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Let the tied interface viscous traction be described by the penalty function, $$\begin{equation} \mathbf{t}^{\tau}=\varepsilon_{t}\mathbf{g}\,,\label{eq:TFI-t-penalty} \end{equation}$$ where $\varepsilon_{t}$ is a penalty factor associated with $\mathbf{t}^{\tau}$ . The penalty factor $\varepsilon_{t}$ is expressed in units of viscosity per length. Therefore, in an auto-penalty scheme, it may be scaled to the ratio of fluid viscosity to element thickness. Similarly, let $$\begin{equation} v_{n}=\varepsilon_{n}\pi=K\varepsilon_{n}\left(J^{\left(2\right)}-J^{\left(1\right)}\right)\,,\label{eq:TFI-v-penalty} \end{equation}$$ where $\varepsilon_{n}$ is a penalty factor associated with $v_{n}$ . Note that $\varepsilon_{n}$ has units of length per viscosity. It follows that $$\begin{equation} \begin{aligned}D\mathbf{t}^{\tau} & =\varepsilon_{t}\left(\left(\mathbf{I}-\mathbf{n}^{\left(2\right)}\otimes\mathbf{n}^{\left(2\right)}\right)\cdot\Delta\mathbf{v}^{\left(2\right)}-\left(\mathbf{I}-\mathbf{n}^{\left(1\right)}\otimes\mathbf{n}^{\left(1\right)}\right)\cdot\Delta\mathbf{v}^{\left(1\right)}\right)\,,\\ Dv_{n} & =K\varepsilon_{n}\left(\Delta J^{\left(2\right)}-\Delta J^{\left(1\right)}\right)\,. \end{aligned} \label{eq:TFI-penalty-linearization} \end{equation}$$ Given these relations, it can be shown that the directional derivatives of the various terms appearing in the integrand of $\delta G_{c}$ are $$\begin{equation} \begin{aligned} & -D\left(J_{\eta}^{\left(1\right)}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{t}^{\tau}\right)=\\ & \varepsilon_{t}J_{\eta}^{\left(1\right)}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\left(\left(\mathbf{I}-\mathbf{n}^{\left(1\right)}\otimes\mathbf{n}^{\left(1\right)}\right)\cdot\Delta\mathbf{v}^{\left(1\right)}-\left(\mathbf{I}-\mathbf{n}^{\left(2\right)}\otimes\mathbf{n}^{\left(2\right)}\right)\cdot\Delta\mathbf{v}^{\left(2\right)}\right) \end{aligned} \,,\label{eq:TFI-G-linearization-v} \end{equation}$$ $$\begin{equation} D\left(v_{n}\left(\delta J^{\left(1\right)}-\delta J^{\left(2\right)}\right)J_{\eta}^{\left(1\right)}\right)=-\varepsilon_{n}KJ_{\eta}^{\left(1\right)}\left(\delta J^{\left(1\right)}-\delta J^{\left(2\right)}\right)\left(\Delta J^{\left(1\right)}-\Delta J^{\left(2\right)}\right)\,,\label{eq:TFI-G-linearization-e} \end{equation}$$ where $J_{\eta}^{\left(1\right)}=\left|\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{g}_{2}^{\left(1\right)}\right|$ .