# Which one would has a greater magnitude, the viscous drag or the weight of the bubble?

An air bubble in a tank of water is rising with constant velocity. The forces acting on the bubble are X, Y and Z as shown.

What describes the three forces?

A) Z is the viscous drag on the bubble, Y is the weight of the bubble, X is the upthrust on the bubble and X = Y + Z.

B) Z is the viscous drag on the bubble, Y is the weight of the bubble, X is the upthrust on the bubble and X > Y + Z.

C) Z is the weight of the bubble, Y is the viscous drag on the bubble, X is the upthrust on the bubble and X = Y + Z.

D) Z is the weight of the bubble, Y is the viscous drag on the bubble, X is the upthrust on the bubble and X > Y + Z.

The solution is choice A. I know that the answer would be one of A or C because the constant velocity means that the acceleration would be zero and hence X = Y+ Z. However, I don't know which one, Z or Y, would have greater magnitude.

Any help would be much appreciated!

Suppose the volume of the bubble is $V = \tfrac{4}{3}\pi r^3$, and the densities of the air and water are $\rho_a$ and $\rho_w$ respectively. Then:

\begin{align} X &= g\rho_w V \\ Y &= g\rho_a V \end{align}

Now let's write down the expression for $Z-Y$ so we can see if this expression is greater or less than zero. We known that $X = Y + Z$, so:

\begin{align} Z - Y &= X - 2Y \\ &= g\rho_w V - 2g\rho_a V \\ &= gV(\rho_w - 2\rho_a) \end{align}

So the viscous drag is greater than the weight if $\rho_w \gt 2\rho_a$ and it's less than the weight if $\rho_w \lt 2\rho_a$.

And unless we know the densities of the air and the water this is as far as we can get. In practice it should be obvious that the relative density of air is less than $0.5$ so $\rho_w \gt 2\rho_a$ and the drag is greater than the weight. But if you replaced the air by e.g. motor oil with a relative density of about $0.9$ then the weight would be greater than the viscous drag.

The reason why the drag can be greater or less than the weight is because the drag is proportional to the bubble velocity, and the bubble velocity depends on the difference between the density of water and the density of whatever is inside the bubble. If we make the density difference very small then the bubble will move very slowly so the viscous drag will be very low.