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 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 $T=f(\rho, D, \mu, V)$.

Line sources of equal strength m = Ua, where U is a reference velocity, are placed at (x, y) = (0, a) and (0, -a). Sketch the stream and potential lines in the upper half plane. Is y = 0 a “wall”? If so, sketch the pressure coefficient

$$C_p = p - p_0/½ρU²$$

along the wall, where

$$p_0$$

is the pressure at (0, 0). Find the minimum pressure point and indicate where flow separation might occur in the boundary layer.

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 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.

The drag characteristics for a newly designed automobile having a maximum characteristic length of 20 ft are to be determined through a model study. The characteristics at both low speed (approximately 20 mph) and high speed (90 mph) are of interest. For a series of projected model tests, an unpressurized wind tunnel that will accommodate a model with a maximum characteristics length of 4 ft is to be used. Determine the range of air velocities that would be required for the wind tunnel if Reynolds number similarity is desired. Are the velocities suitable? Explain.

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.

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.

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?

The normal stress $\sigma$ in a beam depends on the bending moment $M$, the distance $y$ from the neutral axis, and the moment of inertia $I$. Relate $\sigma$ to the other variables if we know that $\sigma$ varies linearly with $y$.

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