Project ForgeFluid Mechanics
Bernoulli's Equation & Flow
GATE 2016
Hard
A steady laminar boundary layer forms over a flat plate with free-stream velocity Uo. The velocity profile is uniform (Uo) at the inlet a-b, and at a downstream location c-d is u = Uo[2(y/δ) - (y/δ)²]. The ratio of the mass flow rate leaving through the horizontal section b-d to that entering through the vertical section a-b is __________.
0.33(± 0.01 accepted)
Explanation
mdot_ab = ρUoδ (uniform inlet). mdot_cd = ρ∫₀^δ Uo[2(y/δ)-(y/δ)²]dy = ρUoδ(1-1/3) = (2/3)ρUoδ. By mass conservation, mdot_bd = mdot_ab - mdot_cd = (1/3)ρUoδ, so the ratio mdot_bd/mdot_ab = 1/3 ≈ 0.33.
Source: GATE ME 2016 Set-1 (2016)
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