Derive an expression for the torque $T$ needed to rotate a disk of diameter $d$ at the angular velocity $\omega$ in a fluid with density $\rho$ and viscosity $\mu$ if the disk is a distance $t$ from a wall. Also, find an expression for the power requirement.

The drag force $F_D$ on a golf ball depends on the velocity, the viscosity, the density, the diameter, the dimple depth, the dimple radius, and the concentration $C$ of dimples measured by the number of dimples per unit area. What expression relates $F_D$ to the other variables?

The drag force $F_D$ on the smooth sphere of fig. falling in a liquid depends on the constant sphere speed $V$, the solid density $\rho_s$, the liquid density $\rho$ and viscosity $\mu$, the sphere diameter $D$, and gravity $g$. Find an expression for $F_D$ using (a) the $M-L-T$ system, and (b) the $F-L-T$ system.

The thrust force $T$, in newtons, from an airplane propeller is a function of propeller diameter $D$, air density $\rho$, air viscosity $\mu$, and airplane speed $V$. Determine a suitable set of dimensionless numbers to describe the relationship $T=f(\rho, D, \mu, V)$.

The drag force acting on a ship is considered to be a function of the fluid density $\rho$, viscosity $\mu$, gravity $g$, ship velocity $V$, and a characteristic length $l$. Determine a set of suitable dimensionless numbers to describe the relationship

$$F_D=f(\rho, \mu, g, V, l)$$

The drag force $F_D$ on a sphere depends on the velocity $V$, the viscosity $\mu$, the density $\rho$, the surface roughness height $e$, the freestream fluctuation intensity $I$ (a dimensionless quantity), and the diameter $D$. Find an expression for $F_D$.

The flow rate $Q$ in an open channel depends on the hydraulic radius $R$, the cross-sectional area $A$, the wall roughness height $e$, gravity $g$, and the slope $S$. Relate $Q$ to the other variables using (a) the $M-L-T$ system, and (b) the $F-L-T$ system.

The velocity of propagation of ripples in a shallow liquid depends on the liquid depth $h$, gravity $g$, surface tension $\sigma$, and the liquid density $\rho$. Find an expression for the propagation velocity $V$. See Fig.

Select an appropriate set of variables that influence the drag force $F_D$ on an airfoil and write the final form in terms of dimensionless parameters.

The pressure drop $\Delta p$ in the pipe of Fig. depends on the average velocity, the pipe diameter, the kinematic viscosity, the length $L$ of pipe, the wall roughness height $e$, and the fluid density. Find an expression for $\Delta p$.

Derive an expression for the velocity of liquid issuing from a hole in the side of an open tank if the velocity depends on the height $H$ of the hole from the free surface, the fluid viscosity and density, gravity, and the hole diameter.

The read-write head for a hand-held music player has a surface area of $0.04 \mathrm{~mm}^2$. The head is held $0.04\ \mu \mathrm{m}$ above the disk, which is rotating at a constant rate of $1800\ \mathrm{rpm}$. Find the torque $\mathbf{T}$ that must be applied to the disk to overcome the frictional shear resistance of the air between the head and the disk. The surrounding air is at standard atmospheric pressure and a temperature of $20^{\circ} \mathrm{C}$. Assume the velocity profile is linear.

It is suggested that the velocity of the water flowing from a hole in the side of an open tank depends on the height $H$ of the hole from the surface, the gravity, and the density of the water. What expression relates the variables?

Find an expression for the average velocity in a smooth pipe if it depends on the viscosity, the diameter, and the pressure gradient $\partial p / \partial x$.