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Asymptotic expansions for eigenvalues of the Stokes system in the presence of small deformable inclusions
  • Abdessatar khelifi,
  • Ahlem Jaouabi
Abdessatar khelifi
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Ahlem Jaouabi
Faculté des Sciences de Bizerte
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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.