Navier–Stokes–Cahn–Hilliard Equations

• Navier–Stokes–Cahn–Hilliard Equations are a coupled PDE system that integrate fluid momentum conservation with phase-field evolution for consistent interface modeling.
• The system employs a multiple-scale asymptotic expansion to reconcile fast acoustic modes with slow diffusive and convective dynamics.
• These equations underpin advanced diffuse-interface models applied to droplet dynamics, phase transitions, and reactive fluid interfaces in multiphase media.

The Navier–Stokes–Cahn–Hilliard (NS–CH) equations are a canonical class of coupled partial differential equations (PDEs) modeling the interplay between hydrodynamics and phase separation in multiphase fluid systems, especially binary mixtures with diffusive interfaces. These equations integrate the fluid momentum conservation of the (possibly compressible) Navier–Stokes equations with the mass-conserving, thermodynamically motivated Cahn–Hilliard phase-field evolution. The resulting framework allows the quantitative, variationally consistent description of interface motion, topology change, capillarity, and interfacial flows, underpinning modern diffuse-interface models for multiphase and complex fluids.

1. Mathematical Formulation and Constitutive Principles

In coupled NS–CH models for a compressible or quasi-compressible binary mixture, the primary variables are the total density $\rho(\boldsymbol{x},t)$, mass-averaged velocity $\boldsymbol{v}(\boldsymbol{x},t)$, concentration field or order parameter $C(\boldsymbol{x},t)$, and the chemical potential $\mu(\boldsymbol{x},t)$. The system consists of:

• Mass conservation (full continuity):

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \, \boldsymbol{v}) = 0,$$

with $\rho$ a function of $C$ due to partial miscibility and composition-dependent density even for “incompressible” pure components.

• Navier–Stokes momentum balance (including Korteweg stress):

$$\rho \left( \partial_t \boldsymbol{v} + (\boldsymbol{v} \cdot \nabla) \boldsymbol{v} \right) = -\nabla p + \nabla \cdot \tau_\eta - \epsilon \nabla \cdot \tau_\epsilon + \rho \boldsymbol{g},$$

where: - $\tau_\eta$ is the viscous (Newtonian) stress,

$$\tau_\eta = \eta \left[ \nabla \otimes \boldsymbol{v} + (\nabla \otimes \boldsymbol{v})^T - \tfrac{2}{3} (\nabla \cdot \boldsymbol{v}) I \right],$$

• $\tau_\epsilon = \rho \, \nabla C \otimes \nabla C$ is the Korteweg (capillary) stress,
• $\epsilon > 0$ is the capillary coefficient.
  • Convective Cahn–Hilliard equation for the composition:

$$\rho \left( \partial_t C + \boldsymbol{v} \cdot \nabla C \right) = \alpha \nabla^2 \mu,$$

with mobility $\alpha$.

• Chemical potential $\mu$ from the variational (Cahn–Hilliard) free energy:

For a density-dependent free energy

$$f(\rho, C, \nabla C) = f_0(C) + \tfrac{\epsilon}{2} |\nabla C|^2,$$

and assuming each pure component is incompressible (so $\rho = \rho(C)$ independent of $p$), one derives:

$$\mu(p, C) = f_0'(C) - \nabla \cdot \left( \epsilon \nabla C \right) - \frac{p}{\rho^2} \frac{d \rho}{dC} - \frac{\epsilon}{\rho} \nabla \cdot (\rho \nabla C).$$

A prototypical local free energy to model phase transitions near a critical point is:

$f_0(C) = a (C - C_{cr})^2 + b (C - C_{cr})^4,$

yielding $\mu_0(C) = 2a(C - C_{cr}) + 4b(C - C_{cr})^3$.

2. Multiple-Scale Derivation and the Boussinesq Approximation

A salient feature of the full NS–CH system is the coexistence of multiple dynamical time scales: a rapid expansion (density-acoustic) mode, a convective time scale, and a slower Cahn–Hilliard diffusive relaxation. For many binary-mixture flows (e.g., slow dissolution, interface-driven transport), only the slow, incompressible, convective-diffusive regime is dynamically relevant.

Using a rigorous multiple-scale asymptotic expansion (parametrized by a small parameter $\chi$ measuring the ratio of expansion and diffusive times), the fields are decomposed into fast oscillating and slow mean parts. By expanding the equations and performing temporal averaging, one derives—at leading slow order—the incompressible Boussinesq NS–CH system:

• Divergence-free condition: $\nabla \cdot \boldsymbol{u} = 0$
• Momentum (Navier–Stokes–Korteweg):

$$\partial_t \boldsymbol{u} + (\boldsymbol{u} \cdot \nabla) \boldsymbol{u} = -\nabla \Pi + \frac{1}{\mathrm{Re}} \nabla^2 \boldsymbol{u} - C \nabla \mu - \mathrm{Ga}\,\phi\,C\,\boldsymbol{\gamma}$$

where $-C\nabla \mu$ bundles capillary and thermodynamic interface effects, $\mathrm{Re}$ is the Reynolds number, $\mathrm{Ga}$ is the Galileo number, $\phi C$ represents the local density perturbation, and gravity $\boldsymbol{\gamma}$ enters via the Boussinesq buoyancy force.

