A spherical oil-drop injected into water flowing through a straight tube will retain its shape as it is transported along the tube. We have seen in Section 2.3 that a homogeneous drop's linear density has a parabolic profile. The animation here illustrates how that profile floats down the tube.
In Example 1 in Section 2.4 we solve the IBVP \begin{align*} &\frac{\partial}{\partial t} \rho(x,t) + 2\frac{\partial}{\partial x} \rho(x,t) = 0 && x >0, \quad t>0, \\ &\rho(x,0) = e^{-x} && x > 0, \\ &\rho(0,t) = \cos t && t > 0. \end{align*} and arrive at $\rho(x,t) = \psi(x-2t)$, where \[ \psi(s) = \begin{cases} e^{-s} & s >0, \\ \cos\Bigl(-\frac12 s) & s < 0 \end{cases} \] This animation shows what $\rho(x,t)$ looks like.
In Section 2.4 we examine the advection of a spherical oil-drop through a water pipe as it flows through an abrupt constriction. This animation shows the evolution of the drop's linear density and its shape as it is squeezes through the constriction. Note how the drop speeds up after passing to the narrow region.
The space-time diagram shows the passage of time, and the region swept by the drop in the process.
Here is the three-dimensional view of the oil-drop as it goes through the stepped pipe. Two small droplets have been added to enhance the visualization.