Fluid-Structure Interaction in a Flexible Channel#
Mathematical Model#
Consider the FSI of a flexible channel with fluid flow. The fluid velocity \(\mathbf{v}\) and structural displacement \(u\) are coupled through the FSI problem.
Structural Equation:
\[\begin{split}
\begin{align*}
\rho_s A_s \frac{\partial^2 u}{\partial t^2} - \nabla \cdot \sigma(u) &= 0 \quad \text{in } \Omega_s \times (0, T) \\
u &= 0 \quad \text{on } \Gamma_{\text{fixed}} \times (0, T) \\
\sigma(u) \cdot \mathbf{n} &= \mathbf{t} \quad \text{on } \Gamma_{\text{interface}} \times (0, T)
\end{align*}
\end{split}\]
Fluid Equation:
\[\begin{split}
\begin{align*}
\rho_f \frac{\partial \mathbf{v}}{\partial t} + \rho_f (\mathbf{v} \cdot \nabla) \mathbf{v} - \nabla \cdot \sigma(\mathbf{v}) &= 0 \quad \text{in } \Omega_f \times (0, T) \\
\mathbf{v} &= \mathbf{0} \quad \text{on } \Gamma_{\text{inlet}} \times (0, T) \\
\sigma(\mathbf{v}) \cdot \mathbf{n} &= p \cdot \mathbf{n} \quad \text{on } \Gamma_{\text{interface}} \times (0, T)
\end{align*}
\end{split}\]
Coupling Conditions:
\[\begin{split}
\begin{align*}
\mathbf{v}(\mathbf{x}, t) &= \mathbf{v}_f(\mathbf{x}, t) \quad \text{on } \Gamma_{\text{interface}} \times (0, T) \\
\sigma(u) \cdot \mathbf{n} &= \sigma(\mathbf{v}) \cdot \mathbf{n} \quad \text{on } \Gamma_{\text{interface}} \times (0, T)
\end{align*}
\end{split}\]
Weak Formulation#
Find \(u \in V_s\) and \(\mathbf{v} \in V_f\) such that
\[\begin{split}
\begin{align*}
\int_{\Omega_s} \rho_s A_s \frac{\partial^2 u}{\partial t^2} \phi_s \, d\Omega &- \int_{\Omega_s} \nabla \cdot \sigma(u) \cdot \nabla \phi_s \, d\Omega = 0 \\
\int_{\Omega_f} \rho_f \frac{\partial \mathbf{v}}{\partial t} \cdot \mathbf{\phi}_f \, d\Omega &+ \int_{\Omega_f} \rho_f (\mathbf{v} \cdot \nabla) \mathbf{v} \cdot \mathbf{\phi}_f \, d\Omega - \int_{\Omega_f} \nabla \cdot \sigma(\mathbf{v}) \cdot \nabla \mathbf{\phi}_f \, d\Omega = 0
\end{align*}
\end{split}\]
for all \(\phi_s \in V_s\) and \(\mathbf{\phi}_f \in V_f\).