Question 12 of 13: Stable Falling Orientation of a Hemisphere
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
04-BS-7 Mechanics of Fluids — National Examination, 2018-May. Three (3) hours duration, closed book. Section A (Calculative, 9 questions, do 7) and Section B (Graphical & Analytical, 4 questions, do 3); every question is answered below regardless of the exam's "do N of M" instruction, so the set is a complete study resource.
Reference texts: White, F.M., Fluid Mechanics (8th ed.) — fluid statics and manometry (Ch. 2), Bernoulli and the energy equation (Ch. 3), viscous flow in ducts and the Moody chart (Ch. 6), flow past immersed bodies and drag (Ch. 7), potential flow and the Magnus effect (Ch. 8), open-channel flow and the hydraulic jump (Ch. 10), turbomachinery and jet propulsion (Ch. 11).
Check — assumptions used across this paper:
Q1's manometer chain is read off the extraction as a two-stage water–mercury–glycerine–(air)–glycerine–mercury system. The enclosed air pocket between the two glycerine columns is treated as weightless (uniform pressure), so only the one described open end is needed to close the hydrostatic chain back to pipe P; the second "opening" is not load-bearing for this calculation.
Q5's wave/hydraulic-jump analysis takes the depth "in front of the wave" (0.15 m, undisturbed, at rest) as the upstream state and "behind the wave" (0.75 m) as the downstream state, per the question's own prose (the raw figure-label ordering in the extraction is a reconstruction and is not used to override the stated text). The classic hydraulic-jump head-loss formula is applied to the given depths, and the swept flow rate uses the measured wave celerity directly — a standard engineering estimate, not a fully momentum-self-consistent bore solution.
Q6's air properties are taken at 20°C (domestic ambient, ρ=1.19 kg/m³) since no duct-air temperature is stated.
Q7(c)'s "discharged at right angles to the initial direction (20° becomes 0°)" is read as: the reverser's exhaust jet is normally angled 20° forward of the fully-radial (right-angle) direction; part (c) removes that forward lean entirely, leaving a purely radial (90° to the engine axis) discharge with zero axial velocity component.
Q8's terminal velocity and Q9's cable drag coefficient are obtained from the Reynolds-number relations the attached charts themselves plot (Morrison's sphere-drag correlation for Q8; the flat subcritical Cd≈1.2 plateau of the smooth-cylinder curve for Q9, since Re≈3×104 falls solidly within it).
Question 12: Stable Falling Orientation of a Hemisphere (5 marks)
Given. Orientation A: concave (open/cup) side facing downward (convex/rounded face leading into the airflow). Orientation B: concave (open/cup) side facing upward (catching the oncoming flow like a parachute canopy).
Falling direction shown by the red arrow (relative airflow approaches from below in each case). A: convex face leads — streamlines separate early off the rim, wake is asymmetric under any tilt → unstable, tumbles. B: concave face leads (parachute-like) — flow stagnates in the cup, any tilt increases pressure on the low side and restores the orientation → stable.
Find. The preferred (stable) falling orientation, with a streamline-based explanation.
Approach. Compare the two orientations' pressure-drag character and, more importantly, how the centre of pressure moves relative to the centre of gravity when each is perturbed from a purely axial fall.
Orientation B (concave side facing the oncoming flow, i.e. facing downward relative to the ground since the hemisphere falls with its cup catching the air from below) is the stable falling orientation. This is exactly the working principle of a parachute canopy, a badminton shuttlecock, or a falling paper cup: the concave face traps and stagnates the oncoming air, producing a large, nearly uniform high-pressure region on the cup's inner surface and a much higher overall drag coefficient than the convex-leading case. The key to STABILITY, not just high drag, is what happens when the hemisphere tips slightly off-axis: the low side of the cup presents more of its concave face square-on to the flow and experiences a larger local pressure rise than the high side, so the resulting moment always acts to rotate the body back toward the axisymmetric, cup-forward orientation — a genuine restoring (self-righting) moment, with the centre of pressure remaining behind (downstream of, relative to the flow) the centre of gravity at every small tilt angle.
Orientation A (convex/rounded face leading) is unstable. The rounded leading face causes the boundary layer to separate asymmetrically the instant the body tips even slightly, since the separation point on a curved convex surface is highly sensitive to the local angle of attack; the resulting wake and pressure distribution then push the centre of pressure to the same side as the tilt rather than opposing it, producing a destabilizing moment that grows the disturbance rather than damping it — the same reason a coin or a falling leaf tumbles rather than falling face-first in a fixed orientation.
