Coupled Surface–Particle Systems

• Coupled surface–particle systems are multiphysics configurations where deformable surfaces interact dynamically with rigid particles, governed by capillary and fluid forces.
• They employ diverse frameworks such as three-way coupling, LBM–DEM, and neural surrogates to quantify complex feedbacks between interfaces and particles.
• Advanced numerical methods, including hybrid Eulerian–Lagrangian discretization, ensure accurate simulation of capillarity, friction, and interfacial dynamics in practical applications.

A coupled surface–particle system is defined as an interacting multiphysics configuration in which particles or rigid bodies couple dynamically with a contiguous (often deformable) surface or interface. These systems exhibit complex feedbacks, where both the bulk flow and the interfacial geometry/molecular state evolve in response to particle dynamics, and vice versa. Coupled surface–particle systems arise in capillarity-dominated solid–fluid interactions, multiphase suspensions with particulate boundaries, field–boundary diffusion problems, optically active nanosystems, and active matter powered by surface field gradients.

1. Governing Frameworks and Physical Models

The formulation of coupled surface–particle systems depends critically on the regime and dominant physics.

• Three-way solid–membrane–fluid coupling: The “three-way” framework introduces a thin virtual membrane of explicit thickness $h$ modeled as a Lagrangian simplicial mesh, which mediates the interaction between an Eulerian volume of incompressible fluid and rigid bodies (Ruan et al., 2021). Surface tension is encoded as total membrane energy $E_c = \sigma A$, leading to capillary forces and stiffness terms. The membrane carries surface mass, couples to the fluid via pressure impulses, and to solids by adhesion and nonpenetration constraints. Governing equations combine Navier–Stokes for the bulk, a shell equation for the membrane, and generalized coordinates for the rigid body.
• LBM–DEM coupled schemes: Two-component fluids are described by lattice Boltzmann methods (LBM), while particle contacts and their interactions (including friction and capillarity) are handled via a discrete element method (DEM). Capillary forces at the three-phase contact line and wettability effects (via geometric boundary conditions) are included (Naga et al., 15 May 2025). Momentum-exchange methods transfer hydrodynamic forces from LBM to the particle DEM.
• Euler–Lagrange and surrogate frameworks: In flow regimes such as hypersonic or rarefied gas–particle mixtures, an Eulerian flow field (solving the compressible Navier–Stokes equations under thermochemical nonequilibrium) couples to Lagrangian particles, with explicit exchange terms accounting for momentum and energy transfer, and special source terms for wall impacts and erosion (Nam et al., 6 Dec 2025).
• Stochastic lattice models: For field–boundary coupling, exclusion processes defined on lattices of different dimension with special exchange rules yield, in the hydrodynamic limit, a PDE system coupling bulk and lower-dimensional “surface” densities via Robin or reaction terms (Alfaro et al., 2024).

2. Numerical Methods and Discretization Techniques

The choice of discretization reflects distinct topological and multiphysics challenges.

• Hybrid Eulerian–Lagrangian discretization: Surface membranes utilize unstructured meshes, refined to at least twice the grid resolution of the fluid, and remeshed dynamically; fluids employ MAC (Marker-and-Cell) grids for pressure and velocity. Coupling operators include interpolation matrices ($W$) for velocities and divergence/gradient operators ($G, G^\top$) (Ruan et al., 2021).
• LBM–DEM and SPH–DEM frameworks: LBM evolves discrete velocity distributions on D3Q19 (or higher) lattices with streaming/collision steps and applies color-gradient, multiphase, or Bingham-rheology extensions. DEM advances rigid grains (possibly complex composites) with non-linear contact and frictional force models. The coupling occurs at bounce-back links, where fluid distributions are reflected and corrected with wall velocities. In SPH–DEM, fluid “particles” discretize the locally-averaged Navier–Stokes equations using smoothing kernels with compact support, and particles are advanced via Newton’s laws, subject to drag and collisional contacts (Leonardi et al., 2015, Robinson et al., 2013).
• Neural surrogate closures: For suspensions of complex-shaped particles, boundary element methods (BEM) with regularized Stokeslets generate training sets for neural-operator closures, mapping local flow gradients and particle geometry to force-multipole responses (stresslet, angular velocity, thrust), which are then embedded as source terms in mesoscale solvers (e.g., FCM) (Laudato, 16 Dec 2025).

