We model the fluid as isothermal and compressible, consistent with our CFD implementation. Thus, the fluid stress may be separated into the elastic pressure $p\left(J^{f}\right)$ , which only depends on the fluid volume ratio $J^{f}$ , and the viscous stress $\boldsymbol{\tau}\left(J^{f},\mathbf{L}^{f}\right)$ , as $$\begin{equation} \boldsymbol{\sigma}^{f}=-p\mathbf{I}+\boldsymbol{\tau}\,.\label{eq:fsi-fluid-stress-split} \end{equation}$$ Recall that there is no dependence on temperature in an isothermal formulation. As done in our previous study, we integrate the mass balance for the fluid to produce $$\begin{equation} \rho^{f}=\frac{\rho_{r}^{f}}{J^{f}}\,,\label{eq:fsi-integrated-fluid-mass} \end{equation}$$ where $\rho_{r}^{f}$ is the fluid density in the reference state (e.g., under ambient pressure) and $J^{f}=\det\mathbf{F}^{f}$ , where $\mathbf{F}^{f}$ is the fluid deformation gradient. We also use the kinematic constraint $$\begin{equation} \frac{\partial J^{f}}{\partial t}+\grad J^{f}\cdot\mathbf{v}^{f}=J^{f}\divg\mathbf{v}^{f}\,,\label{eq:fsi-fluid-kinematic constraint} \end{equation}$$ to relate the fluid volume ratio $J^{f}$ to its velocity $\mathbf{v}^{f}$ , in the spatial frame.

The mesh through which the fluid flows is defined on the solid component of the mixture. Therefore, we define the relative velocity between the fluid and solid as $$\begin{equation} \mathbf{w}=\mathbf{v}^{f}-\mathbf{v}^{s}\,.\label{eq:fsi-fluid-relative-velocity} \end{equation}$$ Since the solid has zero volume fraction, this expression is the same as the flux of fluid relative to the solid. We choose to define the nodal degrees of freedom in the mixture domain to be the relative fluid velocity $\mathbf{w}$ , the fluid dilatation $e^{f}=J^{f}-1$ , and the solid displacement $\mathbf{u}^{s}$ , which is related to the solid velocity via $\mathbf{v}^{s}=\dot{\mathbf{u}}^{s}$ , with the dot operator denoting the material time derivative following the solid. (For notational convenience, we will continue using $J^{f}$ in the equations below, instead of $e^{f}$ ). Now, $$\begin{equation} \mathbf{v}^{f}=\mathbf{w}+\mathbf{v}^{s}\,,\label{eq:fsi-fluid-velocity} \end{equation}$$ from which it follows that the fluid velocity gradient is $$\begin{equation} \mathbf{L}^{f}=\mathbf{L}^{w}+\mathbf{L}^{s}\,,\label{eq:fsi-fluid-velocity-gradient} \end{equation}$$ where $\mathbf{L}^{w}=\grad\mathbf{w}$ and $\mathbf{L}^{s}=\grad\mathbf{v}^{s}$ . The fluid acceleration may now be rewritten in terms of the FEA degrees of freedom as $$\begin{equation} \mathbf{a}^{f}=\dot{\mathbf{w}}+\dot{\mathbf{v}}^{s}+\left(\mathbf{L}^{w}+\mathbf{L}^{s}\right)\cdot\mathbf{w}\,,\label{eq:fsi-fluid-acceleration-redux} \end{equation}$$ where $$\begin{equation} \begin{aligned}\dot{\mathbf{v}}^{s} & =\frac{\partial\mathbf{v}^{s}}{\partial t}+\mathbf{L}^{s}\cdot\mathbf{v}^{s}\\ \dot{\mathbf{w}} & =\frac{\partial\mathbf{w}}{\partial t}+\mathbf{L}^{w}\cdot\mathbf{v}^{s} \end{aligned} \label{eq:fsi-vs-w-material-derivs} \end{equation}$$ are the material time derivatives of the solid and relative fluid velocities, in the frame following the solid. We conveniently use this material time derivative (instead of the material time derivative following the fluid) since we define the mixture domain mesh on the solid.

Similarly, the kinematic constraint relating $J^{f}$ and $\mathbf{v}^{f}$ may be rewritten as $$\begin{equation} \dot{J}^{f}+\grad J^{f}\cdot\mathbf{w}=J^{f}\divg\left(\mathbf{w}+\mathbf{v}^{s}\right)\,,\label{eq:fsi-fluid-kinematic-redux} \end{equation}$$ where $$\begin{equation} \dot{J}^{f}=\frac{\partial J^{f}}{\partial t}+\grad J^{f}\cdot\mathbf{v}^{s}\,.\label{eq:fsi-dilatation-material-deriv} \end{equation}$$ Finally, as done routinely in our studies of biphasic and multiphasic materials, we find it convenient to substitute the kinematic identity $$\begin{equation} \divg\mathbf{v}^{s}=\frac{\dot{J}^{s}}{J^{s}}\,,\label{eq:fsi-solid-kinematic-constraint} \end{equation}$$ where $J^{s}=\det\mathbf{F}^{s}$ and $\mathbf{F}^{s}=\mathbf{I}+\Grad\mathbf{u}^{s}$ is the solid deformation gradient.

In summary, the governing equations for the FSI mixture domain are $$\begin{equation} \begin{aligned} & \divg\boldsymbol{\sigma}^{s}=\mathbf{0}\\ & \rho^{f}\mathbf{a}^{f}=\divg\boldsymbol{\sigma}^{f}+\rho^{f}\mathbf{b}\\ & \frac{1}{J^{f}}\left(\dot{J}^{f}+\grad J^{f}\cdot\mathbf{w}\right)=\divg\mathbf{w}+\frac{\dot{J}^{s}}{J^{s}} \end{aligned} \label{eq:fsi-governing-equations} \end{equation}$$ where $\mathbf{a}^{f}$ is given in (2.12.1-10) above.