Fluid–fluid and –solid interaction problems: Variational principles revisited

We systematically review some unified variational principles for a strong interaction problem in both a stratified fluid region and a fluid–solid region. The problem is described by a general Lagrangian formulation for an anisotropic elastic solid region, either surrounded by an incompressible non-Newtonian fluid region or surrounding the fluid region. In the first part, we express the fundamental equations of the regular fluid and solid regions in differential form. Then, we deduce the variational principles respectively from the principle of virtual power and the principle of virtual work for the fluid and solid regions. The physics principles are modified through an involutory transformation together with a dislocation potential. In the second part, we similarly establish some multi-field variational principles for a stratified fluid of two or more distinct fluid layers of different thicknesses and mass densities. In the third part, we derive the variational principles for the interior and exterior interaction problems in a fluid region with a surface piercing solid, within either a rigid or an elastic structure. The variational principles, which operate on all the field variables lead to the fundamental equations of the regions, including the interface conditions, as their Euler–Lagrange equations. Some special cases of the variational principles are given.