The motion of long bubbles in tubes

A long bubble of a fluid of negligible viscosity is moving steadily in a tube filled with liquid of viscosity μ at small Reynolds number, the interfacial tension being σ. The angle of contact at the wall is zero. Two related problems are treated here.In the first the tube radius r is so small that gravitational effects are negligible, and theory shows that the speed U of the bubble exceeds the average speed of the fluid in the tube by an amount UW, where $W \simeq 1\cdot 29(3 \mu U|\sigma)^{\frac {2}{3}}\;\;\; as\;\;\; \mu U|\sigma$ (This result is in error by no more than 10% provided $\mu U |\sigma \; \textless \;5 \times 10^{-3}\rightarrow 0$). The pressure drop, P, across such a bubble is given by $P \simeq 3\cdot 58(3\mu U|\sigma)^{\frac {2}{3}}\sigma|r \; \; \;as\; \; \; \mu U|\sigma \rightarrow 0$ and W is uniquely determined by conditions near the leading meniscus. The interface near the rear meniscus has a wave-like appearance. This provides a partial theory of the indicator bubble commonly used to measure liquid flowrates in capillaries. A similar theory is applicable to the two-dimensional motion round a meniscus between two parallel plates. Experimental results given here for the value of W agree well neither with theory nor with previous experiments by other workers. No explanation is given for the discrepancies.In the second problem the tube is wider, vertical, and sealed at one end. The bubble now moves under the effect of gravity, but it is shown that it will not rise at all if $\rho gr^2| \sigma \; \textless \; 0 \cdot 842,$ where ρ is the difference in density between the fluids inside and outside the bubble. If $0 \cdot 842 \; \textless \; 1 \cdot 04,$ then $\rho gr^2| \sigma - 0 \cdot842 \simeq 1 \cdot 25 (\mu U|\sigma)^{\frac {2}{9}} + 2 \cdot 24(\mu U|\sigma)^{\frac {1}{3}},$ accurate to within 10%. Experiments are adduced in support of these results, though there is disagreement with previous work.