12 fluid air flow 164 kreyszig we know that

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Unformatted text preview: in the two-dimensional case. Since T satisfies Laplace's equation, it is also called the heat potential, and is, as usual, the real part of a complex potential F(z) = T(x, y) + ifl(x, y) The equipotentials T = const are called isotherms and the curves § = const are heat flow lines. The conjugate derivative F'(z) gives the heat flow vector field, measured in units of energy per unit time. Examples 36 (A) Temperature between parallel plates Going back to Topic 9 on around p. 28, we find that the complex potential is just linear: F(z) = Az + B where A and B can be found from the temperatures of the two plates and their distance apart. (B) Insulated Hot Wire A Hot Wire (500º) of radius 1 mm. in the center inside a cool cylinder (60º) or radius 100 mm. on the outside: Again going back to Topic 9, we use F(z) = ALn(z) + B Looking at the real part: 500 = Aln(1) + B = B 60 = Aln(100) + B = Aln(100) + 500 So we get A ‡ -95.54 and so F(z) = -95.54Ln(z) + 500 This gives the temperature as the real part: T(x, y) = -95.54 ln r + 500, where r is...
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This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

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