MadSci Network: Physics

Re: How fast does an air bubble ascend underwater?

Area: Physics
Posted By: Jason Goodman, Graduate Student, Massachusetts Institute of Technology
Date: Tue Jun 3 13:36:47 1997
Area of science: Physics
ID: 863802993.Ph

It depends on the size of the bubble. By playing around with bubbles in water, you can see for yourself that small bubbles rise very slowly, while large bubbles go much more quickly. Champagne bubbles have speeds of millimeters per second, while a bubble the size of a baseball may rise at several meters per second.

viscosity is a measure of the sludginess of a fluid. Water has very low viscosity, while molasses or syrup have high viscosity. Viscous fluids don't like to flow around things. A bubble rising through a liquid obeys the same laws as a ball falling through the air: the force of gravity is counteracted by the drag force exerted by the water or air. The drag force is large when the fluid is very viscous or when the bubble or ball is moving very quickly. A bubble in water has almost the same balance of drag and gravity force as a bubble in syrup, but the viscosity of the syrup is higher, so the bubble must rise slower.

But wait, it gets even more complicated! The drag force behaves in two different ways, depending on the speed of the bubble or ball. For very small bubbles, we can show that the speed of the bubble is

  u = 1/3 a^2 g/nu 
where a is the radius of the bubble, g is the acceleration of gravity (980 cm/s), and nu is the "kinematic viscosity" of water (0.14 cm^2/s).

This only holds for bubbles less than a millimeter across. When they get bigger than that, a "boundary layer" forms, which changes the way the water flows around the bubble, and the speed of the bubble is closer to

  u = 1/9 a^2 g/nu 

This holds for bubbles as big as 1 cm across. When they get bigger than that, they're no longer perfect spheres. Instead, they become flattened and lens-shaped, with the upper surface domed and the lower surface ragged, rather like an umbrella. These bubbles behave very differently: their speed is about

  u = 2/3 sqrt(g/R)
where R is the radius of curvature of the spherical top of the bubble and sqrt() denotes the square root. Notice that nu, the viscosity, does not appear in this equation. Notice also that in the "small bubble" equations, bigger bubbles rise faster, but in the "big bubble" equation, bigger bubbles rise slower! Which suggests a fun project: get some swim goggles and a waterproof stopwatch and take a trip to the pool. Dive to the bottom and blow bubbles of different sizes and see how long it takes them to get to the surface. Which size goes fastest? I'm betting it's the 1-2 cm ones, but I'm not sure.

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