Volume of a Partial Sphere

Date: 01/07/98 at 18:50:46
From: Thomas Lee
Subject: Volume of a partial sphere

If I take a spherical container filled with fluid and then introduce a 
bubble of gas into the container (which displaces the fluid so the 
bubble's volume remains unchanged), what would be the surface of the 
sphere which is in contact with the bubble (assuming the bottom 
meniscus of the bubble is flat - i.e. that part which is not in 
contact with the container) given the volume/radius of the sphere and 
the volume of the bubble?

I am asking this because I will be giving a talk on the use of 
intra-ocular gases in eye surgery.

Date: 01/08/98 at 08:35:32
From: Doctor Jerry
Subject: Re: volume of a partial sphere redux

Hi Thomas Lee,

As I understand the problem, you have a sphere of radius a (this is 
the container) into which is put a (small) bubble of gas, say of 
radius b.  In my visualization, this small bubble rises to the top of 
the container and deforms to something fitting against the sphere on 
top and having a flat bottom. I take it that the physics of the 
situation makes this at least approximately true.

You ask "what would be the surface of the sphere which is in contact 
with the bubble." My eye was drawn to the word "surface," hoping that 
you meant surface and not volume.  Of course, either could be 
calculated.

So that I can make my terminology clear, imagine a circle of radius a 
(a cross-section of the container). Draw a horizontal line ABOVE the 
center of this circle. Suppose the horizontal line is h units from the 
top. I stress above since formulas may change if the horizontal line 
is below the center. I assumed above since the bubble is probably 
relatively small. Let me know if this isn't the case.

The chunk of the container lying above the plane corresponding to the 
line is called a spherical segment. Its volume is V=(pi*h^2/3)(3a-h) 
and its (top) surface area is S = 2*pi*a*h.

You said that the bubble doesn't change volume as it deforms. Okay, so 
whatever its shape, its volume is (4/3)*pi*b^3. So,

   (4/3)*pi*b^3 = (pi*h^2/3)(3a-h), 

where a and b are known. We can solve for h (it's just a quadratic) 
and, then, knowing h we can calculate S = 2*pi*a*h.

I'd be grateful if you would send a follow-up message.  
For one thing, I'd like to see if you agree with my calculations and 
my assumption about the size of the bubble. I'd also appreciate more 
detail about the circumstances of this calculation. I'm asking since 
I'm co-authoring a calculus book and would like to use this example as 
a project in the book. So, if I can attach some genuine application 
words and background, all the better. 

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/