\subsection{Hints of linear sloshing theory}
An overview on the theory of sloshing is presented, proposing as main reference \cite{Liquid_sloshing_dynamics:_theory_and_applications} as it can be applicable both to free or forcing liquid free-surface motion. The fluid field equation is obtained for the general case of a tank moving in space along any trajectory. The container's walls are considered rigid and impermeable, while the flow is assumed inviscid, incompressible and initially irrotational. If the field is gravitational, then the hypothesis on the flow could also be considering negligible the capillary or the surface tension.\\
The reference systems implied are exposed in Figure \ref{fig:theory_1}: $Oxyz$ is the moving Cartesian coordinate frame that's attached to the container, while $OX'Y'Z'$ is the stationary Cartesian coordinate frame.
\begin{figure}[H]
    \centering
        \includegraphics[scale=0.50]{immagini/theory/reference_systems.png}
        \caption{Moving liquid container showing inertia and moving coordinates \cite{Liquid_sloshing_dynamics:_theory_and_applications}}
        \label{fig:theory_1}
\end{figure}
Euler equation of motion is written as follows:
\begin{equation}
    \frac{\partial}{\partial t}\textbf{q}+(q\cdot \nabla)\textbf{q}=-\frac{1}{\rho}\nabla P- \nabla(gZ')
\end{equation}
where \textbf{q} is the fluid velocity, $\partial\textbf{q}/\partial t$ is the local acceleration of the flow measured by a fixed observer, $(\textbf{q}\cdot\nabla)\textbf{q}$ is the convective term of acceleration, $P$ is the fluid pressure, $\rho$ is the fluid density, $gZ'$ is the gravitational potential. For irrotational flow $\nabla\times\textbf{q}=0$ and exists the velocity potential function:
\begin{equation}
    \textbf{q}=-\nabla\Phi
\end{equation}
The potential flow function leads to the Laplace's equation:
\begin{equation}\label{eq:Laplace_equation}
    \nabla^2\Phi=0
\end{equation}
Besides, the hypothesis of potential flow leads to the following form of the pressure equation:
\begin{equation}\label{eq:pressure_equation}
    \nabla\left(\frac{P}{\rho}+\frac{1}{2}q^2+gZ'-\frac{\partial\Phi}{\partial t}\right)=0
\end{equation}
in the $OX'Y'Z'$ stationary Cartesian reference frame. Let consider $\textbf{V}_0$ the velocity of $O$ of $Oxyz$ frame relative to $O'$ of the $OX'Y'Z'$ frame. The moving frame is such that the undisturbed free surface coincides with the $Oxy$ plane. Eq. \ref{eq:pressure_equation} is equal to
\begin{equation}
    \nabla\left(\frac{P}{\rho}+\frac{1}{2}q^2+gZ'-\frac{\partial\Phi}{\partial t}+\textbf{V}_0\cdot\nabla\Phi\right)=0
\end{equation}
written in terms of the total fluid velocity related to the fixed coordinates.\\
The kinematic free-surface condition is imposed, so the vertical velocity of a fluid particle on the free surface is equal to the vertical velocity of the same free surface:
\begin{equation}
    -\frac{\partial\Phi}{\partial z}=\frac{\partial\eta}{\partial t}+\textbf{q}_{rel}\cdot\eta
\end{equation}
where $z=\eta(r,\theta,t)=\eta(x,y,t)$ is the vertical velocity of the fluid particle.
The boundary conditions at the wall and the bottom are applied in the Cartesian and cylindrical coordinates:
\begin{equation}
    -\left. \frac{\partial \Phi}{\partial z} \right\vert_{z=-h}=\Ddot{Z}_0, -\left. \frac{\partial \Phi}{\partial x} \right\vert_{x=a}=\Ddot{X}_0
\end{equation}
\begin{equation}
    -\left. \frac{\partial \Phi}{\partial z} \right\vert_{z=-h}=\Ddot{Z}_0, -\left. \frac{\partial \Phi}{\partial r} \right\vert_{r=R}=\Ddot{X}_0cos\theta
\end{equation}
When the hypothesis of the potential flow is respected, then the previous set of equations is the starting point for describing the sloshing problem. The nonlinearity in the boundary value problem only exists in the free-surface boundary conditions for $z=\eta$.\\
Since the sloshing problem requires the extraction of the natural frequencies, a modal analysis is required: this kind of proposed approach works if the potential function is obtained analytically in a closed form. The variational formulation associated with the Rayleigh-Ritz method (as a method for approximating eigenvalues) is another valid approach to the modal problem. It starts with the Hamilton's principle statement:
\begin{equation}\label{eq:Hamilton_principle}
    \delta I=\delta\int^{t_2}_{t_1}(T-V)dt=0
\end{equation}
where $T$ and $V$ are the kinetic and potential energy of the system. The parenthesis $(T-V)$ stands for the Lagrangian function $L=T-V$ that must be minimized by means of the Hamilton's principle. This formulation is very powerful if compared to the proposed previous one because it brings in one statement the fluid equations and the associated boundary conditions. This advantage is typical of the variational formulation's approaches.\\
The variational formulation leads to the following frequency equation:
\begin{equation}
    \vert A_{mn}-\lambda B_{mn}\vert=0
\end{equation}
where $\lambda=\omega^2/g$ and contains the frequency information, while $A_{mn}$ and $B_{mn}$ are symmetrical matrices and are obtained starting from the Hamilton's statement in Eq. \ref{eq:Hamilton_principle} and the application of a Rayleigh-Ritz method \cite{Liquid_sloshing_dynamics:_theory_and_applications}.
