Two identical spherical balloons have been inflated with air to different sizes. They are now
connected together with a hollow leak proof tube, what will be the direction of flow of air-from
bigger to smaller balloon or vice versa. What if we perform experiment with soap bubbles of
different sizes? Explain.
Answers
Answer:
Explanation:
This is a relatively famous problem called ``Two-balloon experiment''. The result is actually a little counter-intuitive.
2.1 Balloon surface tension with Hook's law
2.1.1 Calculations
Suppose there is a perfectly spherical balloon with a rubber with the initial radius of r0, that is, when there is no tension on the rubber (Left panel in Fig.2). There is no excess pressure over the atmospheric pressure of 1 atm in the air inside the balloon at this stage.
The amount of the tension in rubber for a certain stretch is proportional to the amount of material of the rubber. Providing the density of the rubber is uniform, and providing the thickness of a sheet of it is uniform (and negligibly small), it is proportional to the area of the rubber. Let us define k as a spring constant for a certain original area and shape of the rubber. Hook's law states that the force of a tension F, when a (ideal, namely linearly elastic) spring is stretched by the length x, is F=kx, using the spring constant k.
Then, if a very narrow strip of the rubber ( PQ¯ in Fig.2), of which the viewing angle from the origin of the coordinates is Δθ, is stretched to the circular length of πr (with the radius r) from the original length of πr0, keeping the perfect spherical profile, the tension force F(r,Δθ) between PQ¯ is,
F(r,Δθ)∝k(πr-πr0)Δθ2π=Ak(r-r0)Δθ,(1)
where A is a non-dimensional constant. Note that the exact value of A is not quite trivial to calculate, and is not discussed here, however it does not affect the argument below.
We now consider the balance of the works by between the rubber tension and (excess) pressure of the air inside, when this balloon of rubber is stretched from the radius of r by an infinitesimal amount of Δr to r+Δr, as it is inflated. The work ( Wp) by the excess pressure ( P(r)) of the air inside over the atmospheric air pressure for the given radius r of the balloon is written as,
Each of tiny strips of the rubber for the angle of Δθ does a work ΔWb of, using its force .
Therefore, the work Wb by the rubber of the entire balloon is obtained by integrating this ΔWb (Eq. 5) over 2π radians as,
Consequently, the air pressure inside the balloon ,
sharply increases as a function of the radius r to a peak, then sharply decreases beyond,
decreases inversely proportional to the radius r, where r is far greater than r0.
Note that the above argument is not exact. In the more strict formulation, based on the James-Guth stress-strain relation, the first part of the equation, which is dominant at r∼r0, is inversely proportional to the negative r6 [1] with a certain offset, as opposed to r2 as derived above. Namely, the shapes of the functions between the strict one and the derived one here are broadly similar, but the former is a much more steep function of r. On the other hand, the second part, which is dominant at a larger r, is the same, and is inversely proportional to r.
Another note is that Hook's law is a good approximation in general for rubber, but only when the rubber is not exceedingly stretched. It is known the surface tension acutely increases again when the rubber is stretched close to its physical maximum limit of stretch, that is, before the breakage.