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.
It is simple to check that ∂v_1/∂y = ∂v_2/∂x and that Overscript[v, ⇀] is generated by a potential; integrating we find that
        ϕ = U (x + R^2x/(x^2 + y^2)).
Indeed,

Clear[U, R] ϕ[x_, y_] := U (x + (R^2 x)/(x^2 + y^2)) ; GradPhi[x_, y_] := {D[ϕ[ξ ... [{"v(x,y)-∇ϕ(x,y) = ", Cell[]}], ,, Simplify[v[x, y] - GradPhi[x, y]]}], ]}]

v(x,y)-∇ϕ(x,y) =  {0, 0}

The graph of the potential is given below.

U = 1/2 ; R = 1 ; Off[Plot3D :: "plnc"] ;     &n ...   (* Turn messages back on *)On[Plot3D :: "gval"] ;    

[Graphics:../HTMLFiles/index_124.gif]

    We can plot the vector field, together with a selection of equipotential curves as follows.

RowBox[{RowBox[{Needs["Graphics`PlotField`"] ;, , Needs["Graphics`Impli ... e {{-3R, 3R}, {-3R, 3R}}}], ]}], ;}]}],         }]

[Graphics:../HTMLFiles/index_126.gif]