Ideal Fluids in Motion — Question 3

Question 3

An instantaneous streamline can differ from a particle path

A spatially uniform but time-dependent flow is $\mathbf v=U\hat i+bt\hat j$, with $U>0$ and $b>0$. A marked parcel starts at the origin at $t=0$. A streamline is everywhere tangent to the velocity field at one chosen instant.

1. Find the streamline through the origin at time $t_0$ and the marked parcel path.

2. Compare their slopes when the parcel is at time $t_0>0$. Why can the path and a streamline be tangent locally without being the same curve?

Question 3 - Solution

An instantaneous streamline can differ from a particle path

(a) Derivation and calculation

At the chosen instant $t_0$,$$\frac{dy}{dx}=\frac{bt_0}{U}\Rightarrow\boxed{y_{\mathrm{streamline}}=\frac{bt_0}{U}x}\quad\text{through the origin}.$$$$\dot x=U,\quad\dot y=bt\Rightarrow x=Ut,\quad y=\tfrac12bt^2.$$$$\boxed{y_{\mathrm{path}}=\frac{b}{2U^2}x^2}.$$

(b) Result and interpretation

At the parcel position $x_0=Ut_0$,$$\left.\frac{dy_{\mathrm{path}}}{dx}\right|_{x_0}=\frac{bx_0}{U^2}=\frac{bt_0}{U}.$$$$y-\tfrac12bt_0^2=\frac{bt_0}{U}(x-Ut_0)$$$$\text{is the instantaneous streamline through the parcel}.$$It is locally tangent to the path, but is not generally the streamline through the origin or the full path history.

Physical check

For $t_0>0$, the origin streamline and the parcel tangent line are parallel with different intercepts.