Flow Around a Cylinder

     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
with parameters
        R - radius of the obstacle, and
        U - velocity of the fluid far from the obstacle
describes two-dimensional flow around a circular obstacle.
    We can see the vector field with the Mathematica commands:

Needs["Graphics`PlotField`"] U = 1/2 ; R = 1 ; v[x_, y_] := ... nbsp;        (* Ensure that the result is shown *)


Changing the parameters R or U simply changes the scales of the resulting graphs. Indeed, if we make the definition
        Overscript[v, ⇀](x, y ; R) = (0, 0)                                                if  x^2 + y^2≤R^2,
        Overscript[v, ⇀](x, y ; R) = U (1 - R^2 (x^2 - y^2)/(x^2 + y^2)^2, R (-2x y)/(x^2 + y^2)^2)      if  x^2 + y^2≥R^2,
then we see that
        Overscript[v, ⇀](x, y ; R) = Overscript[v, ⇀] (x/R, y/R ; 1).
Changing U, on the other hand, simply dilates the size of the resulting vectors. We can see both of these effects graphically in the following animations. First we adjust U.

Needs["Graphics`Arrow`"] Needs["Graphics`PlotField`"] Off[General :: spell ...          (* Turn on spelling warnings *)  

                Varying U

While here we adjust R.

Needs["Graphics`Arrow`"] Needs["Graphics`PlotField`"] Off[ ... p;                

                Varying R