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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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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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