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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

    $\displaystyle \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

    $\displaystyle \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

    $\displaystyle \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

    $\displaystyle 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

    $\displaystyle {\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$ .]


next up previous
Next: Two-Dimensional Incompressible Inviscid Flow Up: Incompressible Inviscid Flow Previous: Irrotational Flow
Richard Fitzpatrick 2016-03-31