## 1 Introduction
The study of the flow separation from the knuckles allows for evaluation of the dynamic pressures in the wetted chine region in planing hulls. This region is import to analyse of the dynamics and dynamics stability of the planing boats. Figure 1 shows the impact of a wedge section with a symmetry entry, where \(\beta\) is the deadrise angle, c is the half wetted beam, z\({}_{0}\) is the wedge immersion, b is the section beam, d is the vertical distance between keel and knuckle, and y and z are the horizontal and vertical axes. The energy equation for a potential flow is:  
\[\frac{\mathrm{P}}{\rho}\ =\ -\frac{\mathrm{d}\varphi}{\mathrm{dt}}-\frac{1}{2} \biggl{(}\frac{\mathrm{d}\varphi}{\mathrm{dy}}\biggr{)}^{2} \tag{1}\]  
where \(\varphi\) is the potential function of the fluid, p is the pressure, and \(\rho\) is the density of the liquid. The potential function equation for the symmetry impact proposed by [PERSON] [2] is:  
\[\varphi\ =\ -\mathrm{w}\sqrt{\mathrm{c}^{2}-\mathrm{y}^{2}} \tag{2}\]  
\[\frac{\mathrm{P}}{\rho}\ =\ \frac{\mathrm{w}\mathrm{c}\mathrm{c}}{\sqrt{ \mathrm{c}^{2}-\mathrm{y}^{2}}}-\frac{1}{2}\frac{\mathrm{w}^{2}\mathrm{y}^{2} }{\mathrm{c}^{2}-\mathrm{y}^{2}} \tag{3}\]  
where \(\,\dot{\mathrm{c}}\) is the variation with the time for the half wetted beam, for a wedge section this parameter is evaluated as:  
\[\mathrm{c}\ =\ \frac{\pi}{2}\frac{\mathrm{wt}}{\mathrm{tan}\ \beta} \tag{4}\]  
\[\dot{\mathrm{c}}\ =\ \frac{\pi}{2}\frac{\mathrm{w}}{\mathrm{tan}\ \beta} \tag{5}\]Figure 2 shows the asymmetrical impact of a wedge section, [PERSON] [3] extended the model developed by [PERSON] [2] for the impact with asymmetrical entry and introduced the asymmetry parameter \(\mu\), \(\phi\) is the heel angle, \(c_{1}\) is the wetted half beam in the right side, \(c_{2}\) is the wetted half beam in the left side, \(\beta_{1}\) and \(\beta_{2}\) are defined as:  
\[\beta_{1} = \beta-\phi \tag{6}\] \[\beta_{2} = \beta+\phi.\]  
The equation for potential flow form [PERSON] [3] is:  
\[\varphi = -w\sqrt{c^{2}-\left(-\mu+y\right)^{2}}. \tag{7}\]  
The pressure in the asymmetry impact is computed as:  
\[\frac{\mathrm{P}}{\mathrm{\rho}} = \frac{\mathrm{w}\left(\dot{c}c+\dot{\mu}\left(-\mu+y\right) \right)}{\sqrt{c^{2}-\left(-\mu+y\right)^{2}}}-\frac{1}{2}\frac{\mathrm{w}^{2 }\left(-\mu+y\right)^{2}}{c^{2}-\left(-\mu+y\right)^{2}} \tag{8}\]  
The parameters of the model are evaluated as:  
\[\mathrm{R} = \frac{\tan(\beta_{2})}{\tan(\beta_{1})} \tag{9}\]  
\[\varphi\ =\ -\mathrm{w}\sqrt{\mathrm{c}^{2}-\mathrm{y}^{2}} \tag{2}\]0 Impact with symmetry entry. \[\varphi\ =\ -\mathrm{w}\sqrt{\mathrm{c}^{2}-\mathrm{y}^{2}} \tag{2}\]1 Impact with asymmetry entry. where  
\[\varphi\ =\ -\mathrm{w}\sqrt{\mathrm{c}^{2}-\mathrm{y}^{2}} \tag{2}\]2  
\[\mathrm{f} = \left\{ \begin{array}{l}0.77975+0.003771\mathrm{R}+0.001876\mathrm{ R}^{2},\mathrm{R}<3\\ 0.76773+0.015024\mathrm{R}+0.000539\mathrm{R}^{2},\mathrm{R}<10\\ 0.80497+0.007208\mathrm{R}+0.000130\mathrm{R}^{2},\mathrm{R}>10\end{array} \right\} \tag{11}\]  
\[\mu_{0}\;=\;\mathrm{f}\frac{\mathrm{R}-1}{\mathrm{R}+1} \tag{12}\]  
\[\mathrm{z_{0}\;=\;wt} \tag{13}\]  
\[\mathrm{c}\;=\;\frac{\pi\mathrm{R}\mathrm{z_{0}}}{\sqrt{\left(1-\mu_{0}^{2} \right)}\left(1-\mu_{0}^{2}\right)\left(\mathrm{R}+1\right)^{2}}\Bigg{(} \frac{1}{\tan\left(\beta_{1}\right)}+\frac{1}{\tan\left(\beta_{2}\right)} \Bigg{)} \tag{14}\]  
\[\mathrm{\dot{c}\;=\;\frac{\pi\mathrm{R}\mathrm{w}}{\sqrt{\left(1-\mu_{0}^{2} \right)}\left(1-\mu_{0}^{2}\right)\left(\mathrm{R}+1\right)^{2}}\Bigg{(} \frac{1}{\tan\left(\beta_{1}\right)}+\frac{1}{\tan\left(\beta_{2}\right)} \Bigg{)} \tag{15}\]  
\[\mu\;=\;\mu_{0}\mathrm{c} \tag{16}\]  
\[\mathrm{\dot{\mu}\;=\;\mu_{0}\dot{c}} \tag{17}\]  
\[\mathrm{c_{1}\;=\;c+\mu} \tag{18}\]  
\[\mathrm{c_{2}\;=\;c-\mu}. \tag{19}\]  
The aim of this paper is to develop a mathematical model of a two-dimensional impact with three degrees of freedom (3 DOF) and apply the results to study planing hulls.