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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.

Situation 1

|    A    |
|         |      V0
|      ====================    <- B
|         |
|         |
|_________|

Situation 2

|    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?

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Your approach using Hagen-Poiseuillie sounds like a good first approximation. You have to think of the limitations though. It assumes laminar flow which depend on the viscosity, pumping rate and geometry. Additionally your liquids might mix up without any pumping because of diffusion. So a realistic estimate will be hard with a simple approach. –  Alexander Jun 28 '12 at 14:05
    
Is it a bird? Is it a plane? No! It's Empirical Numerical Model man here to save us from certain doom! :) –  Colin K Jun 28 '12 at 19:27
    
Seriously though, if you only need to make a prediction for practical use, and you don't need your fitted model parameters to have physical significance, then any old fit will work. You could even just record some data and then interpolate it for future predictions. –  Colin K Jun 28 '12 at 19:31
    
@Colin: I was afraid of that =) I distrust my current data and wanted to check against theoretical predictions. –  Jens Jun 29 '12 at 5:27
    
Oh, I see. In that case an empirical model obviously won't work! I didn't suggest it because I thought a theoretical model was impossible; I just misunderstood your goal. –  Colin K Jun 29 '12 at 11:45
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