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Microscopic derivation of a one-dimensional lubrication model with roughness

Aline Lefebvre-Lepot, Muhammed Ali Mehmood, Charlotte Perrin, Ewelina Zatorska

TL;DR

The work develops a rigorous bridge from a one-dimensional microscopic model of inertial spheres with lubrication forces and a rough-layer repulsion to a macroscopic continuum description. By scaling the particle size and count, the authors derive a coupled PDE system with a singular viscosity and a soft-congestion pressure, plus a transport equation for a flow-advected critical density ρ*. The analysis combines energy estimates, BV bounds, and compactness arguments to establish convergence of (ρ^ε,u^ε,ρ^{*,ε}) to a weak solution (ρ,u,ρ*), and identifies the precise limits w = ∂xu/(1−ρ) and G^ε → (ρ/ρ*)^γ, ensuring the limit model faithfully captures congestion dynamics. Numerical experiments via Physics-Informed Neural Networks illustrate congestion formation, the role of ρ*, and the impact of γ, highlighting the model’s practical relevance for dense suspensions and crowd-like flows.

Abstract

We derive a hydrodynamic model for the motion of inertial particles with a spherical hard core, interacting through lubrication forces and pairwise repulsive forces. The repulsion arises from the assumption that each particle is surrounded by a thin rough layer of reduced permeability. We prove that, as the number of particles tends to infinity (and their size tends to 0), the microscopic dynamics converges to a macroscopic hydrodynamic model in which congestion effects are encoded directly into the macroscopic interaction forces, depending on a local critical density transported by the flow. In particular, we extend the work of Lefebvre-Lepot and Maury where non-inertial particles, submitted to only a lubrication force were considered, and present the convergence proof when inertial effects and roughness are taken into account.

Microscopic derivation of a one-dimensional lubrication model with roughness

TL;DR

The work develops a rigorous bridge from a one-dimensional microscopic model of inertial spheres with lubrication forces and a rough-layer repulsion to a macroscopic continuum description. By scaling the particle size and count, the authors derive a coupled PDE system with a singular viscosity and a soft-congestion pressure, plus a transport equation for a flow-advected critical density ρ*. The analysis combines energy estimates, BV bounds, and compactness arguments to establish convergence of (ρ^ε,u^ε,ρ^{*,ε}) to a weak solution (ρ,u,ρ*), and identifies the precise limits w = ∂xu/(1−ρ) and G^ε → (ρ/ρ*)^γ, ensuring the limit model faithfully captures congestion dynamics. Numerical experiments via Physics-Informed Neural Networks illustrate congestion formation, the role of ρ*, and the impact of γ, highlighting the model’s practical relevance for dense suspensions and crowd-like flows.

Abstract

We derive a hydrodynamic model for the motion of inertial particles with a spherical hard core, interacting through lubrication forces and pairwise repulsive forces. The repulsion arises from the assumption that each particle is surrounded by a thin rough layer of reduced permeability. We prove that, as the number of particles tends to infinity (and their size tends to 0), the microscopic dynamics converges to a macroscopic hydrodynamic model in which congestion effects are encoded directly into the macroscopic interaction forces, depending on a local critical density transported by the flow. In particular, we extend the work of Lefebvre-Lepot and Maury where non-inertial particles, submitted to only a lubrication force were considered, and present the convergence proof when inertial effects and roughness are taken into account.
Paper Structure (28 sections, 22 theorems, 203 equations, 13 figures)

This paper contains 28 sections, 22 theorems, 203 equations, 13 figures.

Table of Contents

  1. Introduction
  2. The microscopic dynamics
  3. Description of forces
  4. The microscopic model
  5. The main convergence result
  6. The macroscopic model
  7. Approximation of initial data (rho0, u0, rho*0)
  8. Approximation of solutions (rho, u, rho*) to the microscopic model for t≥0
  9. The main theorem
  10. Analysis of the microscopic system
  11. Construction of initial data and existence of a local-in-time solution
  12. Extension to a global solution
  13. Toward the macroscopic system
  14. Obtaining a PDE representation
  15. Uniform bounds on the macroscopic variables
  16. ...and 13 more sections

Key Result

Theorem 3.2

Let $T>0$ and let $f \in W^{1,\infty}([0,T] \times I)$, $(\rho_0, \rho_0^\star) \in [W^{1,\infty}(I)]^2$ satisfying $\rho_0 \in [\delta, \overline{\rho}]$ and $\rho^\star_0 \in [\delta, 1]$ for some $0 < \delta < \overline{\rho} < 1$ be given. Moreover, suppose we have $u_0 \in H^1_0(I)$. 1. There e 2. There exists a unique solution $t \mapsto (\mathbf{q}^{\varepsilon}(t), \mathbf{u}^\varepsilon(t

Figures (13)

  • Figure 1: Discrete particles moving along the horizontal axis.
  • Figure 2: Two rough solid particles
  • Figure 3: $\rho^\varepsilon$ and $\rho^{\star,\varepsilon}$ at time $t \ge0$
  • Figure 4: $u^\varepsilon$ at time $t\ge0$
  • Figure 5: $w^\varepsilon$ and $G^\varepsilon$ at time $t$
  • ...and 8 more figures

Theorems & Definitions (47)

  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5: Notation
  • Proposition 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3: Discrete energy estimate
  • ...and 37 more