# Feel++

## Fluid structure coupling conditions

In order to have a correct fluid-structure model, we need to add to the solid model and the fluid model equations some coupling conditions :

\frac{\partial \boldsymbol{\eta_{s}} }{\partial t} - \boldsymbol{u}_f \circ \mathcal{A}_{f}^t = \boldsymbol{0} , \quad \text{ on } \Gamma_{fsi}^* \times \left[t_i,t_f \right] \quad \boldsymbol{(1)}

\boldsymbol{F}_{s} \boldsymbol{\Sigma}_{s} \boldsymbol{n}^*_s + J_{\mathcal{A}_{f}^t} \hat{\boldsymbol{\sigma}}_f \boldsymbol{F}_{\mathcal{A}_{f}^t}^{-T} \boldsymbol{n}^*_f = \boldsymbol{0} , \quad \text{ on } \Gamma_{fsi}^* \times \left[t_i,t_f \right] \quad \boldsymbol{(2)}

\boldsymbol{\varphi}_s^t - \mathcal{A}_{f}^t = \boldsymbol{0} , \quad \text{ on } \Gamma_{fsi}^* \times \left[t_i,t_f \right] \quad \boldsymbol{(3)}

\boldsymbol{(1)}, \boldsymbol{(2)}, \boldsymbol{(3)} are the fluid-struture coupling conditions, respectively velocities continuity, constraint continuity and geometric continuity.

### Fluid structure coupling conditions with 1D reduced model

For the coupling conditions, between the 2D fluid and 1D structure, we need to modify the original ones \boldsymbol{(1)},\boldsymbol{(2)}, \boldsymbol{(3)} by

\dot{\eta}_s \boldsymbol{e}_r - \boldsymbol{u}_f = \boldsymbol{0} \quad \boldsymbol{(1.2)}

f_s + \left(J_{\mathcal{A}_f^t} \boldsymbol{F}_{\mathcal{A}_f^t}^{-T} \hat{\boldsymbol{\sigma}}_f \boldsymbol{n}^*_f\right) \cdot \boldsymbol{e}_r = 0 \quad \boldsymbol{(2.2)}

\boldsymbol{\varphi}_s^t - \mathcal{A}_f^t = \boldsymbol{0} \quad \boldsymbol{(3.2)}

### Variables, symbols and units

 Notation Quantity Unit \boldsymbol{u}_f fluid velocity m/s \boldsymbol{\sigma}_f fluid stress tensor N/m^2 \boldsymbol{\eta}_s displacement m \boldsymbol{F}_s deformation gradient dimensionless \boldsymbol{\Sigma}_s second Piola-Kirchhoff tensor N/m^2 \mathcal{A}_f^t Arbitrary Lagrangian Eulerian ( ALE ) map dimensionless

and

\boldsymbol{F}_{\mathcal{A}_f^t} = \boldsymbol{\mathrm{x}}^* + \nabla \mathcal{A}_f^t

J_{\mathcal{A}_f^t} = det(\boldsymbol{F}_{\mathcal{A}_f^t})