Mathematical Models for Fluid-Solid Interaction and Their Numerical Solutions.

摘要

This thesis considers various mathematical modeling approaches including ALE used currently for fluid-solid interaction and the associated computational methodologies for obtaining numerical solutions of the resulting initial value problems. The validity of the mathematical models for fluid-solid interaction is established based on: (i) consistency in the use of continuum mechanics principles and concepts (ii) 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 inherently built in adaptivity.;The thesis establishes that: (i) fluid-solid interaction must be intrinsic in the mathematical model(s) (ii) for (i) to hold, the mathematical models for fluid and solid must have the same description, either Eulerian with transport or Lagrangian or Eulerian without transport (iii) since fluids require Eulerian description with transport, a similar description for solid matter (hypo-elastic solid) indeed provides mathematical models for fluid-solid interaction in which the fluid-solid interaction is intrinsic in the mathematical models (iv) the mathematical models for solid matter in Lagrangian description or in Eulerian description without transport and for fluid in Eulerian description with transport can never interact due to fundamental differences in their derivations and the basic assumptions employed in Eulerian description with transport for fluids that precludes material point displacements which are intrinsically present in the Lagrangian descriptions and Eulerian descriptions without transport and are needed for interaction of the fluid with the solid. The mathematical models for solid matter in Lagrangian description, Eulerian description without transport and for fluids in 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 fluid-solid interaction.;The thesis presents numerical studies for simple model problems to demonstrate various issues discussed in (i)-(iv) and ultimately establishes the possible mathematical models and their limitations within the current knowledge of continuum mechanics that provide correct fluid-solid interaction.