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