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})