Flow Around a Cylinder

    The flow around a cylinder is described by the vector field
        Overscript[v, ⇀](x, y) = U(1 - R^2 (x^2 - y^2)/(x^2 + y^2)^2, R (-2x y)/(x^2 + y^2)^2) defined for (x^2 + y^2)^(1/2) ≥R.
To find the streamlines, we set α = U and look for solutions of the differential equations
        x ' (t) = 1 - R^2 (x^2(t) - y^2(t))/(x^2(t) + y^2(t))^2, <br />y ' (t) = R (-2x(t) y(t))/(x^2(t) + y^2(t))^2 .
Mathematica is unable to find an analytic solution to this system of differential equations.

Clear[R] ; DSolve[{x '[t] 1 - R^2 (x[t]^2 - y[t]^2)/(x[t]^2 + y[t]^2)^2, y '[t] R (-2 x[t] y[t])/(x[t]^2 + y[t]^2)^2}, {x[t], y[t]}, t]

DSolve[{x^′[t] 1 - (R^2 (x[t]^2 - y[t]^2))/(x[t]^2 + y[t]^2)^2, y^′[t]  -(2 R x[t] y[t])/(x[t]^2 + y[t]^2)^2}, {x[t], y[t]}, t]

However, Mathematica is quite able to solve the system numerically. We can plot a selection of streamlines together with a selection of equipotential curves as follows.

Needs["Graphics`PlotField`"] ; Needs["Graphics`ImplicitPlot`"] ; U = 1/2 ; ... ge {{-3R, 3R}, {-3R, 3R}}}], ]}], ;}],         }]

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