Suppose that a tube is immersed in a liquid and that air flow through the tube creates a bubble at its end, as shown in Fig. P9.7. Beginning at t = 0, when the bubble radius is R₀, the volume flow rate of air is held constant at Q. This is done in a gravity-free environment. Thus, the bubble remains spherical and static pressure variations are absent, even though the bubble is not necessarily small. Assume that the motion in the liquid is purely radial.

(a) Determine the radius R(t) of the growing bubble, assuming that R₀ and Q are given.

(b) Find v_r in the liquid for all r≥R(t).

(c) Determine P in the liquid for all r≥R(t), assuming that P = P_∞ far from the bubble. 

(d) Derive a general expression for the bubble pressure P_B(t), including the effects of surface tension, viscosity, and inertia.

Transcribed Image Text:

Air PB(t) Figure P9.7 Air bubble growing at the end of a tube. R(t) Liquid