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.

我们使用流体结构界面上的摩擦类型的非线性边界条件，研究了不可压缩粘性牛顿流体与弹性结构之间的稳态流体结构相互作用问题。此条件也称为Tresca滑移边界条件，当流体剪切应力的切向分量达到某个阈值函数时，允许流体在界面上滑动。控制方程是$2d$ 流体的斯托克斯方程，取决于结构位移和 $\mathrm{1个}d$结构的Euler–Bernoulli模型。通过设计基于Schauder不动点定理的证明，我们证明了该非线性耦合问题的一个弱解。理论结果将通过一个数值例子来说明。