(a) A 125 mm diameter vertical cylinder rotates concentrically inside a fixed cylinder of diameter 130 mm. Both cylinders are 325 mm long. Find the dynamic viscosity of the liquid that fills the space between the cylinders, if a torque of 0·92 Nm is

Question

(a) A 125 mm diameter vertical cylinder rotates concentrically inside a fixed cylinder of diameter 130 mm. Both cylinders are 325 mm long. Find the dynamic viscosity of the liquid that fills the space between the cylinders, if a torque of 0·92 Nm is required to maintain a speed of 70 r.p.m. (10 marks)

(b) Calculate the friction drag on a flat plate 15 cm wide and 45 cm long placed longitudinally in a stream of oil of relative density 0·925 and kinematic viscosity 0·9 stoke, flowing with a free stream velocity of 6 m/s. Also find the thickness of the boundary layer and shear stress at the trailing edge. Take density of water 1000 kg/m³. (10 marks)

(c) A large tank as shown in the above figure has a vertical pipe 70 cm long and 2 cm in diameter. The tank contains an oil of density 920 kg/m³ and viscosity 1·5 poise. Find the discharge through the pipe when the height of the oil level of the tank is 0·80 m above the pipe inlet. (10 marks)

(d) A field density test was conducted by core-cutter method and the following data was obtained:
Weight of empty core-cutter = 23 N
Weight of soil and core-cutter = 50 N
Dimensions of the core-cutter dia = 90 mm and height = 180 mm
Weight of wet sample for moisture determination = 55×10⁻² N
Weight of oven dry sample = 52×10⁻² N
Specific gravity of soil grains = 2·70
Determine its dry density, void ratio and degree of saturation. (10 marks)

(e) Two plate load tests were conducted at a site – one with a 300 mm square plate and other with a 600 mm square test plate. For a settlement of 25 mm the loads were found to be 21·6 kN and 64·8 kN respectively in the two tests. Determine the allowable bearing pressure of the sand and the load which a square footing 1·5 m×1·5 m can carry with the settlement not exceeding 25 mm. (10 marks)

UPSC Answer Check · Accepted Answer

Calculate requires systematic numerical problem-solving across all five sub-parts. Allocate approximately 20% time to each sub-part (a-e) as they carry equal marks. Begin each sub-part by stating the governing formula, substitute values with proper unit conversions, show intermediate calculations, and conclude with final answers in correct SI units. For sub-part (c), note that the figure is not provided—state reasonable assumptions about the pipe configuration (likely vertical with tank at top). No separate introduction or conclusion is needed; present each sub-part clearly labelled with complete working.
- Sub-part (a): Apply viscous torque formula T = μ(2πR³ωL)/h for concentric cylinder viscometer; convert rpm to rad/s and solve for dynamic viscosity μ
- Sub-part (b): Determine Reynolds number to confirm laminar flow; apply Blasius solution or appropriate flat plate boundary layer equations for drag, boundary layer thickness δ, and wall shear stress τw
- Sub-part (c): Apply Hagen-Poiseuille equation for laminar pipe flow considering hydrostatic head as driving pressure; verify laminar assumption with Reynolds number
- Sub-part (d): Calculate bulk density from core-cutter data, moisture content from oven-drying, then derive dry density, void ratio using Gs, and degree of saturation
- Sub-part (e): Apply plate load test scaling laws for sandy soils (settlement proportional to plate width); extrapolate from 300mm and 600mm plates to 1.5m footing using Terzaghi's settlement relationship

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies governing principles for all five sub-parts: viscous flow between rotating cylinders (a), boundary layer theory on flat plate (b), laminar pipe flow with hydrostatic head (c), phase relationships in soil mechanics (d), and plate load test scaling for sands (e); no conceptual errors in formula selection	Correct principles for most sub-parts but minor errors in identifying flow regime or using inappropriate formulas for one sub-part (e.g., using turbulent flow equations for clearly laminar conditions)	Major conceptual errors in multiple sub-parts such as using Bernoulli instead of viscous flow equations, confusing kinematic and dynamic viscosity, or applying clay consolidation theory to sand settlement
Numerical accuracy	20%	10	All calculations accurate to appropriate significant figures; correct unit conversions throughout (mm to m, rpm to rad/s, stokes to m²/s, poise to Pa·s); final answers with proper units and realistic magnitudes	Minor arithmetic errors in 1-2 sub-parts or inconsistent significant figures; correct methodology but slips in calculation (e.g., radius vs diameter, missing factor of 2 or π)	Multiple calculation errors, wrong order of magnitude results, missing unit conversions, or answers without units indicating poor numerical control
Diagram quality	15%	7.5	Clear free-body diagrams for (a) showing torque and velocity distribution, boundary layer sketch for (b) with velocity profile, pipe flow schematic for (c) with datum and head annotations, and labeled core-cutter for (d); diagrams aid problem visualization	Basic sketches provided for 2-3 sub-parts but lacking labels or incomplete; diagrams present but not effectively used to explain the physics	No diagrams or completely irrelevant sketches; failure to visualize the physical setup leads to incorrect problem interpretation
Step-by-step derivation	25%	12.5	Each sub-part shows complete derivation from first principles or standard formula with clear statement, substitution of values with units, intermediate steps, and final answer; logical flow with cross-checks (e.g., Re verification before applying laminar formulas)	Derivations present but with gaps or jumps in logic; some sub-parts show only formula and answer without intermediate steps; missing verification checks	Disorganized working with no clear structure; formulas stated without derivation or explanation; impossible to follow the calculation logic or identify errors
Practical interpretation	20%	10	Interprets results physically: comments on viscosity magnitude for typical oils in (a), assesses whether boundary layer is laminar/turbulent in (b), notes discharge practicality in (c), evaluates soil as loose/medium/dense in (d), and discusses safe bearing pressure for foundation design in (e) with reference to IS codes	Brief mention of practical significance for 2-3 sub-parts but superficial; answers are purely numerical without connecting to engineering judgment	No interpretation of results; answers remain abstract numbers without physical meaning or engineering relevance; misses opportunity to validate answers against typical ranges

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