3. Capillarity, Surface Forces, and Wetting Physics

Surface–particle systems often operate near force-balance regimes where capillarity dominates over gravity or inertia. Several mechanisms are central:

• Explicit surface tension via membrane energy: Capillary forces are computed from the first variation of the membrane area, and their discrete implementation involves forces proportional to cross-products of mesh edge vectors and triangle normals. The block capillary stiffness is assembled across triangle elements (Ruan et al., 2021).
• Diffuse-interface and contact-line models: In LBM, continuum surface force (CSF) terms and order-parameter advection equations capture the presence of sharp or diffuse interfaces. Geometric boundary conditions allow control of contact angles by manipulating the interface normal vector at the wall (Naga et al., 15 May 2025).
• Capillary adhesion and Cheerios effect: Capillary attraction between floating solids is resolved both numerically—recovering force amplitudes that match the Young–Laplace law, $\Delta p = \sigma \kappa$—and analytically in terms of interaction integrals along contact perimeters. Surface tension-driven effects (e.g., floating of high-density objects, Cheerios clustering) are quantitatively reproduced (Ruan et al., 2021).
• Friction-capillarity competition: The interplay between contact friction and capillary adhesion sets regimes for particle removal in droplet cleaning: a critical friction-to-capillary ratio $F_f/F_\text{max} \approx 1$ determines transition from “push–pull” to “enter–exit” collision regimes for particle pickup by moving drops (Naga et al., 15 May 2025).

4. Dynamical Coupling and Solver Architectures

Robust solution strategies are required to address the tightly coupled and often ill-conditioned nature of the surface–particle system:

• Monolithic constraint solves: Fully coupled, symmetric but indefinite systems join incompressibility, surface-membrane, and rigid-body constraints. A prediction–correction strategy forecasts rigid/membrane positions, resolves nonpenetration by projection, applies external and capillary forces, assembles the block system, and solves via diagonally preconditioned conjugate gradient to high accuracy ($\sim 10^{-4}$) (Ruan et al., 2021).
• Partitioned multi-solver coupling: For multiphysics at extreme conditions, independent fluid and particle solvers advance on their own time steps, exchanging data at regular intervals via coupling libraries (e.g., preCICE), with source and sink terms for both bulk and surface. Variable-fidelity chemistry and adaptive drag/Nusselt correlations are supported on a per-domain or per-trajectory basis (Nam et al., 6 Dec 2025).
• Meshless approaches: Combined SPH–DEM methods allow simulation around arbitrarily complex surfaces and capture both dilute and dense packing, with fluid–particle forces implemented by interpolated drag and pressure/viscous stress as well as full action–reaction updates on both phases (Robinson et al., 2013).

5. Validation, Quantitative Results, and Physical Phenomena

Validation benchmarks and physical studies span a range of domains:

