Q7
(a) Find the complete integral of z(p²-q²) = x-y; p≡∂z/∂x, q≡∂z/∂y. (15 marks) (b) Find the unique polynomial of degree 2 or less which fits the following data: x : 0 1 3 f(x) : 1 3 55 Also obtain the bound on the truncation error. (15 marks) (c) Show that for an incompressible steady flow with constant viscosity, the velocity components u(y) = (U/h)y - (hy/2μ)(dp/dx)(1-y/h) v = 0 = w, with p = p(x), satisfy the equation of motion in the absence of body force. Given that U, h and dp/dx are constants. (20 marks)
हिंदी में प्रश्न पढ़ें
(a) z(p²-q²) = x-y; p≡∂z/∂x, q≡∂z/∂y का पूर्ण समाकल प्राप्त कीजिए। (15 अंक) (b) घात 2 या 2 से कम का वह अद्वितीय बहुपद, जो आँकड़ों x : 0 1 3 f(x) : 1 3 55 पर ठीक बैठता है, प्राप्त कीजिए। क्षण त्रुटि पर परिबंध भी प्राप्त कीजिए। (15 अंक) (c) दर्शाइए कि अचर विस्कांशता के एक असंपीड्य अपरिवर्ती प्रवाह के लिए वेग घटक u(y) = (U/h)y - (hy/2μ)(dp/dx)(1-y/h) v = 0 = w, p = p(x) के साथ, पिण्ड बल की अनुपस्थिति में गति के समीकरण को संतुष्ट करते हैं। यह दिया गया है कि U, h और dp/dx अचर हैं। (20 अंक)
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How this answer will be evaluated
Approach
Solve all three sub-parts systematically, allocating approximately 30% time to part (a) on PDE complete integral using Charpit's method, 30% to part (b) on Lagrange interpolation with error bound derivation, and 40% to part (c) on Navier-Stokes verification. Begin with clear identification of the method for each part, show complete derivations with all intermediate steps, and conclude with boxed final answers for each sub-part.
Key points expected
- Part (a): Correct application of Charpit's auxiliary equations to find complete integral of z(p²-q²) = x-y, including proper parameterization and final form with arbitrary constants
- Part (b): Construction of Lagrange interpolation polynomial of degree ≤2 using given data points (0,1), (1,3), (3,55), with explicit polynomial expression
- Part (b): Derivation of truncation error bound using the formula |f(x)-P₂(x)| ≤ M₃|x(x-1)(x-3)|/6 where M₃ bounds |f'''(ξ)|
- Part (c): Verification that u(y) satisfies continuity equation (∂u/∂x + ∂v/∂y = 0) for incompressible flow with v=w=0
- Part (c): Substitution of velocity profile into x-momentum Navier-Stokes equation, showing balance between pressure gradient and viscous terms with dp/dx constant
- Part (c): Verification that y-momentum and z-momentum equations are satisfied with v=w=0 and p=p(x) only
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies Charpit's method for (a), Lagrange interpolation for (b), and Navier-Stokes equations for (c); writes proper auxiliary equations, difference table setup, and momentum equations with correct assumptions | Identifies methods correctly but makes minor errors in initial setup such as wrong Charpit equations, incorrect basis for interpolation, or missing incompressibility condition | Wrong method selection (e.g., using Lagrange multipliers for PDE), confuses interpolation types, or applies Euler instead of Navier-Stokes equations |
| Method choice | 20% | 10 | Executes Charpit's method with proper dx/P = dy/Q = dz/(pP+qQ) = -dp/(X+pZ) = -dq/(Y+qZ) for (a); uses efficient Lagrange formula for (b); selects appropriate simplified Navier-Stokes form for (c) | Correct method but inefficient execution such as solving (a) by trial, using divided differences instead of Lagrange for (b), or not exploiting parallel flow simplification for (c) | Attempts separation of variables for (a), solves normal equations for polynomial fit in (b), or uses full 3D Navier-Stokes without simplification for (c) |
| Computation accuracy | 20% | 10 | Flawless algebra: correct complete integral z = a(x+y) + (1/a)(x-y) + b or equivalent for (a); exact polynomial 8x² - 6x + 1 for (b); precise derivative calculations showing μ(d²u/dy²) = dp/dx for (c) | Minor computational slips such as sign errors in Charpit solution, arithmetic error in polynomial coefficients, or algebraic mistake in viscous term evaluation | Major errors: wrong integration in Charpit, incorrect polynomial degree or coefficients, or fundamental error in derivative calculation for momentum balance |
| Step justification | 20% | 10 | Explicitly states Charpit's auxiliary equations and their derivation; shows Lagrange basis polynomial construction; justifies why convective terms vanish and explains Couette-Poiseuille flow physics in (c) | Shows key steps but omits justification for parameter choices, skips derivation of error bound formula, or asserts momentum balance without showing term-by-term evaluation | Jumps to answers without derivation, missing crucial steps like verification of arbitrary constants in complete integral, or no explanation of truncation error derivation |
| Final answer & units | 20% | 10 | Clear boxed final answers: complete integral with two arbitrary constants for (a); P₂(x) = 8x² - 6x + 1 with error bound expression for (b); explicit verification statement with physical interpretation of Couette and Poiseuille contributions for (c) | Correct answers but poorly formatted, missing arbitrary constants in (a), or incomplete error bound expression, or missing physical interpretation in (c) | Missing final answers, wrong form (particular instead of complete integral), incorrect polynomial, or failure to conclude verification in (c) |
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