Demonstration #2: Streamlines

    Another important family of curves associated with a vector field are its streamlines. Mathematically, the streamlines are curves whose tangents are everywhere parallel to the vector field; intuitively they are the paths that a particle in the fluid will actually follow. If Overscript[v, ⇀](x, y) = (v_1(x, y), v_2(x, y)) is a vector field, and if Overscript[r, ⇀](s) = (x(s), y(s)) is a parametric curve of a streamline, then there must be a nonzero constant of proportionality α so that
        x ' (s) = α v_1(x(s), y(s))
        y ' (s) = α v_2(x(s), y(s))
Thus, if we can solve the two ordinary differential equations, we can find the streamlines.
    We also notice that modifying the parameter α only changes the speed along the parametric curve; indeed, we can use the re-parametrization Overscript[x, ~](t) = x(α t), Overscript[y, ~](t) = y(α t).FormBox[Cell[], TraditionalForm]In particular, the geometric shape of the streamline is independent of the choice of α.
    It is worth noting at this point that the preceding discussion is only valid for vector fields Overscript[v, ⇀](x, y) that are constant in time. If we allow the vector field to depend on time, so that Overscript[v, ⇀] = Overscript[v, ⇀](x, y, t) then the streamlines remain the curves whose tangents are everywhere parallel to the vector field; however these no longer represent the actual particle paths, which are called pathlines.

Flow Around a Cylinder

Point Vortex Flow Around a Cylinder

Flow Around a Cylinder with Circulation

Flow in a Corner

Flow in a Closed Channel

Flow Around a Rankine Half-Body