[Paper Review] Derivation of a two-phase flow model with two-scale kinematics, geometric variables and surface tension using variational calculus

Pierre Cordesse, Samuel Kokh+3 more|arXiv (Cornell University)|Oct 31, 2019

Lattice Boltzmann Simulation Studies13 references4 citations

TL;DR

This paper derives a novel two-phase flow model using variational calculus that captures both large-scale interface dynamics and small-scale kinematics and surface tension effects through geometric variables—volume fraction, interfacial area density, and mean curvature. By extending the Least Action Principle to include two-scale surface tension and non-spherical, polydisperse bubbles, the model provides a unified, geometrically consistent framework for complex two-phase flows without assuming spherical symmetry.

ABSTRACT

The present paper proposes a two-phase flow model that is able to account for two-scale kinematics and two-scale surface tension effects based on geometric variables at small scale. At large scale, the flow and the full geometry of the interface may be retrieved thanks to the bulk variables, while at small scale the interface is accurately described by volume fraction, interfacial area density and mean curvature, called the geometric variables. Our work mainly relies on the Least Action Principle. The resulting system is an extension of a previous work modeling small scale pulsation in which surface tension was not taken into account at large or small scale. Whereas the original derivation assumes a cloud of monodispersed spherical bubbles, the present context allows for polydispersed, non-spherical bubbles. The resulting system of equations solely involves small scale geometric variables, thus contributing in the construction of a unified model describing both large and small scales.

Motivation & Objective

• To develop a unified Eulerian two-phase flow model that captures both large-scale interface dynamics and small-scale geometric effects.
• To extend previous models by incorporating surface tension effects at both large and small scales, not just at the small scale.
• To account for polydisperse, non-spherical bubbles rather than assuming monodispersed spherical bubbles.
• To use geometric variables—volume fraction, interfacial area density, and mean curvature—as the sole small-scale descriptors, enabling closure of the two-scale system.
• To derive a consistent set of governing equations via the Least Action Principle, ensuring thermodynamic consistency and conservation laws.

Proposed method

• Formulates a Lagrangian energy functional that includes kinetic energy, internal energy, and surface energy contributions at both large and small scales.
• Applies the Least Action Principle to derive the governing partial differential equations from variations of the action functional with respect to state variables and their spatial and temporal derivatives.
• Uses Weyl’s Tube Formula to relate the small-scale geometry of gas inclusions (diffeomorphic to spheres) to interfacial area density and mean curvature.
• Introduces a two-scale parametrization where large-scale interface position is captured via the volume fraction gradient, while small-scale interface properties are described by Σ (interfacial area density) and H (mean curvature).
• Imposes constraints on the evolution of geometric variables, including the transport equations for α, Σ, and h (normal displacement), ensuring consistency with interface kinematics.
• Derives the equations of motion by computing variations of the action with respect to the flow map and geometric variables, leading to momentum, mass, and geometric evolution equations.

Experimental results

Research questions

• RQ1 How can surface tension effects be consistently modeled at both large and small scales in a two-phase flow system?
• RQ2 What geometric variables are necessary and sufficient to describe the small-scale interface morphology in polydisperse, non-spherical bubbly flows?
• RQ3 How can the Least Action Principle be extended to include two-scale kinematics and capillarity in a variational framework?
• RQ4 What is the relationship between volume fraction, interfacial area density, and mean curvature in a small-scale interface description?
• RQ5 Can a single, unified model describe both large-scale interface motion and small-scale pulsation effects, including surface tension, without assuming spherical symmetry?

Key findings

• The derived model is the first to consistently account for surface tension effects at both large and small scales using only small-scale geometric variables: volume fraction, interfacial area density, and mean curvature.
• The model successfully extends previous work by removing the assumption of monodispersed spherical bubbles, enabling the description of polydisperse, non-spherical bubbles.
• The governing equations are derived via the Least Action Principle, ensuring thermodynamic consistency and conservation of mass, momentum, and energy.
• The relationship between interfacial area density Σ, mean curvature H, and volume fraction α is preserved through the geometric constraints derived from Weyl’s Tube Formula.
• The model provides a closed system of equations that depend only on the geometric variables at the small scale, enabling a unified description of two-phase flow across scales.
• The derivation establishes a variational foundation for future models of complex two-phase flows, such as spray formation and atomization, where both large-scale interface deformation and small-scale capillary effects are critical.