Many years ago - being a fourth year student at the Eindhoven University of Technology (

Independent variables are the volume flows of air and water, also called air flow $F_L$ and water flow $F_W$. Dependent variables are the frequency $f$ of the bubbles and the bubble volume $V_L$. (It is theoretically possible, but not very practical to consider instead the volume of "water bubbles" $V_W$.) As can be easily shown, there is a simple relationship between the frequency and the bubble volumes: $V_L = F_L/f$ and $V_W = F_W/f$.

The intention has been to also determine experimentally whether there exists a relationship between the independent and dependent variables. Meanwhile we know that such measurement is a form of

Air (mm^{3}/s) |
Water (mm^{3}/s) |
Frequency (1/s) |

44 | 56 | 31 |

62 | 50 | 35 |

66 | 45 | 35 |

78 | 45 | 37 |

140 | 41 | 43 |

149 | 42 | 44.5 |

29 | 70 | 25 |

18 | 79 | 20 |

9 | 98 | 14.5 |

6 | 133 | 11 |

21 | 177 | 42 |

18 | 180 | 35 |

14 | 199 | 30 |

7 | 205 | 19 * |

9 | 232 | 22 |

1 | 235 | 2.4 |

33 | 145 | 47 |

49 | 135 | 57 |

55 | 135 | 62 |

62 | 124 | 60 |

66 | 117 | 65 |

91 | 117 | 80 |

95 | 152 | 80 |

96 | 109 | 75 |

99 | 108 | 73 |

120 | 112 | 90 |

122 | 92 | 76 |

145 | 115 | 92 |

146 | 98 | 88 |

157 | 98 | 97 |

169 | 100 | 95 |

Subsequently it was expected that I should be able explain the measurements, by
devising a **mathematical model**. The phenomenon of bubble formation has been
put on film as well, in order to be able to study it better. The following pictures
are snapshots from that film.

**Series 1**

**Series 2**

**Series 3**

In retrospect, I am certain that he must have understood perfectly well what is essential and what not with designing a mathematical model. Let us start but assume that the formation of the air bubbles is completely independent of the surface tension, and thus independent of any possible curvature of the surfaces. Then it is in fact not so important what exactly the form the bubble surface is (such in stark contrast to what I had always thought).

The following is a sketch of the T-piece, containing the more "accurate" image of a growing bubble, as seen from one side:

When the bubbles are not required to be round, then we can imagine for our convenience that they are instead, please do not panic:

What is happening here is beyond the ability of a simple measuring instrument. And the solution requires a minimum of physics knowledge. What is really needed, however, is some kind of engineering ingenuity, and quite some guts. In short: a human being is needed here. We can no longer speak of abstraction, we must talk about the creation of an idea in someone's mind:

So the idea is that first the bubble is pressed by the air flow against the other side of the water channel (stage 1-3), and then by the water flow across the width of the air duct is cut off (phase 4-6). On the basis of this idealized picture it is possible to set up a meaningful calculation:

- Time required for phase $(1)-(3) \;=\;$ diameter water-channel / air-speed
- Time required for phase $(4)-(6) \;=\;$ diameter air-channel / water-speed

Here: $A =$ flow area, $d =$ channel diameter. Therefore $V_0 =$ sort of "initial" volume, where $V_0 = A.d$ .

The total time needed for bubble formation is : $\;t_{1-6} = t_{1-3} + t_{4-6} = 1/f = V_0 ( 1/F_L + 1/F_W )$ .

Hence the volume of the air bubbles is calculated by: $$ V_L = F_L/ f = V_0 ( 1 + F_L/F_W ) \qquad \mbox{: MATHEMATICAL MODEL !} $$ The only yet unknown quantity in this formula is the initial volume $V_0$ . This "initial volume" can be determined in two independent ways. The first way is based on a rough estimate: the volume must be something in between a sphere and a cylinder.

Given the fact that the diameter of the capillaries is $1$ mm, we find for the cylinder: $\pi/4.d^2.d = 0.785$ . And the sphere: $\pi/6.d^3 = 0.52$ . Hence: $0.52 < V_0 < 0.78$ .

Secondly, we can try to match the the model with the measurements. Let $x=F_L/F_W$ and $y=V_L$. A least squares adjustment gives: $$ \sum_{i=1}^{N} \{V_0 [1 + x_i] - y_i \}^2 = \mbox{minimal} $$ where $N$ is the number of measuring points. This is a quadratic function in $V_0$: $$ A V_0^2 - 2 B V_0 + C = \mbox{minimal} $$ With: $ A = \sum_{i=1}^{N} (1+x_i)^2 $ ; $ B = \sum_{i=1}^{N} (1+x_i) y_i $ ; $ C = \sum_{i=1}^{N} y_i^2 \; $ . It is well known that the minimum of this parabola, with $V_0$ as an independent variable, is found for : $V_0 = B/A$ . Or: $$ V_0 = \frac{\sum_{i=1}^{N} (1+x_i) y_i}{\sum_{i=1}^{N} (1+x_i)^2} $$ This gives from the measurements a value of $V_0 = 0.6934510 \approx 0.7 \, mm^3$ : the experimental value is indeed in between the rough theoretical values.

We have derived the fundamental natural law for the formation of bubbles in a T-piece: $V_L = V_0 (1 + F_L/F_W)$. A law that convinces by simplicity and elegance. There's a certain charm associated with such formulas, people say, and one wonders what the the deepest ground is of this charm. In case of the bubbles the answer is clear: the beauty of the formula found is due only to the dramatic simplification that we have applied. The beauty does not occur because we have grasped the reality in all its details, but precisely because we have dispensed of most aspects of reality, not because we have pursued the whole truth, but precisely because we have not pursued that truth. The paradox is that we nevertheless have come closer to "the truth". After all, the result is quantitative, agrees well with the experiments, in short: is exactly what one would expect of a (small) piece of science.

Why so much attention has been given to a relatively inane application like this one? Yes, I did talk about the fundamental law of physics for the formation of bubbles in a T-piece, but that will no doubt be intended as a metaphor. This author certainly is not going to say that a formula for silly bubbles, nice as it is, has the same status as, for example, the Laws of Newton, or the Lorenz Transformations? On the contrary; that's precisely what this author intends to say. One of my key points is that, as one gets deeper about it, no substantial distinction can be made between "fundamental" and "applied" research. If we are a bit further, I will gradually show that this distinction is increasingly difficult to sustain, and finally has to go away. Albeit one can defend that $\,F = m\cdot a\,$ has a broader scope than $\,V_L = V_0 (1 + F_L/F_W)$ , and in that sense is more fundamental. But I will keep my stand that it refers to a gradual, not to an essential difference:

- One cannot draw a line somewhere between the laws of physics and say: on this side everything is exact, on the other side everything is an approximation