocal CHNS: For models with convolution-type nonlocal free energy, existence of global weak solutions is proved in both 2D and 3D (Colli et al., 2011, Frigeri et al., 2013), with uniqueness of strong solutions and regularization after finite time in 2D for regular potentials. For singular (e.g., logarithmic) potentials, existence and exponential stability of stationary solutions as well as attractor theory are available (Biswas et al., 2018). The energy identity plays a critical role in constructing absorbing sets and global attractors in 2D, and the Chepyzhov–Vishik trajectory attractor in 3D under nonautonomous forcing (Frigeri et al., 2011).
• Dynamic boundary conditions and bulk-surface coupling: Thermodynamically consistent bulk-surface CHNS models and dynamic boundary conditions have been analyzed with respect to global existence, uniqueness (in 2D), and the presence of compact attractors (Giorgini et al., 2022, You et al., 2016).

Collectively, these results establish a rigorous PDE foundation for CHNS-type models across a broad class of settings and nonlinearities.

4. Thermodynamic Structure and Model Variants

The CHNS system is derived from a variational free-energy principle and dissipative mechanism, leading to energy-dissipation identities and entropy production (Freistuhler et al., 2013, Zaidni et al., 16 Feb 2024, Lam et al., 2017):

• Energetics: The total free energy---kinetic energy plus Ginzburg–Landau free energy---decreases monotonically under physical dissipation by viscosity and mobility, modulo any source/sink terms (e.g., mass transfer, chemotaxis, or external forces).
• Thermodynamically consistent generalizations: Modern variants employ metriplectic (Hamiltonian plus metric) formalisms to systematically encode arbitrary dissipative processes alongside conservation laws. The metriplectic 4-bracket formalism produces models that preserve the noncanonical Poisson bracket (Hamiltonian structure) for reversible processes and ensures dissipation of Casimir invariants (entropy) by construction (Zaidni et al., 16 Feb 2024).
• Chemotaxis, mass transfer, and active matter: CHNS systems with mass transfer, nutrient/chemical coupling, and active stresses have been developed for biological flows and active emulsions (Lam et al., 2017, Padhan et al., 12 Mar 2025). These models introduce additional source terms and modify the energetic structure to reflect coupling with additional fields (e.g., chemical free energy, active stress tensors).

5. Computational Approaches and Numerical Analysis

Numerical methods for CHNS systems must reconcile stiffness due to high-order (biharmonic) operators, conservation properties, and stability across parameter regimes:

• Energy-stable schemes: Temporal discretizations (e.g., convex splitting for Cahn–Hilliard, projection or pressure-correction for Navier–Stokes) are constructed to satisfy discrete analogs of the continuous energy law, ensuring unconditional stability (Han et al., 2014, Feng et al., 2018, Diegel et al., 2016).
• Implicit-explicit (IMEX) schemes: Efficient linearly implicit or IMEX time-stepping methods leverage explicit treatment of convective terms and implicit treatment of the stiff Cahn–Hilliard and viscous terms, yielding stable large-time-step algorithms for both incompressible and compressible CHNS (Mulet, 30 Mar 2024).
• High-performance solvers and adaptivity: Scalable solvers for large-scale problems employ multigrid or domain decomposition within finite element or finite difference frameworks, often on adaptive meshes for resolving interfacial structures and turbulent flows (Khanwale et al., 2021).
• Numerical validation: Schemes are validated on canonical problems including spinodal decomposition, Rayleigh–Taylor and Kelvin–Helmholtz instabilities, droplet coalescence, and rising bubble benchmarks, confirming energy dissipation, mass conservation, and quantitative agreement with theoretical predictions or established benchmarks.

6. Physical and Modeling Significance

The CHNS framework captures a wide array of two-phase and multiphase flow phenomena with consistent treatment of interface dynamics and hydrodynamics:

• Interface modeling: The phase-field variable provides a mathematically convenient and physically justified means to incorporate interfacial tension (capillarity), coarsening/growth, and topological changes (merging, breakup) without explicit interface tracking (Padhan et al., 12 Mar 2025).
• Compressibility and varying density: The CHNS framework accommodates both incompressible and compressible mixtures, with appropriate modifications to continuity and momentum equations, and is well-suited to addressing multiphase flows with strong density or viscosity contrasts (Freistuhler et al., 2013, Liang et al., 2020).
• Bulk-surface and dynamic boundary effects: Extended models represent surface diffusion, dynamic contact angles, and mass exchange between bulk and boundaries, yielding realistic descriptions of wetting phenomena and interfacial flows in confined or complex geometries (Giorgini et al., 2022).
• Generalizations: The CHNS system forms the backbone of advanced multiphase models, including ternary and multicomponent fluids, active fluids, and systems exhibiting chemotactic or reactive behaviors, enabling application in soft matter, biological flows, and materials science (Padhan et al., 12 Mar 2025).

In summary, the Cahn–Hilliard–Navier–Stokes system constitutes a comprehensive, physically consistent PDE model for multiphase fluid dynamics, unifying energy-based interface modeling and viscous hydrodynamics. The CHNS framework supports robust mathematical analysis, diverse numerical discretizations, and sophisticated physical modeling extending from laminar to turbulent and active regimes (Freistuhler et al., 2013, Liang et al., 2020, Feng et al., 2018, Han et al., 2014, Diegel et al., 2016, Zaidni et al., 16 Feb 2024, Padhan et al., 12 Mar 2025).