When an air bubble of radius $r$ rises from the bottom to the surface of a lake,its radius becomes $\frac{5r}{4}$. Taking the atmospheric pressure to be equal to $10 \ m$ height of water column,the depth of the lake would approximately be ....... $m$ (ignore the surface tension and the effect of temperature).

An air bubble of volume $1\,cm^3$ rises from the bottom of a lake $40\,m$ deep to the surface at a temperature of $12^{\circ}C$. The atmospheric pressure is $1 \times 10^5\,Pa$,the density of water is $1000\,kg/m^3$,and $g = 10\,m/s^2$. There is no difference in the temperature of water at the depth of $40\,m$ and on the surface. The volume of the air bubble when it reaches the surface will be $..........\,cm^3$. (in $,cm^3$)

The vertical limbs of a $U$ shaped tube are filled with a liquid of density $\rho$ up to a height $h$ on each side. The horizontal portion of the $U$ tube having length $2h$ contains a liquid of density $2\rho$. The $U$ tube is moved horizontally with an acceleration $g/2$ parallel to the horizontal arm. The difference in heights in liquid levels in the two vertical limbs,at steady state,will be:

$A$ spherical raindrop evaporates at a rate proportional to its surface area. The differential equation involving the rate of change of its radius $r$ with time $t$ is ... (where $k$ is a positive constant)

$A$ liquid is kept in a cylindrical vessel which is rotated along its axis. The liquid rises at the sides. If the radius of the vessel is $0.05\,m$ and the speed of rotation is $2\,rev/s$,the difference in the height of the liquid at the center of the vessel and its sides will be .............. $cm$ $(\pi^2 = 10)$.

Write two parts of fluid mechanics and what studies are made in them?