[Paper Review] Dynamics of several rigid bodies in a two-dimensional ideal fluid and convergence to vortex systems

Olivier Glass, Franck Sueur|arXiv (Cornell University)|Oct 7, 2019

Navier-Stokes equation solutions24 references4 citations

TL;DR

This paper studies the dynamics of multiple rigid bodies in a 2D incompressible ideal fluid, proving that as the size of some bodies shrinks to zero under different inertia scalings, their motion converges to a system of point vortices, while larger bodies follow Newtonian dynamics. The key result establishes convergence to a coupled fluid-vortex system with precise asymptotic behavior via novel normal form analysis and uniform fluid velocity estimates.

ABSTRACT

We consider the motion of several solids in a bounded cavity filled with a perfect incompressible fluid, in two dimensions. The solids move according to Newton's law, under the influence of the fluid's pressure, and the fluid dynamics is driven by the 2D incompressible Euler equations, which are set on the time-dependent domain corresponding to the cavity deprived of the sets occupied by the solids. We assume that the fluid vorticity is initially bounded and that the circulations around the solids may be non-zero. The existence of a unique corresponding solution, \\`a la Yudovich, to this system, up to a possible collision, is known. In this paper we identify the limit dynamics of the system when the radius of some of the solids converge to zero depending on how, for each body, the inertia is scaled with the radius. We obtain in the limit some point vortex systems for the solids converging to particles and a form of Newton's law for the solids that have a fixed radius; for the fluid we obtain an Euler-type system. This extends earlier works to the case of several moving rigid bodies. A crucial point is to understand the interaction, through the fluid, between small moving solids, and for that we use some normal forms of the ODEs driving the motion of the solids in two steps: first we use a normal form for the system coupling the time-evolution of all the solids to obtain a rough estimate of the acceleration of the bodies, then we turn to some normal forms that are specific to each small solid, with an appropriate modulation related to the influence of the other solids and of the fluid vorticity, to obtain some precise uniform a priori estimates of the velocities of the bodies, and then pass to the limit.

Motivation & Objective

• To analyze the limit dynamics of multiple rigid bodies in a 2D incompressible ideal fluid as their radii tend to zero.
• To extend previous results on single-body limits to the multi-body case, where fluid-mediated interactions between solids complicate the analysis.
• To identify distinct limiting behaviors depending on how the inertia of each small body scales with its radius—resulting in point vortex or Newtonian dynamics.
• To develop a new analytical framework based on normal forms and uniform fluid velocity estimates to handle the coupled fluid-solid system.
• To rigorously pass to the limit and establish convergence of the fluid and solid dynamics to a well-defined vortex system and Euler-type equations.

Proposed method

• Use of Yudovich-type existence theory for the fluid-solid system on a time-dependent domain with bounded initial vorticity and non-zero circulations.
• Introduction of a two-step normal form reduction: first for the coupled system of solids, then individual normal forms for each small solid to estimate accelerations.
• Application of shape derivative estimates for Kirchhoff potentials and reflected circulation stream functions in the presence of small solids.
• Derivation of uniform estimates for the fluid velocity field relative to the solids, refining the reflection method for div/curl systems with prescribed circulations.
• Use of modulated energy estimates to control the evolution of the system under the scaling limits.
• Compactness arguments and time-local convergence analysis to pass to the limit in the fluid and solid equations on compact time intervals.

Experimental results

Research questions

• RQ1 How does the motion of multiple rigid bodies in a 2D incompressible fluid behave in the limit as their radii tend to zero?
• RQ2 What are the limiting dynamics for small solids when their inertia scales differently with size—specifically, for heavy versus light bodies?
• RQ3 How do fluid-mediated interactions between multiple moving solids affect the convergence to a vortex system?
• RQ4 Can the standard single-body limit techniques be extended to the multi-body case, or are new analytical tools required?
• RQ5 What role does the uniform control of fluid velocity relative to the solids play in establishing convergence to the vortex limit?

Key findings

• The limit dynamics of small rigid bodies converging to zero radius depend on their inertia scaling: they converge to point vortices if light, or follow Newtonian dynamics if heavy.
• For solids of fixed size, the limit system satisfies a form of Newton’s law coupled to the 2D Euler equations for the fluid.
• The fluid velocity field converges uniformly to a solution of the 2D Euler equations in the complement of the limiting solid positions.
• The convergence holds on any compact time interval $[0,T]$ where the limit system remains collision-free and the solids maintain a positive distance from the fluid vorticity support.
• The proof relies on a novel two-step normal form reduction and uniform estimates of the fluid velocity with respect to the solids’ positions and radii.
• The analysis confirms that the limiting system is well-posed under the same conditions as the original fluid-solid system, provided no collisions occur in the limit.