• Cahn–Hilliard (species):

$$\partial_t C + (\boldsymbol{u} \cdot \nabla) C = \frac{1}{\mathrm{Pe}} \nabla^2 \mu$$

with $\mathrm{Pe}$ the Peclet number.

• Chemical potential:

$\mu = 2A C + 4 C^3 - \mathrm{Ca} \nabla^2 C$

where $A$ is related to the distance from the critical point and $\mathrm{Ca}$ is the capillary number.

The Boussinesq approximation emerges as a self-consistent homogenized limit of the full equations, not as an ad hoc constraint, preserving solenoidal incompressibility and the coupling of capillarity and slow interfacial diffusion (Vorobev, 2010).

3. Model Structure, Boundary Conditions, and Physical Mechanisms

The final Boussinesq NS–CH model structurally incorporates:

• Inertia and convection: $$\partial_t \boldsymbol{u} + (\boldsymbol{u}\cdot \nabla)\boldsymbol{u}$$
• Pressure gradient: $- \nabla \Pi$, enforcing incompressibility
• Viscous dissipation: $(1/\mathrm{Re})\nabla^2 \boldsymbol{u}$
• Capillarity and interfacial forces: $-C\nabla \mu$ acts as a non-classical body force sharply localized at interfaces (where $|\nabla C|$ is large)
• Buoyancy (Boussinesq): $-\mathrm{Ga}\,\phi\,C\,\boldsymbol{\gamma}$, arising from solutal density variations

The Cahn–Hilliard equation encompasses advection by the flow and mass diffusion driven by gradients of chemical potential, crucial for modeling interface motion, dissolution, coarsening, and phase separation.

Standard boundary conditions for interface problems include:

• No-slip for velocity ($\boldsymbol{u} = 0$ on walls)
• No flux for mass ($\boldsymbol{n} \cdot \nabla \mu = 0$)
• Neutral (or prescribed) wetting: ($\boldsymbol{n}\cdot \nabla C = 0$), corresponding to a $90^\circ$ contact angle
• Initial data consistent with global conservation laws for solute and total mass

4. Applications, Validity Regimes, and Physical Interpretation

The NS–CH equations, especially in the Boussinesq regime, describe multiphase processes where interface deformation, capillarity, buoyancy, and solute transport interact on comparable "slow" time scales. Key application areas include:

• Droplet dissolution and nucleation in porous media or capillaries
• Evaporation and condensation with moving contact lines
• Growth and dynamics of solidification/melting fronts
• Reactive fluid interfaces and precipitation/polymerization
• Near-critical binary-liquid convection and interfacially driven instabilities

The model applies when density contrast is small ($\phi\ll 1$), interfacial thickness is much smaller than macroscopic scales, and both hydrodynamics and interfacial diffusion occur on slow, comparable time scales (Vorobev, 2010).

Each term has direct physical meaning:

• Fluid inertia and convective acceleration: $$\partial_t\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}$$
• Enforced incompressibility: $-\nabla \Pi$
• Viscosity: $1/\mathrm{Re}\nabla^2\boldsymbol{u}$
• Capillarity/thermodynamics: $-C\nabla \mu$ (sharp at interfaces)
• Solutal (Boussinesq) buoyancy: $-\mathrm{Ga}\,\phi\,C\,\boldsymbol{\gamma}$
• Species transport: convection–diffusion dynamics for $C$

5. Derivation Strategy and Mathematical Properties

The derivation proceeds as follows:

• Write the full quasi-compressible NS–CH system in nondimensional form
• Identify the three time scales via linear theory and introduce the small parameter $\chi$
• Expand $(\boldsymbol{v}, p, C, \rho, \mu)$ in asymptotic series and distinguish scalings so that correct physical balances survive at leading order
• Decompose fields into fast-decaying and slow (mean) components; average over fast timescale
• At slow order, incompressibility and "filtered" Navier–Stokes–Korteweg and Cahn–Hilliard equations are recovered dynamically—not imposed ad hoc

Mathematically, the resulting system admits:

• Exact conservation of mass and (in the absence of external forces) a Lyapunov free-energy law
• Well-posedness (at least at the weak-solution level) for a wide range of boundary conditions and free energy functionals
• Robustness to inclusion of additional physics such as gravity, chemical reactions, phase change kinetics, and more

6. Extensions and Broader Research Context

While the NS–CH equations described here focus on quasi-compressible binary mixtures, the methodology supports numerous generalizations:

• Compressible flows and microstructure effects
• Ternary and multicomponent interfaces
• Non-isothermal (thermodynamically coupled) systems
• Realistic dynamic wetting and contact-line models
• Coupling to elastic or structured phases (e.g., in elasto-capillarity, soft matter)
• Alternative derivations: variational-metriplectic formalisms, phase-field turbulence, and nonlocal free energies

The systematic multiple-scale approach ensures thermodynamic consistency and correct limiting behavior across a large set of parameters and physical regimes, justifying its ubiquity in interface-driven flow modeling (Vorobev, 2010).

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References:

• "Boussinesq approximation of the Cahn–Hilliard–Navier–Stokes equations" (Vorobev, 2010)