A fluid-structure model coupling the Navier-Stokes equations and the Lamé system (Q401297)

The authors study the model of interaction between a fluid flow and an elastic solid. The solid parallelepiped \(\Omega_S\) floats into the parallelepiped \(\Omega\). The domain \(\Omega_F=\Omega\setminus\overline{\Omega_S}\) is occupied by the fluid. The domains \(\Omega_S\) and \(\Omega_F\) are time depended. The velocity of a fluid \(u(x,t)\) and the pressure \(p(x,t)\) satisfy to the Navier-Stokes equations NEWLINE\[NEWLINE \begin{cases} \frac{\partial u}{\partial t}-\text{div}\,\mathbb{T}(u,p) u+u\cdot\nabla u=0,\quad \text{div}\,u=0,\quad x\in\Omega_F(t),\;t>0, \\ u(x,0)=u_0(x),\quad x\in\Omega_F(0), \end{cases}NEWLINE\]NEWLINE where \(\mathbb{T}(u,p)\) is the stress tensor NEWLINE\[NEWLINE \mathbb{T}(u,p)=\nu(\nabla u+(\nabla u)^T)-p\mathbb{I}. NEWLINE\]NEWLINE The displacement \(w(x,t)\) of the elastic solid satisfies to the Lame system NEWLINE\[NEWLINE\begin{cases} \frac{\partial^2 w}{\partial t^2}-\text{div}\,\sigma(w) =0,\quad x\in\Omega_S(t),\;t>0, \\ w(x,0)=0,\quad \frac{\partial w}{\partial t}(x,0)=w_1,\quad x\in\Omega_S(0), \end{cases} NEWLINE\]NEWLINE where \(\sigma(w)\) is the elastic stress tensor NEWLINE\[NEWLINE \sigma(w)=\lambda\,\text{trace}\,\varepsilon(w)\mathbb{I}+ 2\mu\varepsilon(w),\quad \varepsilon(w)=\frac{1}{2}(\nabla w+(\nabla w)^T). NEWLINE\]NEWLINE On the fluid-solid interface the continuity of velocities and of the Cauchy stress forces are fulfilled. Function \(u\) satisfies to the Dirichlet boundary conditions on the exterior boundary, \(u\) and \(w\) satisfy to the periodic boundary conditions.NEWLINENEWLINEIt is proved that the problem admits a unique strong local in time solution. The result is obtained by successive approximation method.