I Have the velocity fields for two vortices that are located at two different points
${\bf{x_1}}(x_1,y_1)$, ${\bf{x_2}}(x_2,y_2)$
$\vec{V_1} = (u_1,v_1) = \frac{\Gamma_1}{2\pi}\frac{1}{(x-x_1)^2 + (y-y_1)^2}(-(y-y_1),(x-x_1))$ $\vec{V_2} = (u_2,v_2) = \frac{\Gamma_2}{2\pi}\frac{1}{(x-x_2)^2 + (y-y_2)^2}(-(y-y_2),(x-x_2))$
I now consider an assumption that vortex one 1 is transported by the velocity of vortex 2. and this leads to a system of ODEs.
$$\frac{dx_1}{dt} = u_2 = \frac{\Gamma_2}{2\pi}\frac{-(y-y_2)}{(x-x_2)^2 + (y-y_2)^2}$$ $$\frac{dy_1}{dt} = v_2 = \frac{\Gamma_2}{2\pi}\frac{(x-x_2)}{(x-x_2)^2 + (y-y_2)^2} $$ $$\frac{dx_2}{dt} = u_1 = \frac{\Gamma_1}{2\pi}\frac{-(y-y_1)}{(x-x_1)^2 + (y-y_1)^2}$$ $$\frac{dy_2}{dt} = v_1 = \frac{\Gamma_1}{2\pi}\frac{(x-x_1)}{(x-x_1)^2 + (y-y_1)^2} $$
I now am trying to solve this system of ODEs using maple.
where I consider x1=(-1,0) and x2=(1,0)
In maple:
u(x,y,t) := (-( y))/(2*Pi*((x-1)^(2)+(y)^(2)));
v(x,y,t):=((x-1))/(2*Pi*((x-1)^(2)+(y)^(2)));
f(x,y,t):= ((y))/(2*Pi*((x+1)^(2)+(y)^(2)));
g(x,y,t):=(-(x-1))/(2*Pi*((x-1)^(2)+(y)^(2)))
ODEs := diff(a(t), t) = u(x(t), y(t), t), diff(b(t), t) = v(x(t), y(t), t), diff(c(t), t) = f(x(t), y(t), t), diff(d(t), t) = g(x(t), y(t), t)
but am now running into the problem of having 4 equations and 6 unknowns.
4 equations are the ODEs, and 6 unknowns are a,b,c,d, x and y.
I am obviously missing something when i'm writing the system of ODE's down in maple could some one help me with this.
I need to sovle this system and plot it to determine the pathlines.
I know the pathlines should be vertical lines for my current u,v,f,g.