Asymptotic expansions for eigenvalues of the Stokes system in the
presence of small deformable inclusions
Abstract
In this article, we provide a rigorous derivation of an asymptotic
formula for the perturbation of eigenvalues associated to the Stokes
eigenvalue problem with Dirichlet conditions and in the presence of
small deformable inclusions. Taking advantage of the small sizes of the
inclusions immersed in an incompressible Newtonian fluid having
kinematic viscosity different from the background one, we show that our
asymptotic formula can be expressed in terms of the eigenvalue in the
absence of the inclusions and in terms of the so-called viscous moment
tensor (VMT). We believe that our results are ambitious tools for
determining the locations and/or shapes of small inhomogeneities by
taking eigenvalue measurements.