Q7
(a) Find the integral surface of the following quasi-linear equation $$(y - \phi) \frac{\partial \phi}{\partial x} + (\phi - x) \frac{\partial \phi}{\partial y} = x - y,$$ which passes through the curve $\phi = 0$, $xy = 1$ and through the circle $x + y + \phi = 0$, $x^2 + y^2 + \phi^2 = a^2$. (15 marks) (b) Integrate $f(x) = 5x^3 - 3x^2 + 2x + 1$ from $x = -2$ to $x = 4$ using (i) Simpson's $\frac{3}{8}$ rule with width h = 1, and (ii) Trapezoidal rule with width h = 1. (15 marks) (c) Let the velocity field $$u(x, y) = \frac{B(x^2 - y^2)}{(x^2 + y^2)^2}, \quad v(x, y) = \frac{2Bxy}{(x^2 + y^2)^2}, \quad w(x, y) = 0$$ satisfy the equations of motion for inviscid incompressible flow, where B is a constant. Determine the pressure associated with this velocity field. (20 marks)
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(a) निम्नलिखित रैखिक-कल्प समीकरण $$(y - \phi) \frac{\partial \phi}{\partial x} + (\phi - x) \frac{\partial \phi}{\partial y} = x - y,$$ का वह समाकल पृष्ठ ज्ञात कीजिए, जो कि वक्र $\phi = 0$, $xy = 1$ और वृत्त $x + y + \phi = 0$, $x^2 + y^2 + \phi^2 = a^2$ से होकर गुजरता है। (15 अंक) (b) (i) चौड़ाई h = 1 के साथ सिम्पसन के $\frac{3}{8}$ नियम, और (ii) चौड़ाई h = 1 के साथ समलंबी (ट्रेपिजॉइडल) नियम का उपयोग करके $f(x) = 5x^3 - 3x^2 + 2x + 1$ का $x = -2$ से $x = 4$ तक समाकलन कीजिए। (15 अंक) (c) मान लीजिए कि वेग क्षेत्र $$u(x, y) = \frac{B(x^2 - y^2)}{(x^2 + y^2)^2}, \quad v(x, y) = \frac{2Bxy}{(x^2 + y^2)^2}, \quad w(x, y) = 0,$$ जहाँ B एक अचर है, अश्यान असंपीड्य प्रवाह के लिए गति समीकरणों को संतुष्ट करता है। इस वेग क्षेत्र से सहचारी (एसोसिएटेड) दाब का निर्धारण कीजिए। (20 अंक)
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How this answer will be evaluated
Approach
Solve all three parts systematically, allocating approximately 30% time to part (a) on Lagrange's auxiliary equations and integral surfaces, 30% to part (b) on numerical integration with proper tabulation for both Simpson's 3/8 and Trapezoidal rules, and 40% to part (c) on applying Euler's equations for inviscid flow to determine pressure distribution. Present each part with clear headings, show all working steps, and verify boundary conditions are satisfied.
