aa r X i v : . [ phy s i c s . pop - ph ] F e b Physics of Beer Tapping – Lower vs. Upper Bottle
Johann Ostmeyer
Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Germany
Abstract “Beer tapping” is a well known prank where a bottle of carbonised liquid strikes anotherbottle of carbonised liquid from above, with the usual result that the lower bottle foams overwhereas the upper one does not. Though the physics leading to the foaming process in thelower bottle has been investigated and well documented, no explanation to date has beenprovided why the upper bottle produces little to no foam. In this article we describe thereasons for the entirely different behaviours of the two bottles.
Recently it was shown in [1, 2] why a bottle of supersaturated carbonised liquid foams over, afterbeing struck from above by another bottle, usually also containing supersaturated liquid. I.e. thegas in the liquid expands and the ensuing foam rises and spills over the container. The beer-drinking community is well versed in such a phenomenon. Such an event, usually performed asa prank, is called ‘beer tapping’. Here the victim of the prank holds an open beer bottle, whilea prankster strikes the top of the beer bottle with the bottom of their own (open) beer bottle.The beer in the lower bottle then foams over the opening and onto the victim’s hands, hopefullyeliciting some good-natured laughs. The physics dictating the foaming of the beer is well describedin [1] and we provide a cursory explanation in section 2 closely following the aforementioned article.Missing from [1, 2] is an explanation for why the prankster leaves the scene relatively unscathedand dry. They also hold an open beer bottle, which they use to strike the victim’s bottle, yet littleto no foam exits their bottle. This outcome is also easily described using the methods of [1], andis the subject of this article.
The most important stages of beer tapping are as follows. The initial shock due to the “tap” inducesoscillations of the radius R of small bubbles already existing in the liquid. These oscillations aredriven by the shock pressure p S ( t ) and follow, to a good approximation, the Rayleigh-Plesset-equation [3] ρR ¨ R + 32 ρ ˙ R − (cid:18) p + 2 σR (cid:19) (cid:18) R R (cid:19) γ + 2 σR + 4 µ ˙ RR = − ( p + p S ( t )) (1)where R ≈ µ m is the typical size of the initial radius of the bubble, ρ = 10 kg / m the densityof water, p = 10 Pa the ambient pressure, σ = 0 . / m the surface tension between CO andwater, γ = 1 . the heat capacity ratio of CO and µ = 10 − kg / m / s the liquid viscosity.When reaching the first minimum in radial size after t min ≈ .
15 ms the bubbles in the lowerbottle collapse [4] according to the model of Brennen [5] into about N fragments of nearly equal1 COMPARISON OF UPPER AND LOWER BOTTLE N = (cid:18) (cid:16) √ − (cid:17)(cid:19) , (2) Γ = ρσ R ( t min ) ¨ R ( t min ) . (3)Each initial bubble, therefore, forms a cloud of N smaller bubbles, where typically N ∼ . Athus increased number of bubbles leads to a larger interface area between gas and liquid. Thisleads to the second stage where the gas within the liquid diffuses into the bubbles of each cloud.The volume of the cloud formed by the bubbles grows in square-root time √ t [6] for about thenext
10 ms .Finally, during the last stage, the cloud of bubbles becomes so large that it quickly rises upbecause of its buoyancy. As it moves, it collects even more gas, which in turn further increasesthe sizes of the bubbles. Cloud growth now scales as t and the cloud takes a form reminiscent ofa mushroom cloud from a nuclear explosion (see the illuminating figures in [7]). If enough largeclouds have been formed, the liquid foams over at approximately . after the “tap”.The interested reader may find a more detailed physical explanation of the foaming process,as well as many interesting phenomena appearing in carbonised beverages, in reference [7]. The last two stages do neither depend on the bottle position, nor on the position of a bubble inthe bottle. Thus it is sufficient to investigate the collapse of a single bubble at a fixed position inorder to understand why the upper bottle (which we now label as U) does not react as vehementlyas the lower bottle (now labelled L). Indeed, it is this inequality between the two bottles whichmakes the prank attractive in the first place.
To do so we first have to understand the time evolution of the shock wave. The pressure inducedby the “tap” can be modelled by a damped oscillation p S ( t ) = p A sin (cid:18) πtT (cid:19) exp (cid:18) − tτ (cid:19) (4)where p A is the initial amplitude set by the strength of the “tap”, T = 0 .
