$\newcommand{\Dev}{\operatorname{Dev}}$ $\newcommand{\dev}{\operatorname{dev}}$ $\newcommand{\grad}{\operatorname{grad}}$ $\newcommand{\Grad}{\operatorname{Grad}}$ $\newcommand{\divg}{\operatorname{div}}$ $\newcommand{\Ei}{\operatorname{Ei}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\Divg}{\operatorname{Div}}$ $\newcommand{\cay}{\operatorname{cay}}$ $\newcommand{\rot}{\operatorname{rot}}$

Let the normal component of the viscous traction be given by $$\begin{equation} t_{n}^{\tau}=\begin{cases} \beta\rho_{r}w_{n}^{2} & w_{n}<0\\ 0 & w_{n}\ge0 \end{cases}\,.\label{eq:fsi-BFS-normal-traction} \end{equation}$$ Then, the contribution of this traction to the virtual external work $\delta W_{ext}$ is $$\begin{equation} \delta G=\int_{\partial\Omega}\delta\mathbf{w}\cdot t_{n}^{\tau}\mathbf{n}\,da\,.\label{eq:fsi-BFSckflow-virtual-work} \end{equation}$$ The linearization of $\delta G$ along an increment $\Delta\mathbf{w}$ in the relative velocity is given by $$\begin{equation} D\delta G\left[\Delta\mathbf{w}\right]=\int_{\partial\Omega}\delta\mathbf{w}\cdot\alpha_{f}\mathbf{K}_{n}\cdot\Delta\mathbf{w}\,da\,,\label{eq:fsi-BFS-linearization} \end{equation}$$ where $$\begin{equation} \mathbf{K}_{n}=\begin{cases} 2\beta\rho_{r}w_{n}\left(\mathbf{n}\otimes\mathbf{n}\right) & v_{n}<0\\ \mathbf{0} & v_{n}\ge0 \end{cases}\,.\label{eq:fsi-BFS-stiffness} \end{equation}$$ Similarly, $$D\delta G\left[\Delta\mathbf{u}\right]=\int_{\partial\Omega_{\eta}}\delta\mathbf{w}\cdot\alpha_{f}t_{n}^{\tau}\left(\mathbf{g}_{1}\times\frac{\partial\Delta\mathbf{u}}{\partial\eta^{2}}-\mathbf{g}_{2}\times\frac{\partial\Delta\mathbf{u}}{\partial\eta^{1}}\right)\,d\eta^{1}d\eta^{2}$$ The discretized form of $\delta G$ is $$\begin{equation} \delta G=\sum_{a}\delta\mathbf{w}_{a}\cdot\int_{\partial\Omega_{\eta}}\mathbf{f}_{a}^{n}\,d\eta^{1}d\eta^{2},\quad\boxed{\mathbf{f}_{a}^{n}=N_{a}t_{n}^{\tau}\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)}\,,\label{eq:fsi-BFS-discretized-work} \end{equation}$$ whereas the discretized form of $D\delta G\left[\Delta\mathbf{v}\right]$ is $$\begin{equation} D\delta G\left[\Delta\mathbf{v}\right]=\sum_{a}\delta\mathbf{w}_{a}\cdot\sum_{b}\left(\int_{\partial\Omega_{\eta}}\mathbf{K}_{ab}^{wu}\,d\eta^{1}d\eta^{2}\cdot\Delta\mathbf{u}+\int_{\partial\Omega_{\eta}}\mathbf{K}_{ab}^{ww}\,d\eta^{1}d\eta^{2}\cdot\Delta\mathbf{w}_{b}\right)\,,\label{eq:BFS-discretized-tangent-1} \end{equation}$$ with $$\begin{equation} \boxed{\mathbf{K}_{ab}^{ww}=\alpha_{f}N_{a}N_{b}\left|\mathbf{g}_{1}\times\mathbf{g}_{2}\right|\mathbf{K}_{n}}\,,\label{eq:fsi-BFS-Kww} \end{equation}$$ and $$\begin{equation} \boxed{\mathbf{K}_{ab}^{wu}=\alpha_{f}N_{a}t_{n}^{\tau}\boldsymbol{\mathcal{A}}\left\{ \frac{\partial N_{b}}{\partial\eta^{2}}\mathbf{g}_{1}-\frac{\partial N_{b}}{\partial\eta^{1}}\mathbf{g}_{2}\right\} }\label{eq:fsi-BFS-Kwu} \end{equation}$$