Fluid Dynamics | Part IB, 2003

State the form of Bernoulli's theorem appropriate for an unsteady irrotational motion of an inviscid incompressible fluid in the absence of gravity.

Water of density ρ\rho is driven through a tube of length LL and internal radius aa by the pressure exerted by a spherical, water-filled balloon of radius R(t)R(t) attached to one end of the tube. The balloon maintains the pressure of the water entering the tube at 2γ/R2 \gamma / R in excess of atmospheric pressure, where γ\gamma is a constant. It may be assumed that the water exits the tube at atmospheric pressure. Show that

R3R¨+2R2R˙2=γa22ρL.R^{3} \ddot{R}+2 R^{2} \dot{R}^{2}=-\frac{\gamma a^{2}}{2 \rho L} .

Solve equation ( \dagger ), by multiplying through by 2RR˙2 R \dot{R} or otherwise, to obtain

t=R02(2ρLγa2)1/2[π4θ2+14sin2θ]t=R_{0}^{2}\left(\frac{2 \rho L}{\gamma a^{2}}\right)^{1 / 2}\left[\frac{\pi}{4}-\frac{\theta}{2}+\frac{1}{4} \sin 2 \theta\right]

where θ=sin1(R/R0)\theta=\sin ^{-1}\left(R / R_{0}\right) and R0R_{0} is the initial radius of the balloon. Hence find the time when R=0R=0.

Typos? Please submit corrections to this page on GitHub.