Exercises

1. Liquid is led steadily through a pipeline that passes over a hill of height $h$ into the valley below, the speed at the crest being $v$ . Show that, by properly adjusting the ratio of the cross-sectional areas of the pipe at the crest and in the valley, the pressure may be equalized at these two places. (Milne-Thomson 1958.)

2. Water of mass density $\rho$ and pressure $p$ flows through a curved pipe of uniform cross-sectional area $S$ , whose radius of curvature is $R$ , at the uniform speed $v$ . Demonstrate that there is a net force per unit length $S\,(p - p_0 + \rho\,v^{\,2})/R$ acting on the pipe, and that this force is everywhere directed away from the pipe's local center of curvature. Here, $p_0$ is atmospheric pressure.

3. Water is held in a right circular conical tank whose apex lies vertically below the center of its base. The water initially fills the tank to a height $h$ above the vertex. Let $a$ be the initial radius of the surface of the water inside the tank. A small hole of area $S$ (that is much less than $\pi\,a^{\,2}$ ) is made at the bottom of the tank. Demonstrate that the time required to empty the tank is at least
   $$\frac{5}{2}\,\frac{\pi\,a^{\,2}}{S}\left(\frac{h}{2\,g}\right)^{1/2}.$$

4. Water is held in a spherical tank of radius $a$ , and initially fills the tank to a height $h < 2\,a$ above its lowest point. A small hole of area $S$ (that is much less than $\pi\,a^{\,2}$ ) is made at the bottom of the tank. Demonstrate that the time required to empty the tank is at least
   $$\frac{2\pi}{S\,(2\,g)^{1/2}}\left(\frac{2}{3}\,a\,h^{\,3/2} - \frac{1}{5}\,h^{\,5/2}\right).$$

5. Water is held in two contiguous tanks whose cross-sectional areas, $A_1$ and $A_2$ , are independent of height. A small hole of area $S$ (where $S\ll A_1$ , $A_2$ ) is made in the wall connecting the tanks. Assuming that the initial difference in water level between the two tanks is $h$ , show that the time required for the water levels to equilibrate is at least
   $$\frac{2}{S}\left(\frac{A_1\,A_2}{A_1+A_2}\right)\left(\frac{h}{2\,g}\right)^{1/2}.$$

6. For a channel of width $W$ , having a discharge rate $Q$ , show that there is a critical depth $h_c$ , where
   $$h_c=\left(\frac{Q^{\,2}}{g\,W^{\,2}}\right)^{1/3},$$
   which must be exceeded before a hydraulic jump is possible.

7. Show that for a stationary hydraulic jump in a rectangular channel, the upstream Froude number ${\rm Fr}_1$ , and the downstream Froude number ${\rm Fr}_2$ , are related by
   $${\rm Fr}_2^{\,2} =\frac{8\,{\rm Fr}_1^{\,2}}{[(1+8\,{\rm Fr}_1^{\,2})^{1/2}-1]^{\,3}}.$$

8. Consider a simply-connected volume $V$ whose boundary is the surface $S$ . Suppose that $V$ contains an incompressible fluid whose motion is irrotational. Let the velocity potential $\phi$ be constant over $S$ . Prove that $\phi$ has the same constant value throughout $V$ . [Hint: Consider the identity $\nabla\cdot(A\,\nabla A)\equiv \nabla A\cdot \nabla A +A\,\nabla^{\,2} A$ .]

9. In Exercise 8, suppose that, instead of $\phi$ taking a constant value on the boundary, the normal velocity is everywhere zero on the boundary. Show that $\phi$ is constant throughout $V$ .

10. An incompressible fluid flows in a simply-connected volume $V$ bounded by a surface $S$ . The normal flow at the boundary is prescribed. Show that the flow pattern with the lowest kinetic energy is irrotational. This result is known as the Kelvin minimum energy theorem. [Hint: Try writing ${\bf v}=-\nabla\phi + \delta{\bf v}$ , where $\phi$ is the velocity potential of the irrotational flow pattern. Let $\nabla\cdot\delta{\bf v}=0$ throughout $V$ , and $\delta{\bf v}\cdot d {\bf S} = 0$ on $S$ . Show that the kinetic energy is lowest when $\delta {\bf v}={\bf0}$ throughout $V$ .]