(a) The resistance force F of a ship is a function of its length L, velocity V, acceleration due to gravity g and fluid properties like density ρ and viscosity μ. Write this relationship in a dimensionless form. (15 marks)

(b) The stream function fo

Question

(a) The resistance force F of a ship is a function of its length L, velocity V, acceleration due to gravity g and fluid properties like density ρ and viscosity μ. Write this relationship in a dimensionless form. (15 marks)

(b) The stream function for a two-dimensional flow is given by ψ = 2xy. Calculate the velocity and velocity potential at point P(2, 3). (15 marks)

(c) A group of nine friction piles is driven through 5 m of clay with unconfined compressive strength of 60 kN/m² followed by 10 m of clay with unconfined compressive strength of 100 kN/m². The piles are in 3 rows and will be 1·00 m centres in a row and the rows will be 750 mm on centres. Each pile has a diameter of 300 mm. If a factor of safety of 2·5 is required, determine the maximum load that can be carried by the group. Take Nc = 9 and unit weight of clay as 16·4 kN/m³. (20 marks)

UPSC Answer Check · Accepted Answer

Solve all three sub-parts systematically, allocating approximately 35% time to part (a) dimensional analysis, 30% to part (b) potential flow calculations, and 35% to part (c) pile group capacity. Begin each part with stating the governing equations, show complete derivations with proper units, and conclude with boxed final answers. For part (c), explicitly state whether block failure or individual pile failure governs.
- Part (a): Apply Buckingham π-theorem correctly with 6 variables and 3 fundamental dimensions to obtain two independent dimensionless groups (Froude number and Reynolds number based forms)
- Part (b): Calculate velocity components u = ∂ψ/∂y = 2x and v = -∂ψ/∂x = -2y, then find velocity magnitude and direction at P(2,3); verify irrotationality and obtain φ = x² - y² + C
- Part (c): Calculate individual pile capacity using Qs = α·c·As with appropriate adhesion factor; determine group efficiency using Converse-Labarre formula or block failure perimeter; compare block failure vs. individual failure modes
- Part (c): Correctly apply spacing parameters (s = 1.0 m, s' = 0.75 m) and group dimensions (2.5 m × 2.0 m block) for efficiency and block failure calculations
- Part (c): Apply factor of safety of 2.5 to the lesser of group capacity or sum of individual pile capacities divided by efficiency

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	25%	12.5	Correctly identifies repeating variables for Buckingham π-theorem in (a); recognizes flow is irrotational with valid potential function in (b); distinguishes between individual pile and block failure mechanisms in (c) with proper Nc application	Minor errors in selecting repeating variables or misses irrotationality check in (b); applies pile formulas correctly but confuses group efficiency with block failure perimeter	Fundamental misunderstanding of dimensional analysis procedure; treats flow as rotational or omits velocity potential; applies end-bearing formula to friction piles or ignores group effects
Numerical accuracy	25%	12.5	Precise calculations: π groups correctly formed as F/(ρV²L²) = f(V/√(gL), ρVL/μ); velocity at P(2,3) as 7.21 m/s at -56.3°; pile group capacity with correct α values (0.6 and 0.5) yielding ~850-900 kN safe load	Correct methodology with arithmetic slips in final values; velocity magnitude correct but direction angle wrong quadrant; pile capacity within 10% of correct value	Order-of-magnitude errors in dimensionless groups; velocity components swapped or sign errors; pile capacity errors >25% from incorrect adhesion factors or perimeter calculations
Diagram quality	10%	5	Clear 3×3 pile group layout showing 1.0 m × 0.75 m spacing with dimensions labeled; velocity vector diagram at point P showing u, v components and resultant; block failure perimeter clearly indicated	Basic pile group sketch without dimensions or unclear spacing notation; omits velocity vector diagram	No diagrams despite visual nature of problems; incorrect pile arrangement (e.g., 2×2 instead of 3×3)
Step-by-step derivation	25%	12.5	Explicit m=3, n=6 → 3 π groups in (a) with full matrix method; velocity potential derivation via integration showing φ = x² - y²; pile capacity with clear As calculation (π×0.3×5 + π×0.3×10) and efficiency formula with θ = tan⁻¹(0.75/1.0)	Skips intermediate steps in π-theorem; jumps to velocity potential without integration constant discussion; pile calculations correct but efficiency formula stated without derivation	Missing derivation steps with only final answers; no dimensional analysis procedure shown; pile capacity stated without showing perimeter or area calculations
Practical interpretation	15%	7.5	Interprets Froude-Reynolds similarity for ship model testing in Indian naval contexts; explains stream function represents flow past corner/stagnation point; discusses when block failure governs (cohesive soils, close spacing) vs. individual failure for pile groups in Ganga basin alluvium	Brief mention of model testing relevance; generic statement about pile groups; no connection to Indian geotechnical practice	No physical interpretation of dimensionless groups; fails to identify flow pattern; ignores practical significance of group vs. individual failure

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