= Constant-tension balloon discharge
{title2=$R^3\ddot R+2R^2\dot R^2=-K,\quad K=\gamma a^2/(2\rho L)$}
A spherical balloon feeding an inviscid uniform tube plug, with entrance excess <pressure> $2\gamma/R$, satisfies $L\dot u=2\gamma/(\rho R)$ and $4\pi R^2\dot R=-\pi a^2u$. Eliminating $u$ yields the displayed equation. Its <first integral> is $R^4\dot R^2+KR^2=C$. From rest at radius $R_0$, $C=KR_0^2$; integration gives $t=R_0^2K^{-1/2}(\pi/4-\theta/2+\sin(2\theta)/4)$, where $\theta=\arcsin(R/R_0)$. The ideal emptying time is $\pi R_0^2/(4\sqrt K)$. Independently prescribed initial flow changes $C$ and therefore this time.
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