# Airplane Vortices Evolution Near Ground
## 2 Mathematical Procedure to Obtain the Velocity and Pressure Fields
 \tag{19}\]  
Remembering the concept of the logarithm of a complex function, once the logarithmic terms from Equation (19) have been multiplied by their conjugate and the real and imaginary terms separated, the resulting reduced potential and stream functions take the form:  
\[\phi^{*}=U_{\infty}X-\frac{\Gamma}{2\pi}\Bigg{[}\text{atan}\frac{Y+Y_{0}}{X-X_ {0}}\Bigg{]}+\frac{\Gamma}{2\pi}\Bigg{[}\text{atan}\frac{Y+Y_{0}}{X+X_{0}} \Bigg{]}+\frac{\Gamma}{2\pi}\Bigg{[}\text{atan}\frac{Y-Y_{0}}{X+X_{0}}\Bigg{]} \tag{20}\]\[\psi^{*}=U_{\infty}Y+\frac{\Gamma}{4\pi}ln((X-X_{0})^{2}+(Y+Y_{0})^{2})-\frac{ \Gamma}{4\pi}ln((X+X_{0})^{2}+(Y+Y_{0})^{2}) \tag{21}\] \[+\frac{\Gamma}{4\pi}ln((X+X_{0})^{2}+(Y-Y_{0})^{2})\]  
The velocity of the vortex located at the initial point \((X_{0};Y_{0})\) is defined as:  
\[V_{X0}=\frac{\partial X_{0}}{\partial t}=\frac{\partial\phi^{*}}{\partial X} \bigg{|}_{X=X_{0};\ Y=Y_{0}}=\frac{\partial\psi^{*}}{\partial Y}\bigg{|}_{X=X _{0};\ Y=Y_{0}} \tag{22}\]  
\[V_{Y0}=\frac{\partial Y_{0}}{\partial t}=\frac{\partial\phi^{*}}{\partial Y} \bigg{|}_{X=X_{0};\ Y=Y_{0}}=-\frac{\partial\psi^{*}}{\partial X}\bigg{|}_{X= X_{0};\ Y=Y_{0}} \tag{23}\]  
Then the resulting equations take the form:  
\[V_{X0}=V_{X}(X_{0},Y_{0})=U_{\infty}+\frac{\Gamma}{4\pi}\Bigg{[}\frac{1}{Y_{0} }-\frac{Y_{0}}{\big{(}X_{0}^{2}+Y_{0}^{2}\big{)}}\Bigg{]}=U_{\infty}+\frac{ \Gamma}{4\pi}\Bigg{[}\frac{X_{0}^{2}}{Y_{0}\big{(}X_{0}^{2}+Y_{0}^{2}\big{)}} \Bigg{]} \tag{24}\]  
\[V_{Y0}=V_{Y}(X_{0},Y_{0})=\frac{\Gamma}{4\pi}\Bigg{[}\frac{X_{0}}{\big{(}X_{0} ^{2}+Y_{0}^{2}\big{)}}-\frac{1}{X_{0}}\Bigg{]}=-\frac{\Gamma}{4\pi}\Bigg{[} \frac{Y_{0}^{2}}{X_{0}\big{(}X_{0}^{2}+Y_{0}^{2}\big{)}}\Bigg{]} \tag{25}\]  
To obtain the terms \(\frac{\partial\phi}{\partial X_{0}}\) and \(\frac{\partial\phi}{\partial Y_{0}}\) it will be necessary to make the derivative of Equation (11), this is straightforward and the outcome is presented below. \[ \begin{split}\frac{\partial\phi}{\partial X_{0}}& =\frac{\Gamma}{2\pi}\Bigg{[}-\frac{Y+Y_{0}}{\big{(}X-X_{0}\big{)} ^{2}+\big{(}Y+Y_{0}\big{)}^{2}}+\frac{Y-Y_{0}}{\big{(}X-X_{0}\big{)}^{2}+ \big{(}Y-Y_{0}\big{)}^{2}}+\\ &\quad-\frac{Y+Y_{0}}{\big{(}X+X_{0}\big{)}^{2}+\big{(}Y+Y_{0} \big{)}^{2}}+\frac{Y-Y_{0}}{\big{(}X+X_{0}\big{)}^{2}+\big{(}Y-Y_{0}\big{)}^{2 }}\Bigg{]}\end{split}  \tag{26}\] \[ \begin{split}\frac{\partial\phi}{\partial Y_{0}}& =\frac{\Gamma}{2\pi}\Bigg{[}-\frac{X-X_{0}}{\big{(}X-X_{0}\big{)} ^{2}+\big{(}Y+Y_{0}\big{)}^{2}}-\frac{X-X_{0}}{\big{(}X-X_{0}\big{)}^{2}+ \big{(}Y-Y_{0}\big{)}^{2}}+\\ &\quad+\frac{X+X_{0}}{\big{(}X+X_{0}\big{)}^{2}+\big{(}Y+Y_{0} \big{)}^{2}}+\frac{X+X_{0}}{\big{(}X+X_{0}\big{)}^{2}+\big{(}Y-Y_{0}\big{)}^{2 }}\Bigg{]}\end{split}  \tag{27}\]  
Then the equation giving the temporal variation of the velocity potential at any point of the domain takes the form:  
\[ \begin{split}\frac{\partial\phi}{\partial t}&=\frac{ \Gamma}{2\pi}\Bigg{(}-\frac{Y+Y_{0}}{\big{(}X-X_{0}\big{)}^{2}+\big{(}Y+Y_{0} \big{)}^{2}}+\frac{Y-Y_{0}}{\big{(}X-X_{0}\big{)}^{2}+\big{(}Y-Y_{0}\big{)}^{2} }+\\ &\quad-\frac{Y+Y_{0}}{\big{(}X+X_{0}\big{)}^{2}+\big{(}Y+Y_{0} \big{)}^{2}}+\frac{Y-Y_{0}}{\big{(}X+X_{0}\big{)}^{2}+\big{(}Y-Y_{0}\big{)}^{2 }}\Bigg{)}\bigg{(}U_{\infty}+\frac{\Gamma}{4\pi}\bigg{[}\frac{X_{0}^{2}}{Y_{0} \big{(}X_{0}^{2}+Y_{0}^{2}\big{)}}\bigg{]}\bigg{)}+\\ &\quad+\frac{\Gamma}{2\pi}\Bigg{(}-\frac{X-X_{0}}{\big{(}X-X_{0} \big{)}^{2}+\big{(}Y+Y_{0}\big{)}^{2}}-\frac{X-X_{0}}{\big{(}X-X_{0}\big{)}^{2 }+\big{(}Y-Y_{0}\big{)}^{2}}+\\ &\quad+\frac{X+X_{0}}{\big{(}X+X_{0}\big{)}^{2}+\big{(}Y+Y_{0} \big{)}^{2}}+\frac{X+X_{0}}{\big{(}X+X_{0}\big{)}^{2}+\big{(}Y-Y_{0}\big{)}^{2 }}\bigg{)}\bigg{(}-\frac{\Gamma}{4\pi}\bigg{[}\frac{Y_{0}^{2}}{X_{0}\big{(}X_{0 }^{2}+Y_{0}^{2}\big{)}}\bigg{]}\bigg{)}\end{split}  \tag{28}\]  
Finally, substituting Equations (14), (15) and (28) into the potential Bernoulli equation (Equation (1)), the relative pressure along the flow domain can be found.