[Paper Review] A 2d spray model with gyroscopic effects

Ayman Moussa, Franck Sueur|arXiv (Cornell University)|Dec 15, 2011

Navier-Stokes equation solutions14 references3 citations

TL;DR

This paper introduces a 2D PDE system coupling a Vlasov-type equation for a spray of particles with an Euler-type equation for an incompressible fluid, where particles experience gyroscopic (lift) forces due to fluid vorticity, and the spray contributes to fluid vorticity. The key contribution is a rigorous mean-field derivation of the system from particle dynamics and the first proof of existence of weak solutions, along with Hamiltonian structure and vanishing mass limit analysis.

ABSTRACT

In this paper we introduce a PDE system which aims at describing the dynamics of a dispersed phase of particles moving into an incompressible perfect fluid, in two space dimensions. The system couples a Vlasov-type equation and an Euler-type equation: the fluid acts on the dispersed phase through a gyroscopic force whereas the latter contributes to the vorticity of the former. First we give a Dobrushin type derivation of the system as a mean-field limit of a PDE system which describes the dynamics of a finite number of massive pointwise particles moving into an incompressible perfect fluid. This last system is itself inferred from a joint work of the second author with O. Glass and C. Lacave, where the system for one massive pointwise particle was derived as the limit of the motion of a solid body when the body shrinks to a point with fixed mass and circulation. Then we deal with the well-posedness issues including the existence of weak solutions. Next we exhibit the Hamiltonian structure of the system and finally, we study the behavior of the system in the limit where the mass of the particles vanishes.

Motivation & Objective

• To model the interaction between a dispersed spray of particles and an incompressible perfect fluid in 2D using a kinetic/fluid coupling.
• To derive the system rigorously as a mean-field limit of a finite system of massive point particles in a fluid, based on a previous model for a single particle.
• To establish the well-posedness of the system, particularly the existence of weak solutions.
• To reveal the Hamiltonian structure of the system and analyze its behavior in the limit of vanishing particle mass.

Proposed method

• Derive the system as a mean-field limit of a PDE system describing N massive point particles in a fluid, using optimal transportation theory and relative entropy methods.
• Use the Biot-Savart operator K to relate vorticity ω and fluid velocity u via u = K[ω + ρ], where ρ is the particle density.
• Establish weak solution existence by compactness arguments in space-time distributions, leveraging bounds on ρ^ε and ω^ε in L∞(Lp) and H^{-m} spaces.
• Apply Ascoli’s theorem and diagonal extraction to prove relative compactness of particle density and fluid vorticity sequences.
• Analyze the Hamiltonian structure by identifying the conserved energy functional and Poisson bracket formulation.
• Study the vanishing mass limit by scaling particle mass and circulation, showing convergence to a limiting system with no inertia.

Experimental results

Research questions

• RQ1 Can a 2D fluid-spray system with gyroscopic (lift) forces be rigorously derived from a particle-based model via a mean-field limit?
• RQ2 What are the well-posedness properties of the resulting fluid-kinetic system, particularly the existence of weak solutions?
• RQ3 Does the system admit a Hamiltonian structure, and how is it related to the energy and Poisson bracket?
• RQ4 How does the system behave when the particle mass tends to zero, and what is the limiting dynamics?

Key findings

• The system is derived as a mean-field limit of a finite particle system using optimal transport and relative entropy techniques, establishing its physical relevance.
• Weak solutions exist for the coupled Vlasov-Euler system (1)–(3), proven via compactness and weak convergence in distribution spaces.
• The system possesses a Hamiltonian structure, with a conserved energy functional and a Poisson bracket formulation, indicating integrable-like behavior.
• In the limit of vanishing particle mass, the system converges to a limiting fluid-kinetic model where particle inertia disappears, and the dynamics are governed solely by fluid vorticity and particle velocity.
• The convergence of the particle density ρ^ε and vorticity ω^ε is established in C⁰_t(D’(R²)) via compactness and diagonal extraction, ensuring the limit system is well-defined.
• The energy dissipation inequality is verified in the limit, confirming the stability of the weak solution under perturbations.