# How do we calculate the speed of an air bubble rising in water?

We all know that an gas bubble expands as it rises through a liquid due to decreasing pressure.But at what speed does it rise? if we make a bubble of unit volume filled with a gas of given density at the bottom of one meter deep liquid container (liquid's density is also known), then how much time will the bubble take to rise? We also have to incorporate the continuously increasing volume of the bubble.

## 2 Answers

As this is marked homework, I'll only give some rough ideas.

You need to find the Volume depending on the height, that's straightforward as you can calculate the Volume from the ideal gas law and the pressure that in turn depends on depth. I would assume equal temeprature between gas and liquid - though that does not always hold. From the Volume the buyoncy is straight forward - so you arrive at a formula that gives you buyoncy in depency of height (or rather depth).

On the other side you need the viscous forces - those are dependent on speed and size of the bubble, so again dependent on depth.

So, with x beeing the height (positive upwards), B buyoncy and D drag forces and m mass of the bubble, you get something like: $$m \ddot x = B(x) - D(\dot x, x)$$

This does not look easy. Also note: I assume that the time the bubble takes to expand is negligable

• You also have to keep in mind what shape the air bubble has. This will also be affected by the surface tension. I think that this will be very hard to calculate for bigger bubbles (for very small bubbles the drag forces will be very small compared to the surface tension and the shape will approximate a sphere). – fibonatic Jun 24 '13 at 12:12
• I agree, @fibonatic. Small bubbles are spherical, so a drag coefficient is known. As the small bubbles rise, they continuously expand, and their shape changes into something that looks more like a pancake. It is also practically certain that as the bubbles grow past a certain size, they break into 2 or 3 smaller bubbles. Unfortunately, I don't have any "feel" for what parameters (surface tension, liquid/vapor densities, etc.) lead to bubble breakup. – David White Aug 10 '16 at 19:00
1. Determine the bubbles buoyant force in the Y direction. This can best be analyzed by looking at the relationship between volume and density of the bubble in relation to the surrounding viscus fluid.

2. Determine the general shape of the bubble, oblong, elliptical shape, and its coefficient of friction with respect to the fluid it will be flowing through. You may need to determine if the bubble will be laminar or turbulent to further investigate the friction factor.

3. Multiply the force created by the buoyant force, by the frictional losses to get your applied force.

4. Once you have applied force, given (Displacement x) (initial condition $V_0$) (Mass) you should be able to figure out the speed, given one of the velocity equations.