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Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Question: A two-dimensional incompressible flow has the velocity potential. = K (x² - y²) + Cln(x² + y²) Where K and C are constants, in this discussion, avoid origin, which is a singularity (infinite velocity). A. Find the sole stagnation point of this flow, which is somewhere in the upper half plate. B. Prove that a stream function exists and then find (x,y), using the following hint. Hint: 2 = (4) tan-¹() f Method of solution A: Method of solution B: ди Әх = K (y²-x²) + C In(y² + x²) ap 2Cx əx y² + x² U = v= - = -2Kx + ap əx = 2Ky + u=0 -2Kx + 2Ky+ 2Cy y² + x² 2Cx y²+x² x=0 v=0 2Cy y² +0²=0 = 0 y = 2C 2Ky + == 0 y ə 2Cy +-(-28x + 2x) + (2xy +243) = y² ду əx ду 2Cx 4Cx² 4Cy² -[-2x + ²+² ³+²³² + 2x + y² + x² (²+3³-0 = [2K = (y² x²)² [u= for [[-2Kx + 2x Jay - (ap + = = -2Kxy + 2C tan-¹ -1 (+) + Constant
Transcribed Image Text:Question: A two-dimensional incompressible flow has the velocity potential. = K (x² - y²) + Cln(x² + y²) Where K and C are constants, in this discussion, avoid origin, which is a singularity (infinite velocity). A. Find the sole stagnation point of this flow, which is somewhere in the upper half plate. B. Prove that a stream function exists and then find (x,y), using the following hint. Hint: 2 = (4) tan-¹() f Method of solution A: Method of solution B: ди Әх = K (y²-x²) + C In(y² + x²) ap 2Cx əx y² + x² U = v= - = -2Kx + ap əx = 2Ky + u=0 -2Kx + 2Ky+ 2Cy y² + x² 2Cx y²+x² x=0 v=0 2Cy y² +0²=0 = 0 y = 2C 2Ky + == 0 y ə 2Cy +-(-28x + 2x) + (2xy +243) = y² ду əx ду 2Cx 4Cx² 4Cy² -[-2x + ²+² ³+²³² + 2x + y² + x² (²+3³-0 = [2K = (y² x²)² [u= for [[-2Kx + 2x Jay - (ap + = = -2Kxy + 2C tan-¹ -1 (+) + Constant
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