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Old Mar 6th 2017, 01:29 AM   #1
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Question pressure of bubbles underwater

I've duelling with this problem for awhile now: what is the pressure inside a cavity with water? What's the difference if I add an air bubble? What if I add an oil bubble instead? Note: surface tension is neglected.
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Old Mar 6th 2017, 11:36 AM   #2
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What is a "cavity with water? " How do you "add an air bubble."
Unclear what you intend.
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Old Mar 6th 2017, 12:22 PM   #3
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By cavity I mean something like a tall vessel (10R X 6R X 2R with R being the radius of the bubble). The problem is, of course, purely imaginary. I consider 3 different scenarios:
a) the vessel is fully filled with water only
b) the vessel has a air bubble located near the bottom of the vessel which is surrunded by water
c) same as b) but I consider oil instead of air
Hope it's more clear now.
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Old Mar 8th 2017, 08:22 AM   #4
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The pressure at each point in the vessel varies with the depth. To find the pressure at each point, calculate the weight, W, of liquid above a rectangle of area A. The pressure at that point is W/A then take the limit as A goes to 0. Similarly for the case of an air bubble or oil bubble except that, inside the bubble, you use the density of air or oil to calculate the weight above that point.
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Old Mar 8th 2017, 11:08 AM   #5
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My interpretation of my problem is the following:

Firstly I will consider pressure as function of depth along centre of cavity (passing through the centre of the bubble for case b) and c).

I agree that pressure increases with depth and should be calculated by P=rho X g X h. The same equation can be applied to the oil bubble, however shouldn't we consider a different equation of state for the air bubble? Shouldn't the pressure inside the air bubble be constante and follow the ideal gas by P = rho x R+ X T (derived from the general equation PV=nRT)?

I'm also assuming that the pressure inside the the bubble should be higher than the water to avoid colapse of the bubble.

This means that we will have 2 points of discontinuity for cases b) and c) at the transition between the water and the bubble.

CASE a) WATER
Top of the cavity: P0 = 0
Bottom of the cavity: P1 = rho_water X g X h

CASE b) WATER + AIR
Top of the cavity: P0 = 0
Just above bubble: P1 = rho_water X g X h1
Top of the bubble: P2 = rho_air x R+ X T
Middle of the bubble: P3 = P2
Middle of the bubble: P4 = P2
Bottom of the cavity: P5 = rho_water X g X h5

CASE c) WATER + OIL
Top of the cavity: P0 = 0
Just above bubble: P1 = rho_water X g X h1
Top of the bubble: P2 = rho_oil X g X h2
Middle of the bubble: P3 = rho_oil X g X h3
Middle of the bubble: P4 = rho_oil X g X h4
Bottom of the cavity: P5 = rho_water X g X h5
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Old Mar 9th 2017, 01:20 AM   #6
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The pressure inside the bubble is not the same as the pressure outside,
this is why bubbles move.

At the middle of the bubble, the pressures on either side of the bubble surface match.
At the top of the bubble, the pressure inside the bubble is higher than the pressure outside.
At the bottom of the bubble, the pressure inside the bubble is lower than the pressure outside.
Note that it is these differences in pressure that push the bubble upwards.
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