Quiz7_8.EGM4313.solutions

Quiz7_8.EGM4313.solutions - EGM 4313 Spring 2011 Quiz 7...

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EGM 4313 NAME:_____________________________ Spring 2011 Open book/Closed notes/No calculators or computers 50 minute time limit Find the complex potential such that the real part solves the steady heat conduction problem in the upper half-plane with a missing hole as shown. The temperature on the x-axis is held at 100 and the temperature of the surface of the hole is held at 200. (Hint: Consider the analytic function ( 29 B f z A z = + . A and B are not restricted to be real.) Solution: ( 29 ( 29 ( 29 2 2 2 2 2 2 cx dy i dx cy c id c id x iy f z a ib a ib a ib x iy x iy x iy x y cx dy dx cy a i b x y x y + + - + + - = + + = + + = + + + + - + + - = + + + + + ( 29 2 2 ,0 100 100, 0 cx dy T a x y c T x a a c x + = + + = + = = = On the circle ( 29 2 2 2 2 2 2 1 1 2 1 1 2 y x y y x x y y - + = ⇒ - + + = ⇒ + = Therefore T on the circle is 100 200 200 2 2 2 dy d d T a a d y = + = + = + = = ( 29 ( 29 200 100 i f z ib z = + + The choice of b is arbitrary and can be taken as zero.
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Quiz7_8.EGM4313.solutions - EGM 4313 Spring 2011 Quiz 7...

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