Terminal velocity of an air bubble rising up in a water column

Here is one question which has amused me a lot and I'm still unable to figure out the correct way to approach this.

What is the terminal velocity of an air bubble (spherical in shape) which rises up in a water column of density \(\rho_w\) and height \(h\) with an initial radius of \(r\). Consider the coefficient of viscosity as \(\eta\) and acceleration due to gravity as \(g\).

The issue is that we can't just apply Stokes law directly cuz, the radius of the air bubble increases up the water column.

So, if you have got the solution or atleast some idea of how to solve this please comment below.

Thanks. .

Note by Ashish Siva
1 year, 6 months ago

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Are we assuming that the temperature of the air bubble remains constant?

Sumanth R Hegde - 1 year, 5 months ago

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Nope, temperature will obviously rise as the bubble gets bigger. But if you have any relevant approach you can consider it constant.

Ashish Siva - 1 year, 5 months ago

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