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Fluid of density rho and viscosity eta flows steadily through a tube of length L and radius R. The flow velocity varies with distance r from the axis as v = v0 (1 - r²/R²). Which of the listed quantities can be correctly determined: (a) the volume flow rate through a cross-section per unit time, (b) the kinetic energy of fluid inside the tube, (c) the friction (viscous) force on the tube from the fluid, (d) the pressure difference across the tube's ends?
- (a) only
- (a) and (b) only
- (a), (b) and (c) only
- (a), (b), (c) and (d)
Correct answer: (a), (b), (c) and (d)
Solution
The parabolic velocity profile is the Poiseuille profile. Integrating v over the cross-section gives the flow rate (a); integrating (1/2) rho v² over the tube volume gives the kinetic energy (b); the wall shear integrated over the inner surface gives the friction force (c); and in steady flow that friction equals the net pressure force, giving the pressure difference (d). All four can be found.
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