Mathematical models for fluid–solid interaction and their numerical solutions

Abstract This paper considers various approaches used currently for the fluid–solid interaction problem and associated computational methodologies. The validity of the mathematical models for fluid–solid interaction is established based on the consistency in the use of continuum mechanics principles and whether the interaction between the solid and the fluid is inherent in the mathematical model or is established external to the mathematical model through interface constraint equations. Computational methodologies are considered from the point of view of unconditional stability, accuracy, and adaptivity of the numerical schemes employed. In particular, the paper establishes that fluid–solid interaction physics must be intrinsic in the mathematical model(s), the mathematical models for fluid and solid must have the same description, either Eulerian with transport, Lagrangian, or Eulerian without transport. Since fluids require the Eulerian description with transport, a similar description for solid matter (hypo-elastic solid) indeed provides a mathematical model for fluid–solid interaction in which the fluid–solid interaction is intrinsic in the mathematical model. The mathematical models for solid matter in the Lagrangian description or in the Eulerian description without transport and for fluids in the Eulerian description with transport can never interact due to fundamental differences in their derivations and the basic assumptions employed. For example, the Eulerian description with transport for fluids precludes material point displacements, which are intrinsically present in the Lagrangian description and the Eulerian description without transport, and they are needed for interaction of the fluid with the solid. The mathematical models for solid matter in the Lagrangian description, the Eulerian description without transport, and for fluids in the Eulerian description with transport are presented to illustrate why fluid–solid interaction is not possible with these mathematical models. The ALE methodologies using the mathematical models in Lagrangian and Eulerian descriptions have been carefully evaluated and are demonstrated to be invalid for a consistent formulation of a fluid–solid interaction problem. Some numerical studies for simple model problems are presented to demonstrate various issues discussed here. The present study establishes the possible mathematical models and their limitations within the current knowledge of continuum mechanics that provide correct model for fluid–solid interaction.