Flow Around a Rankine Half-Body

    The Rankine half-body is the curve that satisfies the equation
        x = -y cot ((2π y)/h) for -h/2≤y≤h/2.
The parameter h describes the width of the half-body. Indeed, as x∞, it is easy to see that y ± h/2, and that total width of the body approaches h. Here is a graph of the Rankine half-body.

h = 1 ; ParametricPlot[{-t Cot[(2π t)/h], t}, {t, -h/2, h/2}, PlotRange { ...  y axes have the same scale *)PlotLabel->"Graph of the Rankine Half-Body"] ;


Note that the Rankine half-body does not pass through the origin; rather its vertex is at the point (-h/(2π), 0); indeed

Clear[h] ; Series[-y Cot[(2 π y)/h], {y, 0, 4}]

-h/(2 π) + (2 π y^2)/(3 h) + (8 π^3 y^4)/(45 h^3) + O[y]^5

    The vector field that described the flow of a fluid around a Rankine half-body is given by
        Overscript[v, ⇀] = U (1 + h/(2π) x/(x^2 + y^2), h/(2π) y/(x^2 + y^2))
with the parameters
        h - which gives the width of the Rankine half-body, and
        U - which gives the velocity of  the fluid far from the body.

Needs["Graphics`PlotField`"] Needs["Graphics`FilledPlot`"] U = 1/2 ; h = 2 ... entity], DisplayFunction$DisplayFunction, PlotRange {{-h, h}, {-h, h}}] ;


    Changing the parameters U or h simply re-scales the resulting vector field.