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When I walk my kid sometimes I put a cut of coffee in a cupholder attached to the stroller. When I push the stroller over a brick pavement it seems to vibrate vertically. That is enough for the coffee to spill, even if the cup is only 3/4 full. It looked like waves were forming in the cup, beating its forward and backward sides, and they quickly grew large enough to spill.

Thus the question: is it possible to quantitively estimate the size of the waves caused by the vibration, so that one could figure out the level one can safely fill the cup to?

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Have you tried different speeds? I ask because it could provide a hint. If changing speeds helps it may well be a resonance effect. –  dmckee Jul 20 at 3:14
    
You could always get a cup with a lid. –  LDC3 Jul 20 at 3:15
    
@LDC3: The cup did have a lid; the coffee spilled through the sipping hole. Regardless of that, I am curious about the physics related to the waves that form inside the cup, even if the answer wouldn't have a practical application. –  Michael Jul 20 at 3:34
    
@dmckee: I did; the spillage occurs even at slow stroller speed. I think it has to do with the hardness of vertical movement, such as a momentary acceleration on the order of g when passing the seem between the bricks, more than the frequency of those vertical movements, and only their frequency depends on stroller speed. –  Michael Jul 20 at 3:37

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up vote 3 down vote accepted

Not a complete answer, but this is a classic "sloshing" problem.

The interaction between the fluid and the container wall, under the influence of the external (periodic) force sets up a (self-reinforcing) and harmful resonance.

This is of immense practical interest: jet-fuel sloshing inside airplane tanks, for instance.

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Thanks! I knew an effect like that had to have a name! Where would I look for references on quantitative analysis of the effect? Something like the dependency of the resonance frequency on the cylinder diameter, things like that? –  Michael Jul 20 at 6:15
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@Michael: IANAE, but sloshing is a highly nonlinear problem (consider that the fluid can detach and form new surfaces, among other sources of nonlinearities). I know people have used Computational Fluid Dynamics (CFD) for this, but plain-vanilla CFD works only if you assume that the tank is rigid. The general problem with a deformable tank is an FSI (fluid-structure interaction) problem, which is a different beast altogether. You might wish to google "CFD + tank + sloshing" for refs. –  user_of_math Jul 20 at 6:27

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