3.6 Exercises [hydraulic circuits top]

1. Consider steady Poiseuille flow through a control volume with a circular cross section with radius R, perimeter P , cross-sectional area A, and differential length dz. Assume a pressure gradient with magnitude  is present. Write the net pressure on the control volume in terms of A, dz, and . Write the wall shear stress in terms of R and . The sum of these forces is zero at equilibrium. Given this, write the relation between R, A, and P .

   For a circle, the relation between R, A, and P  follows directly from geometry, and the analysis above was not necessary. However, for a channel of unknown geometry but known A and P , the above analysis allows us to derive the hydraulic radius r_h.

   Repeat the above analysis for a cross-section of unknown geometry but known A and P . To do this analysis, you will have to assume that the surface stress is uniform, and define it as τ = . What is the value of r_h in terms of A and P ?

   This analysis shows that the key approximation associated with using the hydraulic radius is the assumption that the wall shear stress is uniform along the perimeter of the channel. For what geometries is this assumption good? For what geometries is this assumption bad?

2. Consider water (η = 1×10^-3 Pa s) in an infinitely rigid tube of circular cross-section with radius a = 10 μm and length L = 10 cm. Thecompressibility β = - of water at standard temperature and pressure is a thermodynamic property and is roughly equal to 5×10^-5atm^-1. Calculate the hydraulic resistance and the compliance of the tube.

3. Most exercises are excluded from this web posting. Follow the links to buy the text at Cambridge or Amazon or Powell's or Barnes and Noble.

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