Fluid-Rigid Body Interaction in Incompressible Viscous Systems

The interaction between an incompressible viscous fluid and an immersed rigid body presents a rich tapestry of physical phenomena and mathematical challenges. At its core, the problem involves the Navier–Stokes equations coupled with Newton’s laws of motion for a non-deformable solid. The fluid exerts viscous stresses and pressure forces on the body, inducing translation and rotation, while the moving obstacle shapes the surrounding flow field. Ensuring well-posedness often requires handling non-linear coupling, moving boundaries, and intricate boundary conditions such as no-slip or Navier slip. Recent theoretical advances have focused on global existence and uniqueness of weak and strong solutions, the treatment of collisions or contact events, and asymptotic regimes including vanishing body size and low Mach-number limits. Control strategies have been devised to drive the system toward desired states, with feedback laws applicable even in complex geometries. Computational approaches, grounded in penalty or fictitious domain methods, facilitate high-fidelity simulations of self-propelled micro-swimmers, biomedical devices and civil structures subjected to fluid flows. Foundational insights into interface stress transmission and contact modelling continue to inform applications across environmental engineering, aeronautics and biomechanics, underscoring the field’s enduring significance and interdisciplinary reach.

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Fluid-Rigid Body Interaction in Incompressible Viscous Systems publication trend

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Technical terms

Navier–Stokes equations: Governing equations for the motion of viscous incompressible fluids, expressing momentum balance and mass conservation.

Incompressible fluid: A fluid whose density remains constant throughout the flow, leading to a divergence-free velocity field.

Rigid body: A solid object that does not deform under applied forces, interacting dynamically with the surrounding fluid via surface stresses.

Navier slip boundary condition: A condition allowing tangential slip between fluid and solid surfaces, proportional to the local shear stress.

Weak solution: A generalised solution of partial differential equations that satisfies integral formulations of the governing laws rather than pointwise expressions.

References

1. On the Motion of a Nearly Incompressible Viscous Fluid Containing a Small Rigid Body. Journal of Nonlinear Science (2023).
2. Existence and uniqueness result for a fluid–structure–interaction evolution problem in an unbounded 2D channel. Nonlinear Differential Equations and Applications NoDEA (2022).
3. Local null controllability of a fluid–rigid body interaction problem with Navier slip boundary conditions. ESAIM Control Optimisation and Calculus of Variations (2021).