• Force and energy benchmarks: Simulations reproduce energy conservation in elastic impacts, terminal velocities and restitution coefficients in settling, and match analytical solutions for drag, stresslet, and rolling/sliding friction (Naga et al., 15 May 2025, Robinson et al., 2013, Laudato, 16 Dec 2025).
• Free-surface and capillarity phenomena: Floating of heavy objects (e.g., paperclips, robots), surface-induced particle attraction (“Cheerios effect”), and surfactant-exacerbated sinking are all matched quantitatively, using integrated capillary and buoyancy force measurements and surface deformation scaling laws (Ruan et al., 2021).
• Self-cleaning by drop motion: Simulated trajectories of particles under droplet impact reproduce experimentally observed force histories and two dominant regimes (“push–pull” and “enter–exit”), identifying friction as necessary for correct dynamical response; frictionless models yield artifacts in rolling direction and removal (Naga et al., 15 May 2025).
• Hydrodynamic limit and field–boundary exchange: In lattice-based SSEP models, rigorous derivation shows how microscale jump processes couple to macroscopic PDE systems (e.g., the field–road model) with accurate Robin-type flux conditions and reaction terms. This bridges disparate dimension constraints at the coupled interface (Alfaro et al., 2024).
• Active matter and field-mediated pairing: In source–inert particle experiments, controlled transitions from rotational to translational pair trajectories are mapped to bifurcations in the dimensionless drag parameter $\eta$; a critical value $\eta_c$ is observed, with scaling of motion curvature consistent with supercritical pitchfork bifurcation (Ishikawa et al., 2021).

6. Advanced Applications and Theoretical Extensions

• High-speed multiphase erosion: Integrated Euler–Lagrange frameworks (e.g., ORACLE+HEGEL) address two-way coupled gas–particle flows, modeling collision heating and surface recession as source terms in complex boundary conditions, validated on Martian entry capsule recession and shock–drag effects in dusty nozzles (Nam et al., 6 Dec 2025).
• Quantum coherent nanostructures: In waveguide-coupled plasmonic systems, coupling between a quantum emitter and a nearby nanoparticle forms hybrid excitons with tunable plasmon transmission/reflection, enabling device-level applications such as quantum switches and single-photon transistors. Transmission amplitude $t(\omega)$ and reflection $r(\omega)$ show sharply controllable crossovers by varying interparticle distance $R$ and size $a$ (Ko et al., 2015).
• Data-driven hydrodynamic closures: Neural-operator surrogates constructed from high-fidelity BEM simulation data deliver $O(1\%)$ accuracy for stresslet and rotation responses, achieving 95th percentile errors below $3\%$ on untrained orientations and flow conditions, and enabling particle-resolved Stokes simulation of microstructure and rheology at population scales unfeasible with classical methods (Laudato, 16 Dec 2025).

7. Limitations, Open Problems, and Best Practices

• Resolution and error control: For SPH–DEM and LBM–DEM, accurate momentum balance requires smoothing lengths or lattice resolutions at least 2–6× particle diameter; failing this, errors in drag, pressure, and interphase force calculations rise sharply (Robinson et al., 2013, Naga et al., 15 May 2025).
• Porosity and friction sensitivity: Accurate modeling of suspension flows with sharp porosity gradients demands careful selection of smoothing parameters to capture hindered settling and particle stratification accurately. Friction coefficients control dynamical response in droplet self-cleaning; incorrect values result in qualitatively incorrect force histories and particle motion (Naga et al., 15 May 2025).
• Surface–bulk dimension mismatches: Rigorous treatment of coupled surface–bulk systems across dissimilar geometries (e.g., curves and domains) requires advanced stochastic methods and replacement lemmas to derive and validate the macroscopic limit, as seen in the field–road diffusion context (Alfaro et al., 2024).
• Interface tracking and remeshing: In capillarity-dominant three-way systems, mesh connectivity must be updated each timestep, using local flips, splits, and collapses to ensure stability and fidelity of the surface representation (Ruan et al., 2021).
• Domain-specific best practices: Recommendations include validating against known benchmarks, conservatively choosing coupling/averaging lengths, applying minimal artificial viscosity, and tuning friction coefficients to match experimental removal thresholds and rolling behavior (Robinson et al., 2013, Naga et al., 15 May 2025).

In summary, coupled surface–particle systems integrate multiphysical modeling of interfaces, solids, and fluids with rigorous numerical and data-driven methods. These frameworks enable quantification and prediction of capillarity-driven phenomena, interface-mediated transport, field–boundary diffusion, and active matter dynamics across a spectrum of scientific and engineering domains.