Bubble Size and Fluid Motion
Name: Nathan W.
I noticed while scuba diving that my larger exhaled
air bubbles rose faster than my smaller ones. Obviously the buoyancy
force must be greater on the larger bubbles, but surely they have
greater resistance too. I am wondering what the relationship (if any)
is between the size of a bubble and its rate of ascension.
The buoyant force on a bubble is proportional to its volume, but the
drag is proportional to its surface area. The ratio of volume to
surface area of a bubble is 4/3*pi*r^3 / 4*pi*r^2 = r/3, so a large
bubble will have a greater buoyant-force/drag ratio than a small bubble.
The exact relationship between size and rate of ascent will be pretty
complicated, because drag forces are strongly dependent on speed.
The relationship between bubble size and rate of rising velocity is very
complicated. It depends upon size, water purity, turbulence, the number of
bubbles/volume present, and a host of other things. Roughly speaking the
rising velocity increases with bubble radius -- about 0.03meters/second
(m/s) for a bubble radius of 1x10^-4 (m) increasing to about 0.25m/s for a
bubble radius of 6x10^-4(m), then decreases to about 0.2(m/s) at a radius of
about 3x10^-3(m), then increases again back to about 0.25(m/s) at a bubble
radius of 1x10^-2(m). This refers to the "terminal" velocities. How quickly
a bubble attains its "terminal" velocity is a whole other problem. In some
regimes bubbles that are too large break up, which raises a whole other set
of problems. Bubbles can be spherical or mushroom shaped depending upon the
various parameters. It is fair to say that such a simple problem as the rise
of a bubble in a fluid is not well understood in general. It is under
certain special conditions, but not in general.
See NEWTON BBS archive:
http://www.newton.dep.anl.gov/askasci/phy00/phy00505.htm for some more info
and further reading.
Buoyancy is related to the volume of the object. Resistance (drag) is
related to the frontal area. In the case of a sphere, these are
proportional to R^3 and R^2, respectively. Buoyancy force wins out: bigger
bubbles move faster.
Dr. Ali Khounsary
Argonne National Laboratory
Large bubbles flatten out as they try to rise fast.
I'm not going to try to figure the net effect of that change, just now.
Small-to-medium bubbles stay roughly spherical in the water.
Buoyancy force is proportional to volume, which goes as diameter cubed.
Drag force at a given speed is proportional to frontal area, the area of a
circle, which goes as diameter squared.
So, yes, as the diameter increases, buoyancy increases faster than drag,
and speed will then increase until drag force is again equal to buoyancy
Drag in turbulent, fast, "mass-like", non-viscous liquids goes as the
square of speed.
So spherical bubbles will rise with speed proportional to the square-root
of their diameter.
Extremely tiny bubbles may have a different power-law: speed linearly
proportional to size.
Most fluids tend to act viscous rather than inertial, at sufficiently
Tiny dust falls slowly in the air, and tiny bubbles rise slowly in water.
Someday I must ask whether a 6-foot-tall skyjumper tends to overtake a
4-foot-tall skyjumper, before they open their chutes.
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Update: June 2012