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Existence and Uniqueness of Solutions to a Fluidstructure Model Coupling the Navier-Stokes Equations and the Lame System

Golmirzaee, Narges | 2016

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 48827 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Hesaraki, Mahmoud
  7. Abstract:
  8. In this thesis, we consider a system containing fluid equations, structure equations, and equations of these two materials’ common interface in three dimensions and on the regular domains. We suppose that the solid which is described by the Lamé system of linear elasticity, moves inside an incompressible viscous fluid in three dimensions, and the fluid obeys the incompressible Navier-Stokes equations in a time-dependent domain. At the fluid–solid interface, natural conditions are imposed, continuity of the velocities and of the Cauchy stress forces. The fluid and the solid are coupled through these conditions. By this interaction, the fluid deforms the boundary of the solid which in turn influences the fluid motion. We prove the existence and uniqueness of local strong solutions by rewriting the coupled system in Lagrangian variables and by using the method of successive approximations. Our analysis relies on new regularity results for the linearized coupled system. In particular, if ũ is the Lagrangian velocity of our system, we have to estimate separately Pũ and (I − P)ũ, where P is the Leray projector for the fluid model. Moreover, we prove new regularity results for the Lamé system, that we are able to establish only in the case when the interface, in the reference configuration, between the fluid and the solid is flat. Due to this restriction, the model is completed by periodic boundary conditions in two directions. The other results may be transposed to more general configurations
  9. Keywords:
  10. Linear Elasticity ; Fluid-Structure Interaction ; Uniqueness Solution ; Incompressible Navier-Stokes Equations ; Lame System ; Strong Solutions

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