Fill in the blanks:
$(i)$ The lines of flow and streamlines coincide with each other in ...... flow.
$(ii)$ The formula for the horizontal velocity of water coming from a hole at the bottom at a height $h$ from the surface of the water is ......
$(iii)$ $1 \ Pa = ...... \ dyne/cm^{2}$
$(iv)$ The relative velocity of two parallel layers of water is $6 \ cm/s$. If the perpendicular distance between the two layers is $0.1 \ mm$,then the velocity gradient will be ......

Drops of liquid of density $d$ are floating half immersed in a liquid of density $\rho$. If the surface tension of the liquid is $T$, then the radius of the drop is

$A$ light semi-cylindrical gate of radius $R$ and length $l$ is pivoted at its midpoint $O$ of the diameter as shown in the figure,holding liquid of density $\rho$. The force $F$ required to prevent the rotation of the gate is equal to:

Two vessels of equal cross-sectional area $A$ are filled with a liquid of density $d$. The height of the liquid in one vessel is $h_1$ and in the other vessel is $h_2$. When both vessels are connected,find the work done by gravity.

$A$ cylindrical container is filled with liquid. When the container is rotated about its axis,the liquid rises along its sides. If the radius of the container is $r$ and the angular velocity of the container is $\omega$ rad/s,what will be the difference in the height of the liquid at the center and at the sides?

$A$ cylindrical furnace has height $(H)$ and diameter $(D)$ both $1 \ m$. It is maintained at temperature $T=360 \ K$. The air gets heated inside the furnace at constant pressure $P_a$ and its temperature becomes $T=360 \ K$. The hot air with density $\rho$ rises up a vertical chimney of diameter $d=0.1 \ m$ and height $h=9 \ m$ above the furnace and exits the chimney. As a result,atmospheric air of density $\rho_a=1.2 \ kg \ m^{-3}$,pressure $P_a$ and temperature $T_a=300 \ K$ enters the furnace. Assume air as an ideal gas,neglect the variations in $\rho$ and $T$ inside the chimney and the furnace. Also ignore the viscous effects. [Given: The acceleration due to gravity $g=10 \ ms^{-2}$ and $\pi=3.14$]
$(1)$ Considering the air flow to be streamline,the steady mass flow rate of air exiting the chimney is . . . . .$g \ s^{-1}$.
$(2)$ When the chimney is closed using a cap at the top,a pressure difference $\Delta P$ develops between the top and the bottom surfaces of the cap. If the changes in the temperature and density of the hot air,due to the stoppage of air flow,are negligible then the value of $\Delta P$ is . . . . .$N \ m^{-2}$.