[Paper Review] Continuum Model for Nanoscale Multiphase Flows

Alexandre M. Tartakovsky|arXiv (Cornell University)|May 21, 2018

Computational Fluid Dynamics and Aerodynamics4 references3 citations

TL;DR

This paper proposes a nonlocal continuum model for nanoscale multiphase flows that incorporates a nonlocal surface tension term via an integral of a molecular-force-like function, enabling accurate capture of curvature-dependent surface tension and interfacial pressure drops. The model recovers the Young-Laplace law at larger scales while resolving mesoscale interfacial physics at molecular scales (≤10 nm), validated through analytical solutions and SPH simulations showing agreement with capillary wave theory and curvature-dependent surface tension trends.

ABSTRACT

We propose a nonlocal model for surface tension. This model, in combination with the Landau-Lifshitz-Navier-Stokes equations, describes mesoscale features of the multiphase flow, including the static (pressure) tensor and curvature dependence of surface tension. The nonlocal model is obtained in the form of an integral of a molecular-force-like function added into the momentum conservation equation. We present an analytical steady-state solution for fluid pressure at the fluid-fluid interface and numerical Smoothed Particle Hydrodynamics solutions that reveal the mesoscopic features of the proposed model.

Motivation & Objective

• To develop a continuum model that captures nanoscale interfacial physics, such as curvature-dependent surface tension and pressure drops across fluid interfaces, which are not resolved by classical Young-Laplace models.
• To address the limitation of the classical Young-Laplace law, which fails at molecular scales (≤10 nm), where surface tension decreases with decreasing curvature radius.
• To incorporate nonlocal effects into the momentum conservation equation using a molecular-force-like integral term, enabling recovery of mesoscale features without full molecular dynamics.
• To validate the model against analytical solutions and Smoothed Particle Hydrodynamics (SPH) simulations, particularly for interfacial fluctuations and static stress tensors.
• To demonstrate that the model accurately reproduces thermally driven capillary wave spectra and Rayleigh instability thresholds, confirming consistency with capillary wave theory.

Proposed method

• The model modifies the Landau-Lifshitz-Navier-Stokes (LLNS) equations by replacing the local surface force with a nonlocal integral term: $\mathbf{F}(\mathbf{x}) = -\int_{\Omega} s(\mathbf{x},\mathbf{y}) f_{\varepsilon}(|\mathbf{x}-\mathbf{y}|) \frac{\mathbf{x}-\mathbf{y}}{|\mathbf{x}-\mathbf{y}|} d\mathbf{y}$, where $f_{\varepsilon}$ is a force shape function with negative values at short distances and positive at long distances.
• The force strength $s(\mathbf{x},\mathbf{y})$ is derived from molecular interactions and ensures the nonlocal term captures the broken symmetry of intermolecular forces at interfaces.
• A semi-analytical steady-state solution is derived for the pressure profile across a fluid-fluid interface, revealing non-linear curvature dependence and anisotropic static stress tensor near the interface.
• Smoothed Particle Hydrodynamics (SPH) simulations are used to numerically solve the modified LLNS equations, including stochastic stress tensors $\mathbf{s}_l = \gamma_l \boldsymbol{\xi}$ satisfying the fluctuation-dissipation theorem.
• The surface tension $\sigma(R)$ is computed from the stress tensor difference $T_N(r) - T_T(r)$ via $\sigma(R) = \int_0^\infty [T_N(r) - T_T(r)] dr$, enabling direct comparison with curvature-dependent trends.
• Capillary wave spectra are analyzed by computing $\langle \hat{\eta}(\mathbf{q})^2 \rangle$ from SPH simulations and comparing with predictions from capillary wave theory (CWT) under gravity and no-gravity conditions.

Experimental results

Research questions

• RQ1 Can a nonlocal continuum model accurately reproduce curvature-dependent surface tension at nanoscale interfaces, where classical models fail?
• RQ2 How does the proposed nonlocal surface tension model affect the static stress tensor and pressure profile across a fluid-fluid interface, particularly near the interface?
• RQ3 To what extent does the model capture thermally driven interfacial fluctuations, including capillary wave spectra, in agreement with capillary wave theory?
• RQ4 Does the model maintain stability against Rayleigh-Taylor instability under gravity, and can it reproduce the critical gravity threshold for instability?
• RQ5 Can the model recover the macroscopic Young-Laplace law for large curvature radii while preserving nanoscale physics at small scales?

Key findings

• The model predicts a decrease in surface tension $\sigma(R)$ with decreasing droplet radius $R$, matching experimental and molecular dynamics observations for $R < 2h$ and $R \leq 10$ nm.
• For droplets with $R = 4.0$, the normalized surface tension $\sigma(R)/\sigma_0$ decreases from its macroscopic value $\sigma_0 = 2.0$ as $R$ decreases, confirming curvature dependence.
• The semi-analytical solution reveals anisotropic static stress tensor near the interface (non-zero $T_\tau$) and isotropic behavior away from the interface, consistent with mesoscale physics.
• SPH simulations show excellent agreement with capillary wave theory: $\langle \hat{\eta}(\mathbf{q})^2 \rangle$ matches theoretical predictions for $|\mathbf{q}| \leq 2\pi/(5h)$ under both gravity and no-gravity conditions.
• The model correctly predicts the onset of Rayleigh-Taylor instability when the gravity $g$ exceeds the threshold $\sigma_0 q_0^2 / (\rho_\beta - \rho_\alpha)$, with stable interfaces observed below and unstable growth above this threshold.
• The nonlocal model successfully recovers the Young-Laplace law for large curvature radii, demonstrating scale bridging between nanoscale and macroscale behavior.