Bubble rate equation?

Question

What is the equation for the speed of a bubble rising in water?

Answer 1

Interesting question with no simple answer.

There are two forces operating on the bubble:

the force of buoyancy (as the bubble has lower density than the surrounding fluid) pushes the bubble up

the drag force due to the viscosity of the fluid as to keep the bubble from speeding up too much.

At any point in along its path, the net bouyant force on the bubble is Fb = mg - dVg where:

g is the gravitational acceleration (~9.8 m/s^2 at the Earth's surface)
m is the mass of the bubble so mg is its weight
d is the density of the fluid
V is the volume of the bubble so dVg is the weight of fluid it displaces.

Since a larger bubble will have a greater buoyant force than a smaller one, a larger bubble will tend to rise faster.

Note, however, that if the bubble is a simple gas bubble, as it rises, the fluid pressure will decrease, so the volume of the bubble will increase as it rises

http://en.wikipedia.org/wiki/Gas_laws

Thus the buoyant force is a function of the depth at which the bubble is at as well as its original size.

As for the viscous drag force, the classic equation is:

Fd = (1/2)(dCAv^2) where:

d is the density of the fluid
C is the drag coefficient of the shape (about 0.1 for a smooth sphere)
A is the cross-sectional area (i.e. pi x radius^2)
v is the velocity of the bubble

http://en.wikipedia.org/wiki/Drag_equation

Since the radius changes with volume, and hence depth, this force is also a function of depth as well as velocity.

To make matters more interesting, even once you have the net force on the bubble, you have to figure out the effective mass of the bubble for the purposes of F = ma.

In particular, the bubble carries some fluid along with it and that fluid must also be accelerated.

The net result is a rather complicated equation that is still valid only for the simple case of relatively slow velocity. Need I say it is still a matter of study? Here is one nice paper on the subject which has an explicit formula for you (as well as many additional references):

http://www.europhysicsnews.com/full/13/article3/article3.html