Linearised coupling of elasticity and Navier-Stokes equations

In the paper we consider a coupled problem of the linearly elastic body immersed in the flowing fluid which is modelled by means of incompressible Navier-Stokes equations. The approach is based on transforming the variable domain occupied by the fluid to the fixed one corresponding to the un-deformed elastic inclusion. The main idea consists in performing the linearisation of this transformation, similarly as in linear elasticity one assumes the smallness of strains. We base our approach on the observation that the gradients of the suitably defined transformation are bounded by traces of strains on the surface of the elastic body