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.

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 $\partial p / \partial x$.

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 $y$.

Include gravity $g$ in the list of variables of Example and determine the final expression that results for $h$.

Find an expression for the centrifugal force $F_c$ if it depends on the mass $m$, the angular velocity $\omega$, and the radius $R$ of an impeller.

Select $d, \mu$, and $V$ as the repeating variables in Example and find an expression for $F_D$. Show that this is equivalent to the expression of example.

Select $l, V$, and $\rho$ as the repeating variables in Example and find an expression for $F_D$. Show that this is an equivalent form to that of example.

Include the density $\rho$ and viscosity $\mu$ of the surrounding fluid and repeat Problem. This would account for the resistance (drag) of the fluid.

Assume that the velocity $V$ of fall of an object depends on the height $H$ through which it falls, the gravity $g$, and the mass $m$ of the object. Find an expression for $V$.

If the velocity $V$ in a fluid depends on the surface tension $\sigma$, the density $\rho$, and a diameter $d$, show that this implies that the Weber number $V^2 d \rho / \sigma$ is constant.

If the velocity $V$ in a fluid flow depends on a dimension $l$, the fluid density $\rho$, and the viscosity $\mu$, show that this implies that the Reynolds number $V l \rho / \mu$ is constant.

If the $F-L-T$ system of units were used, what would the dimensions be on each of the following? (a) Mass flux (b) Pressure (c) Density (d) Viscosity (e) Work (f) Power (g) Surface tension

Write Bernoulli's equation in dimensionless form by dividing Eq. by $V_1^2$ and multiplying by $g$. Express the equation in a form similar to that of equation expressed earlier.