The study case's container can be seen as an excited annular tank partially filled with liquid at a certain height, $h$. The following hypothesis made are:
\begin{enumerate}
    \item the fluid is incompressible, so $\nabla\rho=0$;
    \item no heat transfer is considered in the fluid during the sloshing;
    \item the volume is laterally moved and is subjected to gravity.
\end{enumerate}
The nomenclature adopted corrected to match with the one of the theory Paragraph \ref{parag:sloshing_theory}. It's known that the equation of mass conservation in Lagrangian form is the following:
\begin{equation}\label{eq:mass_conservation}
    \frac{D\rho}{Dt}=-\rho(\nabla\cdot\textbf{q})
\end{equation}
where $\textbf{q}=(q_x,q_y,q_z)^T$ is the fluid velocity in a Cartesian reference frame. Given the hypothesis of incompressible and stationary fluid, it's true that $\frac{D\rho}{Dt}=0$, so the mass conservation equation is written as follows:
\begin{equation}
    \nabla\cdot\textbf{q}=0,\;\left(\frac{\partial q_x}{\partial x}+\frac{\partial q_y}{\partial y}+\frac{\partial q_z}{\partial z}\right)=0
\end{equation}
The momentum conservation equation is written as follows:
\begin{equation}\label{eq:momentum_conservation}
    \frac{\partial\textbf{q}}{\partial t}+\textbf{q}\times\nabla\textbf{u}=-\frac{1}{\rho}\nabla p+\nu\nabla^2\textbf{q}+\frac{1}{\rho}\textbf{F}
\end{equation}
where, according to the hypothesis about volumetric forces, $\textbf{F}=\rho(a_x,0,-g)^T$, with $g=9.81m/s^2$ gravity acceleration along z-direction.\\
The surface tension needs to be formulated because the sloshing problem is under investigation. It is a quantity affecting the movement of the fluid on its boundary, so it should be considered as a boundary condition.\\
The conservation equations are scaled as it follows:
\begin{equation}\label{eq:mass_conservation_dimensionless}
    \nabla^*\cdot\textbf{q}^*=0
\end{equation}
\begin{equation}\label{eq:momentum_conservation_dimensionless}
    \frac{U}{\tau}\frac{\partial\textbf{q}^*}{\partial t^*}+\frac{U^2}{L}\textbf{q}^*\times\nabla^*\textbf{q}^*=-\frac{U^2}{L}\nabla^*p^*+\mu\frac{U}{\rho L^2}\nabla^{*^2}\textbf{q}^*+g\textbf{a}^*
\end{equation}
where the physical quantity and maths operator $\nabla$ has been made dimensionless by means of: \textit{U}, the characteristic system velocity, $\tau$, the characteristic timescale, and \textit{L} is the characteristic length scale.\\
The \textit{Strouhal number} $St=\frac{L}{U\tau}$, the \textit{Reynolds number} $Re=\frac{UL}{\nu}=\frac{UL\rho}{\mu}=[m^2/s]$, and the $Fr=\frac{U^2}{gL}$ are introduced in the dimensionless momentum equation. In the current study case, \textit{L} is the radius \textit{R} of the cylinder, while the characteristic time $\tau$ of sloshing phenomenon is the inverse of pulsation $\tau=\omega^{-1}$.\\
Momentum equation can be written in the canonical form for the convection:
\begin{equation}\label{eq:momentum_equation_characteristic_numbers}
    St\frac{\partial\textbf{q}^*}{\partial t^*}+\textbf{q}^*\times\nabla^*\textbf{q}^*=-\nabla^*p^*+\frac{1}{Re}\nabla^{*^2}\textbf{q}^*+\frac{1}{Fr}\textbf{a}^*
\end{equation}
Momentum conservation in Eq. \ref{eq:momentum_equation_characteristic_numbers}, can be re-write once again in the  canonical form for non-dimensional acceleration $\textbf{a}^*$ as follows:
\begin{equation}\label{eq:momentum_equation_characteristic_numbers_Bond}
    \Lambda\frac{\partial\textbf{u}^*}{\partial t^*}+\frac{1}{Fr}\textbf{u}^*\times\nabla^*\textbf{u}^*=-\frac{1}{Fr}\nabla^*p^*+\frac{Fr}{Re}\nabla^{*^2}\textbf{u}^*+\textbf{a}^*+\frac{1}{Bo}\kappa^*\nabla^*\alpha
\end{equation}
where $\Lambda=\frac{U}{\tau g}$. Eq. \ref{eq:momentum_equation_characteristic_numbers_Bond} takes into account the \textit{surface tension} by \textit{Bond number}. It indicates the relative effect within gravitational forces and tension of the surface of interface, and it's defined as:
\begin{equation}
    Bo=\frac{\rho g L^2}{\sigma}
\end{equation}
where $\sigma$ is the \textit{surface tension} and, as always, $L$ indicates the hydraulic diameter of reference.\\
According to the technical report of reference \cite{ALFRED_Mockup}, Bond number it's negligible when compared to the other physical phenomena taking place.\\
If a sinusoidal excitation along \textit{x-axis} side is considered, as displacement $X=Asin(\omega t)$, and the velocity $U$ is equal to excitation velocity $U=\frac{\partial X}{\partial t}\sim A\omega$, so the characteristic numbers are the following:
$$
    St=\frac{L}{U\tau}=\frac{R}{A}
$$
$$
    Re=\frac{AR\omega}{\nu}
$$
$$
    Fr=\frac{A^2}{R}\frac{\omega^2}{g}
$$