I have a bottle filled with a fluid A, and a tube of volume $V_0$ filled with a fluid B leading away from it. When I now start pumping fluid A out of the bottle, I am interested in the concentration $c(V)$ of fluid A in fluid B at the end of the tube after the volume $V$ exited the tube.
In my second situation (I am afraid there are two), after having pumped $V_0$, a valve at position $V_1$ switches and the pump is reversed. The mixed fluids are now pumped into a second tube of volume $V_2$. Again, I'd like to know what drips out of there.
| A | | | V0 | ==================== <- B | | | | |_________|
| A | | | V0 | =========++========== <- B | | V1 || | | || V2 |_________| || ^- B
I've tried using the Hagen-Poiseuille equation (or its proof as given in the Wikipedia), but that grows ugly fast. Is there a simple(r) way to approach this?