Existence of a weak solution to a steady 2D fluid-1D elastic structure interaction problem with Tresca slip boundary condition. (English) Zbl 1540.35281
Summary: We study a steady state fluid-structure interaction problem between an incompressible viscous Newtonian fluid and an elastic structure using a nonlinear boundary condition of friction type on the fluid-structure interface. This condition, also known as Tresca slip boundary condition, allows the fluid to slip on the interface when the tangential component of the fluid shear stress attains a certain threshold function. The governing equations are the \(2D\) Stokes equations for the fluid, written in an unknown domain depending on the structure displacement, and the \(1D\) Euler-Bernoulli model for the structure. We prove that there exists a weak solution of this nonlinear coupled problem by designing a proof based on the Schauder fixed-point theorem. The theoretical result will be illustrated with a numerical example.
MSC:
| 35Q30 | Navier-Stokes equations |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
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