Initial-boundary Value Problems for Solid–fluid Composite Structures

We investigate a three-dimensional mixed initial-boundary value problem arising in the dynamical solid–fluid interaction theory. A 3D domain occupied by an incompressible and viscous Stokes fluid may be bounded or unbounded, while a domain occupied by an elastic body immersed in the fluid is assumed to be bounded. On the basis of the results obtained for an elastic inclusion of an arbitrary geometrical shape, we derive a special model and analyze in detail the case when an elastic inclusion is a thin prismatic shell, in particular a plate of variable thickness. Here, we apply I. Vekua’s dimension reduction method in the elastic part which reduces 3D solid–3D fluid interaction problems to the 2D solid–3D fluid interaction problems and which is important from the practical point of view since it takes into account intrinsic differences of the dimensions of solid and fluids part. The main goal of the paper was to study the strain–stress state of the elastic part under the action of the Stokes flow. The corresponding mechanical model is described mathematically as a transmission problem for the linear Stokes system and the dynamical Lamé equations in the corresponding domains with appropriate initial conditions along with the boundary and interface conditions. For 3D solid–3D fluid dynamical interaction problems, we prove the uniqueness and existence theorem. Further, considering the case when the elastic inclusion is a thin prismatic shell of variable thickness, we apply the N = 0 approximation of Vekua’s hierarchical model for the elastic field in the solid part. In contrast to the usual classical streamline conditions, in the case under consideration, on the cut surface, there appear non-local boundary conditions. We prove unique solvability of the non-classical boundary value problem that leads to the existence results for the solid–fluid interaction problem with a thin elastic inclusion.