Dynamics | Part IA, 2004

A small spherical bubble of radius a containing carbon dioxide rises in water due to a buoyancy force ρgV\rho g V, where ρ\rho is the density of water, gg is gravitational attraction and VV is the volume of the bubble. The drag on a bubble moving at speed uu is 6πμau6 \pi \mu a u, where μ\mu is the dynamic viscosity of water, and an accelerating bubble acts like a particle of mass αρV\alpha \rho V, for some constant α\alpha. Find the location at time tt of a bubble released from rest at t=0t=0 and show the bubble approaches a steady rise speed

U=29ρgμa2U=\frac{2}{9} \frac{\rho g}{\mu} a^{2}

Under some circumstances the carbon dioxide gradually dissolves in the water, which leads to the bubble radius varying as a2=a02βta^{2}=a_{0}^{2}-\beta t, where a0a_{0} is the bubble radius at t=0t=0 and β\beta is a constant. Under the assumption that the bubble rises at speed given by ()(*), determine the height to which it rises before it disappears.

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