\subsection{Normal modes in an Annular tank}
Hints on the annular tank natural frequencies are presented because the this geometry fits with the simpler structure of primary pool of a nuclear reactor. Indeed, the inner circular cylinder of the container should represents a simplification of the primary core of the pool. An explanatory sketch is in Figure \ref{fig:theory_2}.
\begin{figure}[H]
    \centering
        \includegraphics[scale=0.70]{immagini/theory/annular_tank.png}
        \caption{Annular tank geometry \cite{Liquid_sloshing_dynamics:_theory_and_applications}: $R$ is the external radius, while $R_{in}$ represents the radius of the internal cylinder}
        \label{fig:theory_2}
\end{figure}
The dominant mode is the first $m=1$, and it performs several fluid forces if compared to the others. If the surface tension is negligible, the natural frequencies are:
\begin{equation}\label{eq:natural_frequency}
    \omega_{mn}^2=\frac{g}{R}\xi_{mn}\tanh(\frac{\xi_{mn}h}{R})
\end{equation}
and it's obtained by solving the determinant $\Delta_m(\xi_{mn})=0$ \cite{Theory_of_the_fluid_oscillations_in_a_circular_cylindrical_ring_tank_partially_filled_with_liquid}. If $k=R_{in}/R$ is the ratio of inner radius to outer radius, the values of the parameter $\xi_{mn}$ for an annular tank and when $m=0,1,2$, $n=0,1,2$ are reported in Table \ref{Tab:theory_1} and represented in Figure \ref{fig:theory_3}.
\begin{table}[H]
\centering
 {\fontsize{10}{10}\selectfont
  \begin{tabular}{|l|llllllllll|}
\hline
 &
  \multicolumn{10}{l|}{$k$, ratio of inner radius $R_{in}$ to outer radius $R$} \\ \hline
{\color[HTML]{242021} (m,n)} &
  \multicolumn{1}{l|}{{\color[HTML]{242021} 0.0}} &
  \multicolumn{1}{l|}{{\color[HTML]{242021} 0.1}} &
  \multicolumn{1}{l|}{{\color[HTML]{242021} 0.2}} &
  \multicolumn{1}{l|}{{\color[HTML]{242021} 0.3}} &
  \multicolumn{1}{l|}{{\color[HTML]{242021} 0.4}} &
  \multicolumn{1}{l|}{{\color[HTML]{242021} 0.5}} &
  \multicolumn{1}{l|}{{\color[HTML]{242021} 0.6}} &
  \multicolumn{1}{l|}{{\color[HTML]{242021} 0.7}} &
  \multicolumn{1}{l|}{{\color[HTML]{242021} 0.8}} &
  {\color[HTML]{242021} 0.9} \\ \hline
{\color[HTML]{242021} (0,1)} &
  \multicolumn{1}{l|}{{\color[HTML]{242021} 3.832}} &
  \multicolumn{1}{l|}{{\color[HTML]{242021} 3.941}} &
  \multicolumn{1}{l|}{{\color[HTML]{242021} 4.236}} &
  \multicolumn{1}{l|}{{\color[HTML]{242021} 4.706}} &
  \multicolumn{1}{l|}{{\color[HTML]{242021} 5.391}} &
  \multicolumn{1}{l|}{{\color[HTML]{242021} 6.393}} &
  \multicolumn{1}{l|}{{\color[HTML]{242021} 7.930}} &
  \multicolumn{1}{l|}{{\color[HTML]{242021} 10.522}} &
  \multicolumn{1}{l|}{{\color[HTML]{242021} 15.738}} &
  {\color[HTML]{242021} 31.429} \\ \hline
(0,2) &
  \multicolumn{1}{l|}{7.016} &
  \multicolumn{1}{l|}{7.331} &
  \multicolumn{1}{l|}{8.055} &
  \multicolumn{1}{l|}{9.104} &
  \multicolumn{1}{l|}{10.558} &
  \multicolumn{1}{l|}{12.625} &
  \multicolumn{1}{l|}{15.747} &
  \multicolumn{1}{l|}{20.969} &
  \multicolumn{1}{l|}{31.431} &
  62.839 \\ \hline
(1,0) &
  \multicolumn{1}{l|}{1.841} &
  \multicolumn{1}{l|}{1.804} &
  \multicolumn{1}{l|}{1.705} &
  \multicolumn{1}{l|}{1.582} &
  \multicolumn{1}{l|}{1.462} &
  \multicolumn{1}{l|}{1.355} &
  \multicolumn{1}{l|}{1.262} &
  \multicolumn{1}{l|}{1.182} &
  \multicolumn{1}{l|}{1.113} &
  1.053 \\ \hline
(1,1) &
  \multicolumn{1}{l|}{5.331} &
  \multicolumn{1}{l|}{5.137} &
  \multicolumn{1}{l|}{4.961} &
  \multicolumn{1}{l|}{5.137} &
  \multicolumn{1}{l|}{5.659} &
  \multicolumn{1}{l|}{6.565} &
  \multicolumn{1}{l|}{8.041} &
  \multicolumn{1}{l|}{10.592} &