24 ms [1] and t representsthe time since the shock first reached the bubble. It turns out that the shock wave will lose mostof its intensity through the reflection at the fluid’s surface (implying that the damping does notbehave as smoothly as modelled in reality) and not while travelling in the fluid [1]. Thus we expecta damping in the order τ ≈ T , i.e. the time needed to reach the surface and be reflected. Howeverwe will find that the damping does not have significant influence on the qualitative behaviour ofthe foaming.The difference between bottles U and L now is only the sign of p A , where p A > in U and p A < in L. This means that bubbles in bottle U are compressed by the shock and oscillatemoderately, whereas bubbles in bottle L first expand and then collapse violently. This can beobserved in figure 1 where we plotted the time evolution of equation (1). The solution of thisordinary differential equation is numerically not challenging and has been performed by the clas-sical Runge-Kutta 4 method. Measurements performed in [1, 2] show good agreement with thenumerical predictions for different initial conditions. COMPARISON OF UPPER AND LOWER BOTTLE . . . . . . . . .
45 0 0 .
02 0 .
04 0 .
06 0 .
08 0 . .
12 0 . R / mm t/ ms p A = p (upper bottle) p A = − p (lower bottle)Figure 1: Radius of a CO -bubble since the first excitation without damping, τ = ∞ . Obtainedvia numerical simulations using Runge-Kutta 4. The black (green) curve shows the behaviour ofa bubble in the upper (lower) bottle exposed to positive (negative) initial pressure by the “tap”. The collapse of the oscillating bubbles is described by the model of Brennen [5] according to whichthe most unstable mode n m = 13 (cid:16) √ − (cid:17) (5)is mainly responsible for the bubble collapse at the first local minimum into N ≈ n m fragments.At the same time, n m gives the factor by which the total surface area of all fragments surpassesthe surface area of the original bubble. As this area is proportional to the amount of CO enteringthe bubbles by diffusion, n m is also a direct indicator for the intensity of the foam formation.We calculated n m for different initial pressures and damping strengths. The results can befound in figure 2. We observe that the most unstable mode grows in both bottles with increasinghit strength. However, as long as the damping is not extremely large, n m grows very fast fordecreasing p A in bottle L ( − p < p A < , the lower bound is needed because no pressure belowzero, i.e. vacuum, is possible), but only slowly for increasing p A in bottle U ( < p A ). In [1] n m ≈ is observed. This means that the damping is quite small in reality and τ & T .We cannot determine the damping quantitatively, but we do not have to, either. It is sufficientto observe that for any realistic damping, n m can be an order of magnitude larger in bottle L thanin bottle U. The only requirement for this to be the case is such a strong “tapping” that p A ≈ − p .Obviously the pressure in bottle U is not limited from above, so an extremely strong hit canproduce p A ≫ p and cause the upper bottle to foam over as well. But such a brute force “tap” isnot of high interest and can incur damage to the bottles. The discontinuities in the region − . < p /p A < are not to be taken to seriously. Here the first localminimum disappears, so that another one is the first, and later appears again. There is no relevant physics hiddenin this strange behaviour. SUMMARY − − . . n m p /p A τ = ∞ τ = 2 Tτ = 0 . T Figure 2: Most unstable mode according to the model of Brennen depending on the initial pressure.The different lines correspond to different damping strengths in eq. (4). The black line has beencalculated in the absence of damping.
Although the precise physics of beer tapping including fluid dynamics are quite complicated, itcan easily be understood why the lower bottle L foams over while the upper one U (usually) doesnot. The “tap” creates a low pressure in bottle L which causes existing bubbles of CO to expandand then collapse into many fragments. The subsequent CO -liquid interface grows and this inturn causes more gas to diffuse into the bubble cluster. It rises upwards and creates the foam. Inbottle U the high initial pressure induces only moderate bubble oscillations such that the collapsedoes not happen or, if it does, it does so not as violently. Either way, there is no rapid cloudgrowth and the prankster leaves the scene dry. Acknowledgements
Huge thanks to Thomas Luu for proofreading and giving a lot of helpful comments. I would alsolike to thank Kathrin Grunthal, Carsten Urbach and every other person I ever drank beer withfor the inspiration and motivation to write this article.
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