Lecture25 - 18.3 Heat Problems Laplaces equation also...

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18.3 Heat Problems Laplace’s equation also governs steady heat problems (cf. Sections 12.5, 12.6): the heat equation is given by T t = c 2 T ,where T [K] denotes the tempera- ture and where c [m 2 s - 1 ] denotes the thermal diFusivity. We have T t 0in steady state, i. e. Laplace’s equation. Therefore in two space dimensions, we may again use methods from complex analysis. In this application, T is also called the heat potential, and it is the real part of the complex heat potential F ( z )= T ( x, y )+ i Ψ( x, y ) . (1040) The curves T const . , are called isotherms and the curves Ψ const . are called heat fow lines. Heat ±ows along these lines from higher to lower tem- peratures. Examples: The examples considered in the previous section can now be rein- terpreted as problems on heat ±ow: the electrostatic potential lines now become isotherms, and the lines of electrical force become lines of heat ±ow. So we can immediately write down the heat potential between two plates, two cylinders and so on. We consider two new examples with diFerent boundary conditions: 1. We consider the ²rst quadrant of the unit disk. The temperature on the straight line segments is prescribed (Dirichlet boundary conditions), but on the arc, the domain is assumed to be insulated, so that n T =0 (Neumann boundary condition). We use polar coordinates ( r, ϑ ), so that n r on the arc. The angular solution from Example 3 before satis²es the Neumann boundary condition on the arc. The complex potential is thus given by F ( z a - ib Log z with real part T ( x, y a + , ϑ =Im(Log z ). Coefficients a, b are determined from the Dirichlet boundary values. 2. We consider the upper half-plane. On the boundary (real axis), the temperature is prescribed for x< - 1and x> 1(twod iFerentva lues) and the domain is insulated on ( - 1 , 1). The upper half of the z -plane is mapped onto a semi-in²nite vertical strip in the w -plane, S := { w C | u ( - π/ 2 ,π/ 2) ,v 0 } , by the inverse of the sine mapping: w = f ( z )=arcs in z.
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This note was uploaded on 11/17/2011 for the course MATH 529 taught by Professor Staff during the Spring '08 term at UNC.

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Lecture25 - 18.3 Heat Problems Laplaces equation also...

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