# The Effect of Porous Media on Wave-Induced Sloshing in a Floating Tank
## 2 Methodology
### Governing Equations and Boundary Conditions  
A two-dimensional rectangular water tank filled with porous media on the rectangular floating platform in waves is shown in Figure 1a. Assuming that the fluid is incompressible, inviscid, and irrotational, the continuity and momentum equations of the fluid can be expressed as:  
\[\
abla.\mathbf{u}=0 \tag{1}\]\[\rho\bigg{[}\frac{\partial\mathbf{u}}{\partial t}+(\mathbf{u}\cdot\
abla)\mathbf{u}\bigg{]}=- \
abla p+\
abla p_{d}+\rho\mathbf{g} \tag{2}\]  
where \(\rho\) is the fluid density, \(\mathbf{u}\) is the fluid velocity, \(p\) is the pressure, \(p_{d}\) is the pressure drop due to the porous media, and \(\mathbf{g}\) is the gravitational acceleration. Note that in fresh water, \(\
abla p_{d}=0\), and nonlinearity arises only from the convective term; when the porous media is involved, the gradient of the pressure drop can be expressed by [44]:  
\[\
abla p_{d}=-\frac{\gamma\mu}{\kappa}(\mathbf{u}-\mathbf{v})-\frac{\gamma c_{F}\rho}{ \sqrt{\kappa}}|\mathbf{u}-\mathbf{v}|(\mathbf{u}-\mathbf{v}) \tag{3}\]  
where \(\mathbf{v}\) is the velocity of the porous media, \(\mu\) is the dynamic viscosity of the fluid, \(\gamma\) and \(\kappa\) are the porosity and permeability of the porous media, respectively, and \(c_{F}\) is a dimensionless constant determined by experiments. Note that if \(\mathbf{u}\) is sufficiently small, the first linear term on the right-hand side of Equation (3) dominates, while the second quadratic function becomes significant as \(\mathbf{u}\) increases and the drag due to solid obstacles is comparable with the surface drag due to friction. In potential flow, the fluid velocity can be expressed by the gradient of the potential function \(\phi\) as:  
\[\mathbf{u}=\
abla\phi \tag{4}\]  
Therefore, the Laplace equation should be satisfied as:  
\[\
abla^{2}\phi=0 \tag{5}\]  
The kinematic boundary condition on the free surface can be expressed as:  
\[\frac{D\mathbf{R}}{Dt}=\mathbf{u} \tag{6}\]  
where \(\mathbf{R}\) is the location of a free-surface particle. For the wave channel, the dynamic boundary condition on the free surface can be expressed as:  
\[\frac{\partial\phi}{\partial t}+\frac{1}{2}|\
abla\phi|^{2}+g\eta=0 \tag{7}\]  
where \(\eta\) is the wave elevation. For the sloshing fluid, the dynamic boundary condition on the free surface can be expressed as:  
\[\frac{\partial\phi}{\partial t}+\frac{1}{2}|\
abla\phi|^{2}+(\phi-\psi)( \alpha_{1}+\alpha_{2}|\
abla\phi-\
abla\psi|)+g\eta=0 \tag{8}\]  
where \(\alpha_{1}=\gamma\mu/\kappa\rho\) and \(\alpha_{2}=c_{F}/\sqrt{\kappa}\) are constants related to the nature of the porous media and \(\psi\) is the velocity potential of the tank. On the wetted boundaries, the impermeable condition is taken as:  
\[\frac{\partial\phi}{\partial n}=\dot{\mathbf{x}}\cdot\mathbf{n} \tag{9}\]  
where \(\mathbf{n}\) denotes the normal vector pointing outward to the fluid boundary and \(\dot{\mathbf{x}}\) is the velocity of the node on the wetted boundary. As a wave propagates along the channel, the radiation condition is applied on the outlet boundary to ensure the waves scatter to infinity as [41]:  
\[\frac{\partial\phi}{\partial n}=-\frac{1}{c}\frac{\partial\phi}{\partial t} \tag{10}\]  
where \(c\) is the wave velocity.