Demonstration #3: Motion by the Flow

    Another way to look at a vector fields is to see how objects placed in the flow deform as they move with the flow. In this demonstration we mark a ring in the fluid, and watch how it moves along with the fluid.
    
    When watching the demonstrations, it is interesting to note that the area of the deformed ring always remains constant. More generally, the area of any closed curve remains constant as it flows along any of these vector fields. Indeed, let C be a smooth closed curve, and let C(t) be the resulting curve after it has flowed with the vector field for a time t, so that C = C(0). For each fixed t, letFormBox[RowBox[{Cell[], Cell[]}], TraditionalForm]
        θ↦ (x(t, θ), y(t, θ)),         θ_0≤θ≤θ_1
be the parametrization of the curve C(t). Then the area A(t) of the curve C(t) at time t is
        A(t) = ∲_C(t) x  dy.
Thus
        A ' (t) = d/dt∫_θ_0^θ_1x∂y/∂θ dθ
so that
        A ' (t) = ∫_θ_0^θ_1 (∂x/∂t∂y/∂θ + x∂^2y/(∂t ∂θ)) dθ.
Integration by parts and the fact that the curve is closed imply
        A ' (t) = ∫_θ_0^θ_1 (∂x/∂t, ∂y/∂t) · (∂y/∂θ, -∂x/∂θ) dθ.
We recognize this form as
        A ' (t) = ∲_C(t) Overscript[v, ⇀] · Overscript[n, ⇀] ds.
Then, using Gauss's Theorem, we see that
        A ' (t) = ∫∫_Ω(t) div Overscript[v, ⇀] dA
where Ω(t) is the region enclosed by the curve C(t). Because div Overscript[v, ⇀] = 0 for all of the vector fields under consideration, we see that A ' (t) = 0, as claimed.

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