Q8
(a) Reduce the partial differential equation ∂²z/∂y² - ∂²z/∂x∂y + ∂z/∂x - ∂z/∂y(1+1/x) + z/x = 0 to canonical form. (15 marks) (b) Compute a root of the equation log₁₀(2x+1) - x² + 3 = 0, in the interval [0, 3], by Regula-Falsi method, correct to 6 decimal places. (15 marks) (c) Determine under what conditions the velocity field u = c(x² - y²), v = -2cxy, w = 0 is a solution to the Navier-Stokes momentum equations. Assuming that the conditions are met, determine the resulting pressure distribution, when z is up and the external body forces are Bₓ = 0 = Bᵧ, Bᵤ = -g. (20 marks)
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(क) आंशिक अवकल समीकरण ∂²z/∂y² - ∂²z/∂x∂y + ∂z/∂x - ∂z/∂y(1+1/x) + z/x = 0 को विहित रूप में समानीत कीजिए। (15 अंक) (ख) मिथ्या-स्थिति (रेगुला-फाल्सि) विधि से अंतराल [0, 3] में, समीकरण log₁₀(2x+1) - x² + 3 = 0 के एक मूल का, दशमलव के 6 स्थानों तक सही, अभिकलन कीजिए। (15 अंक) (ग) ज्ञात कीजिए कि किन शर्तों के अंतर्गत वेग क्षेत्र (velocity field) u = c(x² - y²), v = -2cxy, w = 0 नेवियर-स्टोक्स संवेग समीकरणों का एक हल है। यह मानते हुए कि शर्तें मान्य हैं, परिणामी दाब बंटन ज्ञात कीजिए, जब z ऊपर है तथा बाह्य पिंड बल Bₓ = 0 = Bᵧ, Bᵤ = -g हैं। (20 अंक)
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How this answer will be evaluated
Approach
Solve this three-part numerical problem by allocating approximately 30% time to part (a) on PDE canonical reduction, 30% to part (b) on Regula-Falsi root-finding, and 40% to part (c) on Navier-Stokes verification and pressure determination. Begin with clear identification of equation types, proceed through systematic derivations and iterative calculations, and conclude with boxed final answers for each sub-part.
Key points expected
- Part (a): Correct classification of the second-order PDE and identification of characteristic curves to transform to canonical form
- Part (a): Proper substitution of new variables and reduction to standard canonical form (parabolic/hyperbolic/elliptic)
- Part (b): Verification that f(0)·f(3) < 0 for root existence and correct Regula-Falsi iteration formula setup
- Part (b): Iterative computation showing convergence to 6 decimal places with clear tabulation of iterations
- Part (c): Verification of continuity equation (∇·u = 0) as necessary condition for Navier-Stokes solution
- Part (c): Substitution into momentum equations to determine constraints on c and fluid properties
- Part (c): Integration of pressure gradients to obtain p(x,y,z) with proper incorporation of body force B_z = -g
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | For (a): correctly identifies A=0, B=-1/2, C=1 and computes discriminant B²-AC=1>0 (hyperbolic); for (b): verifies f(0)=3>0, f(3)≈-5.2<0 confirming root in [0,3]; for (c): confirms ∇·u = 0 showing incompressible flow | Correct basic setup for two parts but minor errors in PDE classification or missing continuity check in (c) | Wrong PDE classification, incorrect interval check in (b), or fails to verify incompressibility condition |
| Method choice | 20% | 10 | For (a): uses characteristic equation dy/dx = [B±√(B²-AC)]/A to find ξ,η; for (b): applies correct Regula-Falsi formula xₙ₊₁ = [aₙf(bₙ)-bₙf(aₙ)]/[f(bₙ)-f(aₙ)]; for (c): employs cylindrical/stream function approach or direct substitution into N-S | Correct method for two parts with minor formula errors or suboptimal approach in one part | Wrong method for canonical reduction, uses bisection instead of Regula-Falsi, or incorrect N-S formulation |
| Computation accuracy | 20% | 10 | For (a): exact transformation to canonical form ∂²z/∂η² + ... = 0; for (b): converges to x≈2.801373 with all 6 decimal places correct; for (c): correct derivatives uₓ=2cx, uᵧᵧ=-2cx, vₓ=-2cy, vᵧ=-2cx and consistent pressure integration | Correct final answers with minor arithmetic errors in intermediate steps or 4-5 correct decimal places in (b) | Major computational errors, wrong canonical form, divergence in iteration, or incorrect pressure gradient integration |
| Step justification | 20% | 10 | Clear justification for each transformation step in (a), explicit iteration table with error bounds in (b), and physical reasoning for pressure boundary conditions in (c) including reference to hydrostatic pressure -ρgz | Adequate working shown but missing some intermediate justifications or unclear error analysis in (b) | Minimal working, jumps between steps without explanation, or missing physical interpretation of results |
| Final answer & units | 20% | 10 | Boxed canonical form for (a); boxed root x=2.801373 (or similar 6dp) for (b); explicit pressure distribution p = -½ρc²(x²+y²)² - ρgz + C with proper units (Pa or N/m²) and physical interpretation for (c) | Correct answers present but poorly formatted, missing units in (c), or incomplete pressure expression | Missing final answers, wrong boxed values, or completely unphysical pressure distribution without integration constant |
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