Key points expected
- Part (a): Formulate Lagrange's auxiliary equations dx/(y-φ) = dy/(φ-x) = dφ/(x-y) and find multipliers to obtain first integrals; apply both boundary conditions (hyperbola φ=0, xy=1 and circle x+y+φ=0, x²+y²+φ²=a²) to determine the integral surface
- Part (b)(i): Apply Simpson's 3/8 rule with h=1 requiring 3n intervals; construct table for f(x)=5x³-3x²+2x+1 from x=-2 to x=4 (7 points), apply weights 1,3,3,2,3,3,1 and compute integral value
- Part (b)(ii): Apply Trapezoidal rule with h=1; use same tabulated values with weights 1,2,2,2,2,2,1 and compare accuracy with exact analytical integration
- Part (c): Verify incompressibility (∂u/∂x + ∂v/∂y = 0), identify this as a 2D dipole flow in potential flow theory; apply Euler equations ∂u/∂t + u∂u/∂x + v∂u/∂y = -1/ρ ∂p/∂x (similarly for y) to obtain pressure field p(x,y)
- Part (c) continued: Integrate to find p = p∞ - ρ/2 (u²+v²) + C/r⁴ terms, or express pressure coefficient; handle singularity at origin appropriately and express final pressure in terms of B, ρ, and r²=x²+y²
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | For (a): correctly writes Lagrange's auxiliary equations and identifies appropriate multipliers (1,1,1 and x,y,φ); for (b): properly sets up interval [-2,4] with h=1 giving 6 subintervals/7 points; for (c): verifies ∇·v=0 and writes correct Euler equation components with proper sign conventions | Sets up most equations correctly but has minor errors in auxiliary equation formulation or interval counting; may miss incompressibility check in (c) or have sign errors in Euler equations | Major setup errors: wrong auxiliary equations, incorrect number of intervals for Simpson's 3/8 rule (must be multiple of 3), or fundamentally wrong governing equations in (c) |
| Method choice | 20% | 10 | For (a): skillfully uses multiplier method to find two independent integrals x+y+φ=c₁ and x²+y²+φ²=c₂; for (b): correctly identifies Simpson's 3/8 rule requires 3n subintervals and applies proper weight pattern; for (c): recognizes potential flow structure and uses Bernoulli's principle or integrates Euler equations appropriately | Uses correct general methods but with suboptimal choices (e.g., attempts direct integration without multipliers in (a), or uses composite rules without checking interval compatibility); may not recognize potential flow simplification in (c) | Wrong method choices: attempts Charpit's method for quasi-linear equation, uses Simpson's 1/3 instead of 3/8, or tries Navier-Stokes instead of Euler equations for inviscid flow |
| Computation accuracy | 20% | 10 | Accurate arithmetic throughout: correct first integrals in (a), precise table values in (b) with f(-2)=-47, f(-1)=-7, f(0)=1, f(1)=5, f(2)=35, f(3)=121, f(4)=313; correct Simpson's 3/8 formula 3h/8[y₀+3y₁+3y₂+2y₃+3y₄+3y₅+y₆] and trapezoidal h/2[y₀+2(y₁+...+y₅)+y₆]; correct partial derivatives and pressure integration in (c) | Minor computational slips: one or two table value errors, arithmetic mistakes in weighted sums, or algebraic errors in partial derivatives; final answers slightly off but method evident | Significant computational errors: wrong function evaluations, incorrect weight applications (e.g., using Simpson's 1/3 weights for 3/8 rule), major errors in pressure integration, or missing singular point treatment |
| Step justification | 20% | 10 | Clear justification for each step: explains why multipliers 1,1,1 and x,y,φ are chosen in (a); shows derivation of weights for Simpson's 3/8 rule from cubic interpolation; justifies use of steady Euler equations and Bernoulli integration in (c); explicitly verifies both boundary conditions in (a) and discusses singularity at r=0 in (c) | Shows most steps but with gaps in reasoning: states multipliers without explanation, presents formulas without derivation, or integrates pressure without explaining path independence; boundary condition verification may be implicit | Minimal or no justification: jumps to answers, presents numerical results without showing working, or gives final pressure expression without derivation from Euler equations; fails to verify boundary conditions |
| Final answer & units | 20% | 10 | Complete final answers: (a) explicit integral surface equation satisfying both conditions; (b) numerical values with clear indication of which rule applies (Simpson's 3/8: 156, Trapezoidal: 168, exact: 156); (c) pressure field p = p∞ - ρB²/(2r⁴) or equivalent with proper dimensional consistency and discussion of stagnation points | Final answers present but incomplete: one part missing, numerical values without labels, or pressure expression without reference pressure; units inconsistent or missing | Missing or wrong final answers: no integral surface equation, unlabeled numerical results, or pressure expression with wrong dependence on r; fails to recognize exact answer matches Simpson's 3/8 for this cubic |
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