# Airplane Vortices Evolution Near Ground
## 2 Mathematical Procedure to Obtain the Velocity and Pressure Fields
Following the procedure just defined, the influence due to the left part of the domain can be represented as:  
\[\phi_{l}(x,y)=\frac{\Gamma}{2\pi}\bigg{[}\text{atan}\bigg{(}\frac{Y+Y_{0}}{X +X_{0}}\bigg{)}-\text{atan}\bigg{(}\frac{Y-Y_{0}}{X+X_{0}}\bigg{)}\bigg{]} \tag{8}\]  
\[\psi_{l}(x,y)=-\frac{\Gamma}{4\pi}\bigg{\{}ln\Big{[}(X+X_{0})^{2}+(Y+Y_{0})^{ 2}\Big{]}-ln\Big{[}(X+X_{0})^{2}+(Y-Y_{0})^{2}\Big{]}\bigg{\}} \tag{9}\]  
After performing the addition of the two corresponding functions, the resulting potential and stream functions are represented as:  
\[\phi(x,y)=\phi_{r}(x,y)+\phi_{l}(x,y) \tag{10}\]  
\[\phi(x,y)=U_{\infty}X+\frac{\Gamma}{2\pi}\bigg{[}-\text{atan}\bigg{(}\frac{Y+ Y_{0}}{X-X_{0}}\bigg{)}+\text{atan}\bigg{(}\frac{Y-Y_{0}}{X-X_{0}}\bigg{)}+ \tag{11}\]  
\[+\text{atan}\bigg{(}\frac{Y+Y_{0}}{X+X_{0}}\bigg{)}-\text{atan}\bigg{(}\frac{Y- Y_{0}}{X+X_{0}}\bigg{)}\bigg{]}\]  
\[\psi(x,y)=\psi_{r}(x,y)+\psi_{l}(x,y) \tag{12}\]\[\psi(x,y) =U_{\infty}Y+\frac{\Gamma}{4\pi}\Big{(}ln\Big{[}(X-X_{0})^{2}+(Y+Y_{ 0})^{2}\Big{]}-ln\Big{[}(X-X_{0})^{2}+(Y-Y_{0})^{2}\Big{]} \tag{13}\] \[\psi_{r}(x,y)=U_{\infty}Y+\frac{\Gamma}{4\pi}\bigg{\{}ln\Big{[}(X-X_{0})^{2}+ (Y+Y_{0})^{2}\Big{]}-ln\Big{[}(X-X_{0})^{2}+(Y-Y_{0})^{2}\Big{]}\bigg{\}} \tag{7}\]0  
From Equation (13) and through the CauchY-Riemann relations (Equations (14) and (15)) the fluid velocity can be found across the entire domain:  
\[ \begin{split} V_{X}=\frac{\partial\psi}{\partial y}& =U_{\infty}+\frac{\Gamma}{2\pi}\Bigg{[}\frac{Y+Y_{0}}{\left(X-X_ {0}\right)^{2}+\left(Y+Y_{0}\right)^{2}}-\frac{Y-Y_{0}}{\left(X-X_{0}\right)^{ 2}+\left(Y-Y_{0}\right)^{2}}+\\ &\quad-\frac{Y+Y_{0}}{\left(X+X_{0}\right)^{2}+\left(Y+Y_{0} \right)^{2}}+\frac{Y-Y_{0}}{\left(X+X_{0}\right)^{2}+\left(Y-Y_{0}\right)^{2} }\Bigg{]}\end{split}  \tag{14}\]  
\[ \begin{split} V_{Y}=-\frac{\partial\psi}{\partial x}& =-\frac{\Gamma}{2\pi}\Bigg{[}\frac{X-X_{0}}{\left(X-X_{0}\right)^{ 2}+\left(Y+Y_{0}\right)^{2}}-\frac{X-X_{0}}{\left(X-X_{0}\right)^{2}+\left(Y-Y _{0}\right)^{2}}+\\ &\quad-\frac{X+X_{0}}{\left(X+X_{0}\right)^{2}+\left(Y+Y_{0} \right)^{2}}+\frac{X+X_{0}}{\left(X+X_{0}\right)^{2}+\left(Y-Y_{0}\right)^{2} }\Bigg{]}\end{split}  \tag{15}\]  
Finally, the last unknown parameter from Equation (1) is the temporal variation of the velocity potential, and it is found by means of the chain rule:  
\[\frac{\partial\phi}{\partial t}=\frac{\partial\phi}{\partial X_{0}}\frac{ \partial X_{0}}{\partial t}+\frac{\partial\phi}{\partial Y_{0}}\frac{ \partial Y_{0}}{\partial t}=V_{X0}\frac{\partial\phi}{\partial X_{0}}+V_{Y0} \frac{\partial\phi}{\partial Y_{0}} \tag{16}\]  
The terms \(\frac{\partial X_{0}}{\partial t}\) and \(\frac{\partial Y_{0}}{\partial t}\) represent the velocity of the right-hand side real vortex. To determine these terms, it is necessary to consider the effect of the other three vortices on it. To do so it is necessary to calculate the reduced potential function, which takes the following form:  
\[f^{*}(z)=\phi^{*}+i\psi^{*}=U_{\infty}Z+\frac{i\Gamma}{2\pi}ln(Z-Z_{0}^{*})- \frac{i\Gamma}{2\pi}ln\bigg{(}\frac{Z-Z_{1}^{*}}{Z-Z_{1}}\bigg{)} \tag{17}\]  
The different positions of the vortices cores are:  
\[Z_{0}^{*}=(X_{0}-iY_{0});Z_{1}=(-X_{0}+iY_{0});Z_{1}^{*}=(-X_{0}-iY_{0});Z=(X+iY) \tag{18}\]  
Substituting Equation (18) in Equation (17) and after rearranging the real and imaginary terms, we obtain:  
\[ \begin{split}\phi^{*}+i\psi^{*}=U_{\infty}X+iU_{\infty}Y+\frac{ i\Gamma}{2\pi}ln((X-X_{0})+i(Y+Y_{0}))\\ -\frac{i\Gamma}{2\pi}ln((X+X_{0})+i(Y+Y_{0}))+\frac{i\Gamma}{2\pi }ln((X+X_{0})+i(Y-Y_{0}))\end{split}