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

At a fluid-solid interface $\Gamma$ , the fluid traction $\mathbf{t}^{f}=\boldsymbol{\sigma}^{f}\cdot\mathbf{n}$ acts on the solid surface, $\mathbf{t}^{s}=-\mathbf{t}^{f}$ , where $\mathbf{n}$ is the outward normal to the fluid surface. The resulting virtual work on the solid domain is $$\delta G=\int_{\Gamma}\delta\mathbf{v}^{s}\cdot\mathbf{t}_{n+\alpha_{f}}^{s}\,da=\int_{\Gamma}-\delta\mathbf{v}^{s}\cdot\boldsymbol{\sigma}^{f}\cdot\mathbf{n}\,da=\int_{\Gamma_{\eta}}-\delta\mathbf{v}^{s}\cdot\boldsymbol{\sigma}^{f}\cdot\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)\,d\eta^{1}d\eta^{2}$$ The linearization of this work is given by $$D\delta G\left[\Delta\mathbf{u}\right]=\int_{\Gamma_{\eta}}-\delta\mathbf{v}^{s}\cdot\alpha_{f}\left[\left(\boldsymbol{\mathcal{C}}^{v}:\mathbf{M}\cdot\grad\Delta\mathbf{u}\right)\cdot\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)+\boldsymbol{\sigma}^{f}\cdot\left(\frac{\partial\Delta\mathbf{u}}{\partial\eta^{1}}\times\mathbf{g}_{2}+\mathbf{g}_{1}\times\frac{\partial\Delta\mathbf{u}}{\partial\eta^{2}}\right)\right]\,d\eta^{1}d\eta^{2}$$ where $$\mathbf{g}_{\alpha}=\frac{\partial\mathbf{x}}{\partial\eta^{\alpha}}$$ are covariant basis vectors on $\Gamma$ . Similarly, $$D\delta G\left[\Delta\mathbf{w}\right]=\int_{\Gamma_{\eta}}-\delta\mathbf{v}^{s}\cdot\alpha_{f}\left[\left(\boldsymbol{\mathcal{C}}^{v}:\grad\Delta\mathbf{w}\right)\cdot\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)\right]\,d\eta^{1}d\eta^{2}$$ and $$D\delta G\left[\Delta J^{f}\right]=\int_{\Gamma_{\eta}}-\Delta J^{f}\delta\mathbf{v}^{s}\cdot\alpha_{f}\left[\left(-p^{\prime}\left(J^{f}\right)+\boldsymbol{\tau}_{J}^{\prime}\right)\cdot\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)\right]\,d\eta^{1}d\eta^{2}$$