Project ForgeFluid Mechanics
Bernoulli's Equation & Flow
GATE 2020
Expert
Consider steady, viscous, fully developed flow of a fluid through a circular pipe of internal diameter D. The velocity profile forms a paraboloid about the pipe centre line, given by V=−C(r²−D²/4) m/s, where C is a constant. The rate of kinetic energy (in J/s) at the control surface, as shown in the figure, is proportional to Dⁿ. The value of n is __________.
8(± 0.1 accepted)
Explanation
KE rate = (ρ/2)∫V³dA = (ρ/2)∫₀^R [C(R²−r²)]³·2πr dr, with R=D/2. Substituting u=r²: this integral evaluates to (πρC³/2)·R⁸/4, i.e. proportional to R⁸ and hence to D⁸ (C is held fixed, independent of D). So n=8.
Source: GATE ME 2020 Session 1 (ME-1) (2020)
This question is part of a 15-day revision cycle — sign in and Project Forge will bring it back the day before you'd forget it.