  \multicolumn{1}{l|}{15.778} &
  31.447 \\ \hline
(1,2) &
  \multicolumn{1}{l|}{8.536} &
  \multicolumn{1}{l|}{8.199} &
  \multicolumn{1}{l|}{8.433} &
  \multicolumn{1}{l|}{9.308} &
  \multicolumn{1}{l|}{10.683} &
  \multicolumn{1}{l|}{12.706} &
  \multicolumn{1}{l|}{15.801} &
  \multicolumn{1}{l|}{21.004} &
  \multicolumn{1}{l|}{31.451} &
  62.847 \\ \hline
(2,0) &
  \multicolumn{1}{l|}{3.054} &
  \multicolumn{1}{l|}{3.053} &
  \multicolumn{1}{l|}{3.035} &
  \multicolumn{1}{l|}{2.968} &
  \multicolumn{1}{l|}{2.842} &
  \multicolumn{1}{l|}{2.681} &
  \multicolumn{1}{l|}{2.516} &
  \multicolumn{1}{l|}{2.363} &
  \multicolumn{1}{l|}{2.227} &
  2.106 \\ \hline
(2,1) &
  \multicolumn{1}{l|}{6.706} &
  \multicolumn{1}{l|}{6.687} &
  \multicolumn{1}{l|}{6.495} &
  \multicolumn{1}{l|}{6.274} &
  \multicolumn{1}{l|}{6.416} &
  \multicolumn{1}{l|}{7.063} &
  \multicolumn{1}{l|}{8.367} &
  \multicolumn{1}{l|}{10.799} &
  \multicolumn{1}{l|}{15.898} &
  31.500 \\ \hline
(2,2) &
  \multicolumn{1}{l|}{9.970} &
  \multicolumn{1}{l|}{9.888} &
  \multicolumn{1}{l|}{9.550} &
  \multicolumn{1}{l|}{9.918} &
  \multicolumn{1}{l|}{11.056} &
  \multicolumn{1}{l|}{12.949} &
  \multicolumn{1}{l|}{15.962} &
  \multicolumn{1}{l|}{21.106} &
  \multicolumn{1}{l|}{31.511} &
  62.918 \\ \hline
\end{tabular}}
\caption{Values of parameter $\xi_{mn}$ for an annular cylindrical tank when $m=0,1,2$ and $n=0,1,2$ \cite{Tables_of_Zeros_of_Cross_Product_Bessel_Functions}}\label{Tab:theory_1}
\end{table}
\begin{figure}[H]
    \centering
        \includegraphics[scale=1]{immagini/theory/annular_tank_modes.png}
        \caption{Modes of free-surface sloshing in annular cylindrical tank \cite{Annular_tank_modes}}
        \label{fig:theory_3}
\end{figure}
\subsection{Hints of weakly non-linear wave regimes}
The linear theory does not take in account the important vertical motion of the free-liquid surface. The weakly non-linear regimes are experienced when the forcing frequency is at/near the resonance one of the dynamic system. The linear theory fails to predict the complexity of the sloshing since the nodal diameters could change; something that usually happens is the combination of modal forms, whom frequency is different from the linear ones. Experimental results of forced sloshing are quickly treated in \cite{Sloshing_dynamics_investigation_by_means_of_non-intrusive_measurement_techniques}, \cite{Experimental_investigation_sloshing} and \cite{Liquid_sloshing_fuel_tank}.\\
Three main sloshing wave regimes are identified:
\begin{itemize}
    \item steady state planar wave regime, which adjusts the linear motion prediction for a frequency away from the resonance condition;
    \item swirling regime, which is characterized by rotational motion waves;
    \item chaos regime, which is characterized by randomness, potentially wave breaking and possible formation of bubbles in the liquid itself.
\end{itemize}
\begin{figure}[H]
    \centering
        \includegraphics[scale=0.60]{immagini/theory/sloshing_regimes.png}
        \caption{Demonstrative representation of the sloshing regimes taken from \cite{Experimental_investigation_sloshing} and \cite{Liquid_sloshing_fuel_tank}: Dimensionless stabilised wave amplitude ($b_a/\lambda$) for harmonic base excitation in one direction at different forcing amplitude ($A_f/R$) and forcing frequency ($\omega/\omega_{11}$); performed test in \cite{Experimental_investigation_sloshing} are the circles, the triangle and square points are extracted from \cite{Liquid_sloshing_fuel_tank}.}
        \label{fig:theory_4}
\end{figure}