Solving the Rising Bubble Problem using Bernoulli's Equation

[LoopQG]

Homework Statement

Consider a spherical bubble of radius R,rising in water. Using Bernoulli's equation show that the rate of rise of the bubble is:

$$U=(2/3) \sqrt(gR)$$

Homework Equations

Bernoulli Equation
Potential Flow

The Attempt at a Solution

I have considered the problem from the bubble's frame so the rate of rise is just the velocity of the uniform flow around the sphere. I know that there is a stagnation point right at the top of the bubble. So Taking Bernoulli's at points on either side of the stagnation point is what I have been doing but I must be setting it up wrong because i don't see where to get that 2/3 from. Any help appreciated.

From Bernoulli's i get:

$$U= \sqrt(2gR(1-cos(\theta)))$$

 

[Tide]

Try integrating over theta. :)

 

[LoopQG]

Do you mean take $$dU/d\theta$$

which equals

$$\sqrt(2gR)(1/2)(1-cos(\theta))^(-1/2) sin(\theta)$$

but then integrating that don't i just get the same thing back?

If not at what point should I consider integrating. I see where you are going with this because the flow velocity is zero at the boundary layers but I just am not sure how to apply it. Thanks for the help!

 

[Tide]

What does theta in your Bernoulli equation stand for?

 

[LoopQG]

The angle from the vertical axis of the sphere, grcos(\theta) is the gravitational force at that point

 

[Tide]

The angle from the vertical axis of the sphere, grcos(\theta) is the gravitational force at that point

So to get the total force you would have to integrate over the entire surface? :)

 

[LoopQG]

Thanks a lot for the help, I ended up assuming a small $$\theta$$ then expnding and I get the right answer.