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See Section 2.11↑ for a review of fluid materials, and [14] for additional details on the FEBio fluid solver. The tied fluid interface is defined between surfaces $\gamma^{\left(1\right)}$ and $\gamma^{\left(2\right)}$ . Due to continuity requirements on the viscous traction and normal fluid velocity, the external virtual work resulting from tractions $\mathbf{t}^{\tau\left(i\right)}$ and normal velocities $v_{n}^{\left(i\right)}$ ($i=1,2)$ may be combined into the tied interface integral $$\begin{equation} \begin{aligned}\delta G_{c} & =\int_{\gamma^{\left(1\right)}}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{t}^{\tau\left(1\right)}da^{\left(1\right)}\\ & -\int_{\gamma^{\left(1\right)}}\left(\delta J^{\left(1\right)}-\delta J^{\left(2\right)}\right)v_{n}^{\left(1\right)}da^{\left(1\right)}\,. \end{aligned} \label{eq696-2} \end{equation}$$ To evaluate and linearize $\delta G_{c}$ , define the covariant basis vectors on each surface as $$\begin{equation} \mathbf{g}_{\alpha}^{\left(i\right)}=\frac{\partial\mathbf{x}^{\left(i\right)}}{\partial\eta_{\left(i\right)}^{\alpha}},\quad\alpha=1,2,\label{eq697-2} \end{equation}$$ where $\mathbf{x}^{\left(i\right)}$ represents the spatial position of points on $\gamma^{\left(i\right)}$ , and $\eta_{\left(i\right)}^{\alpha}$ represent the parametric coordinates of that point. The unit outward normal on each surface is then given by $$\begin{equation} \mathbf{n}^{\left(i\right)}=\frac{\mathbf{g}_{1}^{\left(i\right)}\times\mathbf{g}_{2}^{\left(i\right)}}{\left|\mathbf{g}_{1}^{\left(i\right)}\times\mathbf{g}_{2}^{\left(i\right)}\right|}\,.\label{eq698-2} \end{equation}$$ Now the contact integral may be rewritten as $$\begin{equation} \begin{aligned}\delta G_{c} & =\int_{\gamma^{\left(1\right)}}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{t}^{\tau}\,\left|\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{g}_{2}^{\left(1\right)}\right|\,d\eta_{\left(1\right)}^{1}d\eta_{\left(1\right)}^{2}\\ & -\int_{\gamma^{\left(1\right)}}\left(\delta J^{\left(1\right)}-\delta J^{\left(2\right)}\right)v_{n}\left|\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{g}_{2}^{\left(1\right)}\right|\,d\eta_{\left(1\right)}^{1}d\eta_{\left(1\right)}^{2} \end{aligned} \,,\label{eq699-2} \end{equation}$$ where $\mathbf{t}^{\tau}\equiv\mathbf{t}^{\tau\left(1\right)}$ and $v_{n}\equiv v_{n}^{\left(1\right)}$ . The linearization $D\delta G_{c}$ of $\delta G_{c}$ has the form $$\begin{equation} D\delta G_{c}=\sum\limits _{i=1}^{2}D\delta G_{c}\left[\Delta\mathbf{v}^{\left(i\right)}\right]+D\delta G_{c}\left[\Delta J^{\left(i\right)}\right]\,.\label{eq700-2} \end{equation}$$